# Give and Take: An End-To-End Investigation of Giveaway Scam Conversion Rates

Enze Liu  UC San Diego 0000-0003-4288-8485<EMAIL_ADDRESS>,  George Kappos
Chainalysis<EMAIL_ADDRESS>,  Eric Mugnier  UC San Diego<EMAIL_ADDRESS>,  Luca Invernizzi  Google<EMAIL_ADDRESS>,  Stefan Savage  UC San
Diego<EMAIL_ADDRESS>,  David Tao  Google<EMAIL_ADDRESS>,  Kurt Thomas
Google<EMAIL_ADDRESS>,  Geoffrey M. Voelker  UC San Diego
<EMAIL_ADDRESS>and  Sarah Meiklejohn  Google<EMAIL_ADDRESS>
###### Abstract.

Scams—fraudulent schemes designed to swindle money from victims—have existed
for as long as recorded history. However, the Internet’s combination of low
communication cost, global reach, and functional anonymity has allowed scam
volumes to reach new heights. Designing effective interventions requires first
understanding the context: how scammers reach potential victims, the earnings
they make, and any potential bottlenecks for durable interventions. In this
short paper, we focus on these questions in the context of cryptocurrency
giveaway scams, where victims are tricked into irreversibly transferring funds
to scammers under the pretense of even greater returns. Combining data from
Twitter,111The platform has been known as X since July 2023, but most of our
analysis pertains to tweets posted before then so we continue to call it
Twitter. YouTube and Twitch livestreams, landing pages, and cryptocurrency
blockchains, we measure how giveaway scams operate at scale. We find that 1 in
1000 scam tweets, and 4 in 100,000 livestream views, net a victim, and that
scammers managed to extract nearly $4.62 million from just hundreds of victims
during our measurement window.

##  1\. Introduction

Since the introduction of the Bitcoin whitepaper in 2008, cryptocurrencies and
blockchain-based technologies have attracted attention from investors,
technologists, researchers, governments, and the general public.
Unsurprisingly, this rapid rise in popularity has also attracted scammers
seeking to exploit the gap between people’s (high) interest in these
technologies and their (low) technical understanding of their operation.
Today, there are many types of scams run within the cryptocurrency ecosystem:
Ponzi and pyramid schemes (vasek2015freelunch, ; vasek2018ponzi, ;
bartoletti2018ponzi, ; chen2018ponzi, ; siu2022bitcointalk, ; kell2023forsage,
); NFT “rug pulls” (roy2023nftscams, ; huang2023rugpulls, ), counterfeiting
(das2022nft, ), airdrop scams (roy2023nftscams, ); pump-and-dump schemes
(hamrick2018pumpdump, ; xu2019pumpdump, ; xia2020covidscams, ); and giveaway
scams (vakilinia2022giveaway, ; li2023giveaway, ; li2023twitter-giveaway, ).

In a giveaway scam, users are enticed to visit a website that promises to give
away a certain number of (cryptocurrency) coins. Often, the website leverages
branding associated with a prominent public figure or company to increase its
perceived legitimacy. To participate, users are instructed to send some coins
to a cryptocurrency address specified on the website and are promised they
will receive double the amount in return. These promised funds fail to
materialize and, given the irreversible nature of cryptocurrency transactions,
there is no way for the user to recover their money.

The existence of giveaway scam sites is well documented both in recent
academic literature (vakilinia2022giveaway, ; li2023twitter-giveaway, ;
li2023giveaway, ) and by online tech journalists (toulas23, ; ahmed23, ).
However, our understanding of how users are lured to these sites, the extent
of victim participation, and the conversion rate that scammers can expect is
largely anecdotal. Indeed, while there is received wisdom that scammers
utilize mainstream “broadcast” style media such as Twitter, the extent to
which such platforms represent the dominant source of such activity is
unclear. Further complicating analysis is the fact that livestreaming
platforms (e.g., YouTube and Twitch) often do not provide a retrospective
record and thus must be analyzed contemporaneously with their use.

In this paper, we present an empirical analysis of cryptocurrency giveaway
scams, conversion rates, and overall revenue. We focus on scams appearing on
Twitter, YouTube, and Twitch, which Li et al. suggest are the most prevalent
platforms for such activity (li2023giveaway, ). From these platforms we
collect posts (retrospectively) and streams (prospectively) and identify and
crawl both existing and new promoted scam giveaway sites. From these, we
extract the cryptocurrency addresses used to solicit victims for funds and
quantify the revenue of these scams; we also explore different aspects of
victim behavior (e.g., how victims pay in) and scammer behavior (e.g., how
scammers promote on Twitter and YouTube as well as cash out). Together this
data provides a broader vantage point for understanding the revenue model of
giveaway scams and potential paths to interventions.

##  2\. Related Work

Our work builds on a number of prior efforts that have investigated
cryptocurrency giveaway scams. Xia et al. (xia2020covidscams, ) document
dozens of giveaway scams while searching for Covid-themed cryptocurrency fraud
reported by a variety of industry and government sources. Phillips and Wilder
(phillips2020tracing, ) analyzed hundreds of giveaways scams (which they refer
to as advanced-fee scams) crowdsourced by victims and researchers on
CryptoScamDB and URLScan. Li et al. (li2023giveaway, ) deployed the
CryptoScamTracker tool, which found thousands of scam domains by examining
Certificate Transparency (CT) log data for a six-month period between January
and June 2022. The authors used a combination of scam-related keywords to
identify potential scam domains, filtering down this list and then validating
manually (i.e., that the site in fact hosts a scam) to obtain their scam
domain corpus.

These efforts generally focus on scam sites themselves, however, and do not
capture the modality by which victims are lured to visit. One recent such
study is Vakilinia’s qualitative analysis of giveaway scams livestreamed over
two days on YouTube (vakilinia2022giveaway, ). While small in scale, this work
helps confirm our understanding of livestreamed scams and thus our methodology
for finding them. Another example is Li et al.’s contemporaneous paper
describing GiveawayScamHunter, which explores giveaway scams specifically
promoted via Twitter Lists (li2023twitter-giveaway, ).

In addition to giveaway scams, the security community has explored a range of
other cryptocurrency-related scams, including high-yield investment scams
(vasek2015freelunch, ; bartoletti2018ponzi, ; vasek2018ponzi, ;
boshmaf2020investigating, ; chen2019exploiting, ; jung2019data, ;
toyoda2019novel, ; chen2018ponzi, ; siu2023get, ; bartoletti2020dissecting, ;
kell2023forsage, ; siu2022bitcointalk, ); fake or high-risk cryptocurrency-
related services (e.g., arbitrage bots (li2023towards, ), exchanges
(vasek2015freelunch, ; moore2013beware, ; xia2020characterizing, ), and new
coins and tokens (gao2020tracking, ; xia2021trade, )); initial coin offering
scams (sapkota2020much, ; zetzsche2017ico, ; liebau2019crypto, ;
tiwari2020future, ; bian2018icorating, ); sextortion scams (paquet2019spams, ;
oggier2020ego, ); technical support scams (acharya2024conning, ); scams and
fraud in non-fungible token markets (kshetri2022scams, ; das2022nft, ); and
cryptocurrency-themed phishing websites (andryukhin2019phishing, ;
holub2018coinhoarder, ; phillips2020tracing, ; he2023txphishscope, ).

Finally, another relevant area of research focuses on understanding the risk
models and security practices of cryptocurrency users (abramova2021bits, ;
alshamsi2019user, ; guanexamining, ; ooi2021embracing, ; frohlich2021don, ;
krombholz2017other, ). These studies help shed light on why or how users fall
for scams in general, but do not specifically target giveaways or similar
scams.

##  3\. Methodology

Here, we describe our various sources of data and explain our methodology for
identifying giveaway scams on social media (Twitter) and livestreams (Twitch
and YouTube).

###  3.1. Data sources

We rely on three pre-existing datasets for our analysis: a snapshot of
Twitter, a corpus of known giveaway scam domains and cryptocurrency addresses,
and a snapshot of blockchain transactions. We add to this a new dataset
derived from monitoring livestreams of giveaway scams and crawling the leads
presented via the streams (discussed in Section 3.2).

#### Twitter dataset

Our Twitter dataset originates from a continuous Internet-wide crawl of public
URLs, in a way that respects robots.txt and other rules for crawlers. For the
purposes of this study, we restrict our analysis to publicly available tweets.
Our dataset covers a period from April 1, 2020 until August 20, 2023. It does
not contain every single tweet, and is biased towards tweets that are more
discoverable (e.g., posted by accounts with a non-zero number of followers).
Access to this data was granted to us only due to the nature of our research
project and its potential to disrupt abusive behavior. In total, our snapshot
contains tens of billions of tweets posted by hundreds of millions of
accounts.

#### Giveaway scam dataset

We rely on a manually curated dataset made available by Li et al.
(li2023giveaway, ) containing 3,863 domains previously identified as
cryptocurrency giveaways over a period from January 1, 2022 to June 29, 2022.
This dataset annotates each domain with the cryptocurrency addresses published
at the time it was crawled. (Websites often use addresses for multiple
different cryptocurrencies.) Starting from this dataset, we identified all
tweets that contained at least one known scam domain, totaling 457,248 tweets
from 33,841 distinct accounts (Table 1). Of the original 3,863 scam domains,
only 361 (9%) appeared on Twitter—a sample of which we show in Figure 1—
indicating that many scams are likely promoted through other means.

#### Blockchain data

We have access to a proprietary service (chainanalysis, ) that aggregates raw
blockchain data and annotates each address indicating (1) the _multi-input
cluster_ (reid2013bitcoin, ; meiklejohn2013fistful, ) it belongs to and (2)
the likely _category_ of the real-world owner of that cluster.222The service
can be configured to provide the name of the cluster owner, but our agreement
meant we had access only to the broader category. These latter annotations are
obtained via regular transactions that this company conducts with known
cryptocurrency services, such as exchanges, and the former annotation avoids
false positives (such as those induced by Coinjoin transactions) using
proprietary heuristics.

Figure 1. Example giveaway scam landing pages promoted via Twitter. Scammers
impersonate popular personalities including Brad Garlinghouse (the Ripple CEO)
and Elon Musk.

###  3.2. Identifying giveaway livestreams

We built a measurement pipeline for identifying and capturing giveaway scams
on the YouTube and Twitch streaming services. These scam livestreams typically
play a pre-recorded video that ultimately directs viewers to a scam page via a
link or QR code. Figure 2 shows such an example, in which the scam page is
promoted both as a QR code and as a link in the chat.

The measurement pipeline frequently polled available livestreams via platform
APIs, recorded them, crawled the pages promoted by the QR codes and links, and
annotated each scam page domain name with blockchain metadata. The design
decisions behind this system are based on a 14-day pilot study conducted July
1–14, 2023. While we crawled both YouTube and Twitch, we did not detect any
giveaway scams during our collection window for Twitch despite past reports of
such scams (twitch-scams2022, ). As a result, we focus our subsequent
discussion and analysis on YouTube, deferring Twitch-specific details to
Appendix B.1.

#### Identifying potential giveaway streams

The measurement pipeline starts data collection by periodically querying the
YouTube API for available streams and associated metadata. Given a set of
keywords, the API returns livestreams that are associated with the keywords
(in a manner similar to Google Search). Our system queried the API at the
maximum rate allowed, which meant retrieving the list of livestreams and
channel metadata (e.g., number of subscribers) every 30 minutes and retrieving
livestream metadata (e.g., concurrent and total viewers) every 7.5 minutes. We
derived our set of search keywords using a combination of popular
cryptocurrency names and those used by Li et al. (li2023giveaway, ) (more
details in Appendix B.2).

Figure 2. Example livestream containing a giveaway scam. The video playing is
of Brad Garlinghouse and the scam website is linked to in both the chat and
the embedded QR code.

#### Stream and chat recording

The system recorded each livestream (using Streamlink (Streamlink:online, ))
and its associated chat (using the YouTube API) to identify linked URLs. In
our pilot study, we observed that URLs either appeared in chat or were
embedded in the video using a QR code. To capture embedded QR codes, the
system recorded videos for two seconds at a time, with snapshots taken every
7.5 minutes. For chats, the system polled the chat API every 7.5 minutes to
obtain the last 70 historical messages (the maximum set by YouTube). Appendix
B.3 provides more information on why two-second samples and 70 historical
messages are sufficient.

#### Extracting URLs and QR code leads

From our livestream data, the system identified potential links to scams by
(1) extracting URLs from chats via regular expressions; and (2) extracting and
interpreting QR codes via visual analysis of the captured video frames using
the opencv and pyzbar Python libraries. We use two libraries to increase the
detection rate.

#### Crawling potential scam URLs

The system crawled each potential scam URL, revisiting both old and newly
discovered websites daily until (1) we reached the end of our collection
period, or (2) fetching the URL resulted in an error three days in a row.
Prior work (li2023giveaway, ) suggested that some giveaway scam sites adopted
anti-bot tactics. We identified four types of cloaking behavior in our pilot
study (Appendix B.4) and counter these behaviors by: (1) using a VPN to mask
the system IP address; (2) spoofing the User-Agent string to appear as a
popular device and browser; (3) having a heuristic module to click through
common front pages; and (4) verifying the crawler with Cloudflare to bypass
anti-bot detection.

#### Validating scam URLs and identifying cryptocurrency addresses

We verified that every final clickthrough or landing page was related to a
giveaway scam by (1) ensuring there was a valid cryptocurrency address
published on the site. In addition, we require that either (2) the landing
page included a set of heuristic keywords related to scams (e.g., “hurry” or
“participate”) or (3) the domain contained a common set of scam keywords. For
(1), we extracted addresses via a regular expression and then validated the
address.333We used coinaddrvalidator (coinaddrvalidator-github:online, ) and
multicoin-address-validator (multicoin-address-validator-npm:online, ), and
considered an address valid if either tool indicated that it was valid. For
(2) and (3), we relied on the same keywords from the CryptoScamTracker tool.
We then manually examined all sites (4,611) that met the requirements. To
ensure that our heuristics do not miss a significant number of domains, we
randomly sampled 389 sites that had only a valid cryptocurrency address and
did not find any additional scam sites.

#### Final dataset

We ran our measurement pipeline from July 24, 2023 to January 21, 2024 (26
weeks), identifying a total of 2,069 livestreams from 1,632 different channels
(Table 1). These streams linked to 343 distinct domains. Our data is
continuous throughout this period apart from 11 days where we experienced
infrastructure outages.444These dates are: 2023-08-15, 2023-08-16, 2023-09-01,
2023-09-28, 2023-10-06, 2023-11-18, 2023-11-19, 2023-12-26, 2023-12-12,
2024-01-06, and 2024-01-21.

Source | Domains | Accounts | Artifacts
---|---|---|---
Twitter | $361$ | $33,841$ | $457,248$
YouTube | $343$ | $1632$ | $2069$
Table 1. The number of objects in our collected datasets. Accounts in the
YouTube dataset correspond to channels, and artifacts are individual tweets or
livestreams.

###  3.3. Limitations

As with any measurement study, there are a number of limitations with our
methodology. Our Twitter dataset may omit some tweets, resulting in us
underestimating the volume of scam tweets or the revenue generated by scams;
likewise, this may happen due to missing or inaccurate cryptocurrency
addresses in the CryptoScamTracker dataset (as indeed we observed manually).
Given that only 9% of the domains previously identified by CryptoScamTracker
overlap with Twitter, it is also clear that many scams are promoted via other
distribution channels—be it social media, email, or messaging. Finally, our
keyword-based matching strategies for YouTube and Twitch likely resulted in
omitting some scams. Despite these limitations, we are able to draw insights
into the promotion of giveaway scams and provide the first estimates on the
conversion rate and revenue of such campaigns.

##  4\. Giveaway Scam Lures

We begin our analysis by examining the volume of tweets and livestreams (i.e.,
“lures”) that ultimately directed victims to scam landing pages; the tactics
used by scammers to make their lures discoverable; and which cryptocurrencies
were targeted by scammers.

Figure 3. The volume of giveaway scam tweets on a given week, between January
1, 2022 and July 7, 2022.

###  4.1. Volume

In total, we observed 457,248 scam tweets and 2,069 scam livestreams. We
present a weekly timeline of scam activity for Twitter in Figure 3 and YouTube
in Figure 4. While scam lures appeared consistently throughout our measurement
window, there are notable bursts of activity. For Twitter, we saw a single
peak in activity in March 2022, with scammers posting a maximum of 90,984
tweets in a single week. On YouTube, we observed a burst in September 2023 and
again over the holiday period of December–January 2024, peaking at 289 streams
and 1,869,399 views in a single week. While these scams are smaller in scale
compared to spam email campaigns involving tens of millions of messages
(kanich2008spamalytics, ), as we show shortly, they are nevertheless highly
profitable.

Figure 4. The volume of giveaway scam streams and views per week, between July
24, 2023 and January 21, 2024.

###  4.2. Discoverability

To reach a broad audience on Twitter, scammers hashtagged their tweets 96% of
the time (e.g., using a cryptocurrency’s name or ticker symbol). This meant
that users searching for the latest news related to popular coins (e.g., #btc,
#eth) might then interact with the scammer’s tweets. While scammers also
pursued alternative audience-generating tactics on Twitter, such as mentioning
users or replying to popular tweets, these strategies occurred in only 0.1%
and 0.3% of tweets respectively.555 For Twitter, our dataset lacks any
information on the number of views per tweet, or the following or follower
counts of accounts. As such, we focus on discoverability. For YouTube,
scammers reached victims using a combination of organic viewership and search
discoverability. Channels used by scammers to host livestreams had a median of
16.8K subscribers, with the single largest channel having 19 million
subscribers—liked a compromised pre-existing channel. Beyond existing
audiences, 93% of livestreams included a cryptocurrency keyword in their
title, their description, or the name of the associated channel.

###  4.3. Currencies

To identify the cryptocurrencies most commonly targeted by scammers, we
searched for relevant keywords in each tweet’s hashtags and each livestream’s
title, channel name, and description, using the names and ticker symbols of
the top 20 coins with the highest market capitalization (e.g., Bitcoin and
btc).666We fetched the coins and their ranking from coinmarketcap.com on
August 7, 2023. For Twitter, of these top 20 coins, Ripple was by far the most
popular choice, matching 91% of the tweets in our dataset, followed by
Ethereum (12%) and Bitcoin (7%). For YouTube, Bitcoin was the most popular
coin (matching 65% of the livestreams), followed by Ethereum (49%) and Ripple
(40%). Totals do not add to 100% as tweets and livestreams could reference
multiple coins. The difference in currencies may be due to changes in tactics
by scammers, with at least a year transpiring between our two datasets.

##  5\. Giveaway Scam Payments

Using our raw blockchain data, we retrospectively analyzed payments to
scammers to explore the revenue they reaped, the conversion rate per lure, the
behavior of victims, and the payment infrastructure relied upon by scammers.

###  5.1. Isolating relevant payments

We focused on the three most popular coins targeted by scammers (per Section
4.3): Bitcoin (BTC), Ethereum (ETH), and Ripple (XRP). For each coin, we
identified all incoming transactions to cryptocurrency addresses that appeared
on a giveaway scam landing page posted to Twitter or YouTube. We filtered the
set of incoming transactions to an identified scam address in two key ways.
First, following Gomez et al. (gomez2023cybercrime, ), we ignored transactions
where the sender is a known scam address, as this clearly represents the
consolidation of funds rather than a payment from an actual victim. Second, we
focused on payments that co-occurred with a lure in order to capture the
lure’s effectiveness in driving victims to the scam. We acknowledge that the
resulting set of payments might both include non-victim transactions (from
unknown scammer addresses) and exclude victim transactions (due to lures that
we were not aware of); nevertheless, we use it as an approximation of victim
payments to the scam. We believe this is a more accurate estimate than
previous studies, which considered all incoming transactions (li2023giveaway,
), but also compare with the total revenue (across all transactions) and leave
as future work a further refinement of our analysis that takes other behaviors
into account.

###  5.2. Twitter scam revenue

The 361 domains promoted by scammers on Twitter used 186 distinct addresses
(with some domains having multiple addresses due to accepting different
cryptocurrencies). After filtering out domains that did not use any BTC, ETH,
or XRP addresses, we were left with 258 domains. Of these, only 121 (47%)
received any incoming transaction; i.e., fewer than half of the domains
received any transaction at all. For the 121 domains that received a
transaction, we compared the transaction times to the times of the tweets
promoting the scam domain. We considered a payment as co-occurring if it
happened within one week of a tweet’s appearance. Despite this generous
window, only 43% of payments (695 of 1,633) fell within it. Of these, we
removed a further 24 payments that came from known scam addresses, resulting
in a final dataset of 671 payments.

Table 2 summarizes the overall co-occurring revenue made across Twitter and
YouTube, normalizing payment values across cryptocurrencies using the average
USD price of each coin on the day in which the payment took place.777We used
historical data from https://finance.yahoo.com/crypto/. In total, we estimate
that Twitter-based giveaway scams yielded $2.7M in revenue. If we include all
incoming transactions to these addresses then the total jumps to $6.6M; as
suggested above, the real revenue likely lies between the two numbers.

Metric | Twitter | YouTube
---|---|---
Payments (co-occurring) | $671$ | $638$
Payments (any) | $1633$ | $2074$
USD revenue (co-occurring) | $2,693,009$ | $1,932,654$
from BTC | $1,269,579$ | $1,422,065$
from ETH | $442,583$ | $266,693$
from XRP | $980,847$ | $243,896$
USD revenue (any) | $6,598,691$ | $4,705,978$
Table 2. Revenue of giveaway scams on Twitter and YouTube livestreams,
considering only co-occurring payments made in BTC, ETH, or XRP. For
completeness, we also include all payments to the addresses, irrespective of
what might explain the origin of the payment.

###  5.3. YouTube livestream scam revenue

Across the 343 domains promoted via YouTube, 342 domains had a BTC, ETH, or
XRP address. Of these domains, 231 (67%) had addresses that received at least
one transaction. As with Twitter, we looked at the timing of each incoming
transaction for livestream addresses. We considered it as co-occurring with a
livestream if the transaction occurred during the stream or up to eight hours
after it ended. Our goal was again to provide a generous window, but the
result was similar: only 34% of payments to the addresses (695 of 2074) fell
within this time interval. We removed another 57 payments from known scam
addresses, resulting in our final dataset of 638 payments.

Table 2 summarizes the overall revenue made by livestream scams. In total, we
estimate that livestream-based giveaway scams yielded $1.9M. If we include all
incoming transactions to these addresses, the total increases to $4.7M.

###  5.4. Victim behavior

#### Conversions

The 671 Twitter-related payments had 528 unique senders, and the 638
livestream-related payments had 399 unique senders. These metrics put into
context the huge volume of lures necessary to attract a victim that ultimately
pays out. For Twitter, the conversion rate of tweets (with associated
cryptocurrency addresses) to victims was 0.12%—or roughly 1 in 1000 tweets
netting a victim. For YouTube, the conversion rate of viewers to victims was
0.0039%—or roughly 4 in 100,000 views netting a victim. We caution comparing
these figures directly as there is an unknown number of viewers per tweet.

#### Payment origins

For the 1,309 payments across Twitter and YouTube, 755 (58%) came from
centralized exchanges;888As we saw only the category of each sender, rather
than the name, we cannot break down such senders into individual exchanges.
the results of Phillips and Wilder suggest that this number would increase if
we used more advanced blockchain analysis to track the source of funds
(phillips2020tracing, ). The prevalence of exchanges is not surprising, as we
would expect many victims of this type of scam to have a low level of comfort
making cryptocurrency transactions directly.

#### Payment distribution

The total revenue was dominated by a small number of transactions: the top 24
Twitter-related payments captured 50% of the value, while the top 164 payments
captured 90% of the value. YouTube scams exhibited a similar phenomenon: 20
payments captured 50% of the value and 147 payments 90% of the value. These
results suggest that the giveaway scam revenue model is significantly
influenced by large transactions, where the goal for promoting the scam is to
cast a wide net and, in doing so, capture a small fraction of victims who will
give large amounts (analogous to the whale profit model for free-to-play
mobile games (udonis-mobile-whales24, )). Such a model notably differs from
the email spam ecosystem, for instance, in which the variance in individual
customer payments is moderate and leads to a business model driven by order
volume from new and repeat customers (pharmleaks:usesec12, ).

###  5.5. Scammer behavior

Across the 1,309 payments, there were only 339 distinct recipients: 68 for
Twitter-related payments and 271 for YouTube-related ones; given the similar
number of payments, this suggests scammers running YouTube campaigns made more
of an effort to cycle between different addresses. Of these, 145 of 166 (87%)
of the BTC addresses were in a multi-input cluster of size one, which suggests
that scammers made an explicit effort to prevent address clustering.

Similarly, when looking at outgoing transactions from these addresses, only 4%
of the recipients (57 of 1,363) were labeled as belonging to a centralized
exchange. Another 87% of recipients were unlabeled, which again suggests the
need for more advanced blockchain analysis (phillips2020tracing, ) to identify
additional exchange interactions. Of the remaining categories associated with
recipients, some suggested meaningful knowledge of how to use cryptocurrency
(e.g., 13 recipients were tagged as ‘token smart contract’ and four as
‘mixing’) and others suggested that the scammers operated within a larger
illicit ecosystem (22 recipients were tagged as ‘scam’ and 13 as ‘sanctioned
entity’).

Altogether, these results suggest a profitable business for a set of
knowledgeable and technically sophisticated scammers who take advantage of a
set of victims with low technical sophistication and knowledge about
cryptocurrencies.

##  6\. Discussion

Based on our findings, we compare the revenue of giveaway scams relative to
past scams and discuss the prospects of interventions to undermine giveaway
scams.

###  6.1. Comparing scam revenue

Previous studies of pharmaceutical spam estimated a conversion rate of
0.00001%, with spam generating $100 per interaction (kanich2008spamalytics, ).
While our conversion rates are not directly comparable (e.g., tweets or
livestreams vs. emails), they are markedly higher than email-based spam: 0.12%
of tweets and 0.0039% of livestream views yielded a victim.

Multiple factors may explain the higher returns for cryptocurrency giveaway
scams. First, the relatively nascent abuse landscape of cryptocurrency may
mean users are less familiar with fraud and its risks as compared to email
spam. Second, the rags to riches narratives surrounding cryptocurrency may
make a headline of “5000 BTC giveaway” more believable, particularly when
there are real headlines such as “Elon Musk will give away 1 million Dogecoin
if you can prove his family own emerald mine” (elon-giveaway, ).

###  6.2. Interventions

A natural question for any scam is where bottlenecks exist that might be ripe
for disruption. Here, one choice appears to be the payment origins of victims,
as at least 58% of victims relied on centralized exchanges. Early detection of
payments to scammers via these exchanges represents a sizable and durable
protection (metamask, ): scammers likely cannot influence the exchange that
victims use, as we suspect most victims have low comfort with using
cryptocurrency directly. In addition, similar to what banks already implement,
exchanges can consider implementing additional warnings when users attempt to
make large transactions, which constitute a large portion of the revenue in
our study. That said, detecting the cryptocurrency addresses engaging in fraud
and informing these exchanges promptly remain challenging and require
collaboration across multiple sectors.

##  7\. Conclusions

In this paper, we provided a first look at cryptocurrency giveaway scams
conducted via social media and livestreams. Scammers collectively earned $2.7M
via Twitter scams and $1.9M via livestreams. These earnings reflect a
conversion rate of 0.12% per tweet and 0.0039% per livestream viewer. We
believe that interventions require a defense-in-depth approach. Apart from
traditional account-based and blocklist-based security measures,
cryptocurrency exchanges appear to be a promising bottleneck for depriving
scammers of the means to transact with victims.

## References

  * [1] How to turn on security alerts.  https://support.metamask.io/privacy-and-security/how-to-turn-on-security-alerts/.
  * [2] E. Abraham.  Elon Musk will give away 1 million Dogecoin if you can prove his family own emerald mine.  https://www.indy100.com/science-tech/elon-musk-emerald-mine-dogecoin-2659890703, Apr. 2023.
  * [3] S. Abramova, A. Voskobojnikov, K. Beznosov, and R. Böhme.  Bits under the mattress: Understanding different risk perceptions and security behaviors of crypto-asset users.  In Proceedings of the 2021 CHI Conference on Human Factors in Computing Systems, pages 1–19, 2021.
  * [4] B. Acharya, M. Saad, A. E. Cinà, L. Schönherr, H. D. Nguyen, A. Oest, P. Vadrevu, and T. Holz.  Conning the crypto conman: End-to-end analysis of cryptocurrency-based technical support scams.  In Proceedings of the 2024 IEEE Symposium on Security & Privacy (S&P), 2024.
  * [5] S. Ahmed.  Linus Tech Tips YouTube channel hacked to promote Elon Musk crypto scam videos, Mar. 2023.  https://www.dexerto.com/entertainment/linus-tech-tips-youtube-channel-hacked-crypto-elon-musk-scam-2094461/.
  * [6] A. Alshamsi and P. Andras.  User perception of Bitcoin usability and security across novice users.  International Journal of Human-Computer Studies, 126:94–110, 2019.
  * [7] A. Andryukhin.  Phishing attacks and preventions in blockchain based projects.  In 2019 international conference on engineering technologies and computer science (EnT), pages 15–19. IEEE, 2019.
  * [8] Anonymized for submission, 2023.
  * [9] M. Bartoletti, S. Carta, T. Cimoli, and R. Saia.  Dissecting Ponzi schemes on Ethereum: identification, analysis, and impact.  Future Generation Computer Systems, 102:259–277, 2020.
  * [10] M. Bartoletti, B. Pes, and S. Serusi.  Data mining for detecting Bitcoin Ponzi schemes.  In Proceedings of the Crypto Valley Conference on Blockchain Technology (CVCBT), 2018.
  * [11] S. Bian, Z. Deng, F. Li, W. Monroe, P. Shi, Z. Sun, W. Wu, S. Wang, W. Y. Wang, A. Yuan, et al.  Icorating: A deep-learning system for scam ICO identification.  arXiv preprint arXiv:1803.03670, 2018.
  * [12] Y. Boshmaf, C. Elvitigala, H. Al Jawaheri, P. Wijesekera, and M. Al Sabah.  Investigating MMM Ponzi scheme on Bitcoin.  In Proceedings of the 15th ACM Asia Conference on Computer and Communications Security, pages 519–530, 2020.
  * [13] W. Chen, Z. Zheng, J. Cui, E. Ngai, P. Zheng, and Y. Zhou.  Detecting Ponzi schemes on Ethereum: Towards healthier blockchain technology.  In Proceedings of the 2018 World Wide Web Conference, 2018.
  * [14] W. Chen, Z. Zheng, E. C.-H. Ngai, P. Zheng, and Y. Zhou.  Exploiting blockchain data to detect smart Ponzi schemes on Ethereum.  IEEE Access, 7:37575–37586, 2019.
  * [15] Christsim.  multicoin-address-validator - npm, 05 2024.
  * [16] D. Das, P. Bose, N. Ruaro, C. Kruegel, and G. Vigna.  Understanding security issues in the NFT ecosystem.  In Proceedings of the ACM Conference on Computer and Communications Security (CCS), 2022.
  * [17] M. Fröhlich, M. R. Wagenhaus, A. Schmidt, and F. Alt.  Don’t stop me now! exploring challenges of first-time cryptocurrency users.  In Designing Interactive Systems Conference 2021, pages 138–148, 2021.
  * [18] B. Gao, H. Wang, P. Xia, S. Wu, Y. Zhou, X. Luo, and G. Tyson.  Tracking counterfeit cryptocurrency end-to-end.  Proceedings of the ACM on Measurement and Analysis of Computing Systems, 4(3):1–28, 2020.
  * [19] G. Gomez, K. van Liebergen, and J. Caballero.  Cybercrime Bitcoin revenue estimations: Quantifying the impact of methodology and coverage.  In Proceedings of ACM CCS, pages 3183–3197, 2023.
  * [20] M. Grguric.  What Are Mobile Game Whales and How to Catch Them?  https://www.blog.udonis.co/mobile-marketing/mobile-games/mobile-games-whales, Mar. 2024.
  * [21] Y. Guan, Y. Yu, T. Sharma, K. Qin, Y. Wang, and Y. Wang.  Examining user perceptions of stablecoins: Understandings and risks.  In Posters at the Symposium on Usable Privacy and Security (SOUPS), 2023.
  * [22] J. Hamrick, F. Rouhi, A. Mukherjee, A. Feder, N. Gandal, T. Moore, and M. Vasek.  The economics of cryptocurrency pump and dump schemes.  SSRN Electronic Journal, 01 2018.
  * [23] B. He, Y. Chen, Z. Chen, X. Hu, Y. Hu, L. Wu, R. Chang, H. Wang, and Y. Zhou.  Txphishscope: Towards detecting and understanding transaction-based phishing on Ethereum.  In Proceedings of the 2023 ACM SIGSAC Conference on Computer and Communications Security, pages 120–134, 2023.
  * [24] A. Holub and J. O’Connor.  Coinhoarder: Tracking a Ukrainian Bitcoin phishing ring DNS style.  In 2018 APWG Symposium on Electronic Crime Research (eCrime), pages 1–5. IEEE, 2018.
  * [25] D. Horetski.  Twitch battles fake Elon Musk streams again with scam Bitcoin giveaways.  https://www.dexerto.com/entertainment/twitch-battles-fake-elon-musk-streams-again-with-scam- bitcoin-giveaways-1803537/, Apr. 2022.
  * [26] J. Huang, N. He, K. Ma, J. Xiao, and H. Wang.  A deep dive into NFT rug pulls, 2023.  https://arxiv.org/pdf/2305.06108.pdf.
  * [27] E. Jung, M. Le Tilly, A. Gehani, and Y. Ge.  Data mining-based Ethereum fraud detection.  In 2019 IEEE international conference on blockchain (Blockchain), pages 266–273. IEEE, 2019.
  * [28] C. Kanich, C. Kreibich, K. Levchenko, B. Enright, G. M. Voelker, V. Paxson, and S. Savage.  Spamalytics: An empirical analysis of spam marketing conversion.  In Proceedings of the Conference on Computer and Communications Security, 2008.
  * [29] T. Kell, H. Yousaf, S. Allen, S. Meiklejohn, and A. Juels.  Forsage: Anatomy of a smart-contract pyramid scheme.  In Proceedings of the 27th Conference on Financial Cryptography and Data Security, 2023.
  * [30] K. Krombholz, A. Judmayer, M. Gusenbauer, and E. Weippl.  The other side of the coin: User experiences with Bitcoin security and privacy.  In Financial Cryptography and Data Security: 20th International Conference, FC 2016, Christ Church, Barbados, February 22–26, 2016, Revised Selected Papers 20, pages 555–580. Springer, 2017.
  * [31] N. Kshetri.  Scams, frauds, and crimes in the nonfungible token market.  Computer, 55(4):60–64, 2022.
  * [32] K. Li, S. Guan, and D. Lee.  Towards understanding and characterizing the arbitrage bot scam in the wild.  Proceedings of the ACM on Measurement and Analysis of Computing Systems, 7(3):1–29, 2023.
  * [33] K. Li, D. Lee, and S. Guan.  Understanding the cryptocurrency free giveaway scam disseminated on Twitter lists.  In 2023 IEEE International Conference on Blockchain (Blockchain), pages 9–16. IEEE, 2023.
  * [34] X. Li, A. Yepuri, and N. Nikiforakis.  Double and nothing: Understanding and detecting cryptocurrency giveaway scams.  In Proceedings of the Network and Distributed Systems Security (NDSS) Symposium, 2023.
  * [35] D. Liebau and P. Schueffel.  Crypto-currencies and ICOs: Are they scams? an empirical study, 2019.  https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3320884.
  * [36] D. McCoy, A. Pitsillidis, G. Jordan, N. Weaver, C. Kreibich, B. Krebs, G. M. Voelker, S. Savage, and K. Levchenko.  PharmaLeaks: Understanding the Business of Online Pharmaceutical Affiliate Programs.  In Proceedings of the 21st USENIX Security Symposium, Bellevue, WA, Aug. 2012.
  * [37] S. Meiklejohn, M. Pomarole, G. Jordan, K. Levchenko, D. McCoy, G. M. Voelker, and S. Savage.  A fistful of bitcoins: Characterizing payments among men with no names.  In Proceedings of the Internet Measurement Conference - IMC ’13, pages 127–140, 2013.
  * [38] T. Moore and N. Christin.  Beware the middleman: Empirical analysis of Bitcoin-exchange risk.  In Financial Cryptography and Data Security: 17th International Conference, FC 2013, Okinawa, Japan, April 1-5, 2013, Revised Selected Papers 17, pages 25–33. Springer, 2013.
  * [39] Nobitex.  Github - nobitex/coinaddrvalidator, 05 2024.
  * [40] F. Oggier, A. Datta, and S. Phetsouvanh.  An ego network analysis of sextortionists.  Social Network Analysis and Mining, 10:1–14, 2020.
  * [41] S. K. Ooi, C. A. Ooi, J. A. Yeap, and T. H. Goh.  Embracing Bitcoin: users’ perceived security and trust.  Quality & Quantity, 55:1219–1237, 2021.
  * [42] M. Paquet-Clouston, M. Romiti, B. Haslhofer, and T. Charvat.  Spams meet cryptocurrencies: Sextortion in the bitcoin ecosystem.  In Proceedings of the 1st ACM conference on advances in financial technologies, pages 76–88, 2019.
  * [43] R. Phillips and H. Wilder.  Tracing cryptocurrency scams: Clustering replicated advance-fee and phishing websites.  In 2020 IEEE international conference on blockchain and cryptocurrency (ICBC), pages 1–8. IEEE, 2020.
  * [44] F. Reid and M. Harrigan.  An analysis of anonymity in the Bitcoin system.  In Security and Privacy in Social Networks, page 197–223, 2013.
  * [45] S. S. Roy, D. Das, P. Bose, C. Kruegel, G. Vigna, and S. Nilizadeh.  Unveiling the risks of NFT promotion scams, 2023.  https://arxiv.org/pdf/2301.09806.pdf.
  * [46] N. Sapkota, K. Grobys, and J. Dufitinema.  How much are we willing to lose in cyberspace? On the tail risk of scam in the market for initial coin offerings.  On the Tail Risk of Scam in the Market for Initial Coin Offerings (November 18, 2020), 2020.
  * [47] G. A. Siu and A. Hutchings.  “Get a higher return on your savings!”: Comparing adverts for cryptocurrency investment scams across platforms.  In 2023 IEEE European Symposium on Security and Privacy Workshops (EuroS&PW), pages 158–169. IEEE, 2023.
  * [48] G. A. Siu, A. Hutchings, M. Vasek, and T. Moore.  “Invest in crypto!”: An analysis of investment scam advertisements found in Bitcointalk.  In Proceedings of the Symposium on Electronic Crime Research (eCrime 2022), 2022.
  * [49] Streamlink.  Streamlink 6.7.4 documentation, 05 2024.
  * [50] M. Tiwari, A. Gepp, and K. Kumar.  The future of raising finance-a new opportunity to commit fraud: a review of initial coin offering (icos) scams.  Crime, Law and Social Change, 73:417–441, 2020.
  * [51] B. Toulas.  Fake cryptocurrency giveaway sites have tripled this year, Sept. 2022.  https://www.bleepingcomputer.com/news/security/fake-cryptocurrency-giveaway-sites-have-tripled-this-year/.
  * [52] K. Toyoda, P. T. Mathiopoulos, and T. Ohtsuki.  A novel methodology for HYIP operators’ Bitcoin addresses identification.  IEEE Access, 7:74835–74848, 2019.
  * [53] I. Vakilinia.  Cryptocurrency giveaway scam with YouTube live stream.  In Proceedings of the IEEE 13th Annual Ubiquitous Computing, Electronics & Mobile Communication Conference (UEMCON), 2022.
  * [54] M. Vasek and T. Moore.  There’s no free lunch, even using Bitcoin: Tracking the popularity and profits of virtual currency scams.  In Proceedings of the 19th Conference on Financial Cryptography and Data Security, 2015.
  * [55] M. Vasek and T. Moore.  Analyzing the Bitcoin Ponzi scheme ecosystem.  In Proceedings of the 5th Workshop on Bitcoin and Blockchain Research, 2018.
  * [56] P. Xia, H. Wang, B. Gao, W. Su, Z. Yu, X. Luo, C. Zhang, X. Xiao, and G. Xu.  Trade or trick? detecting and characterizing scam tokens on Uniswap decentralized exchange.  Proceedings of the ACM on Measurement and Analysis of Computing Systems, 5(3):1–26, 2021.
  * [57] P. Xia, H. Wang, X. Luo, L. Wu, Y. Zhou, G. Bai, G. Xu, G. Huang, and X. Liu.  Don’t fish in troubled waters! characterizing coronavirus-themed cryptocurrency scams.  In Proceedings of the Symposium on Electronic Crime Research (eCrime 2020), 2020.
  * [58] P. Xia, H. Wang, B. Zhang, R. Ji, B. Gao, L. Wu, X. Luo, and G. Xu.  Characterizing cryptocurrency exchange scams.  Computers & Security, 98:101993, 2020.
  * [59] J. Xu and B. Livshits.  The anatomy of a cryptocurrency pump-and-dump scheme.  In Proceedings of the 28th USENIX Security Symposium, pages 1609–1625, 2019.
  * [60] D. A. Zetzsche, R. P. Buckley, D. W. Arner, and L. Föhr.  The ICO gold rush: It’s a scam, it’s a bubble, it’s a super challenge for regulators.  University of Luxembourg Law Working Paper, pages 17–83, 2017.

##  Appendix A Ethics

Our work relies largely on public APIs (e.g., those provided by YouTube and
Twitch) and public data (e.g., blockchain data and public tweets). We do not
seek to deanonymize individuals involved in the transactions we explore, and
aim to reduce this risk by obtaining only the category of cryptocurrency
services as attribution tags, rather than the name of individual services.
Finally, we conducted all of our analysis retrospectively, which means we did
not have the opportunity to report the identified scams as they were being
promoted and thus potentially help their victims.

##  Appendix B Stream Retrieval Pilot Study

In this section, we provide more details on our pilot study conducted July
1-14, 2023. We start by discussing our attempt to identify scam livestreams on
Twitch (Section B.1). Our search yielded no results, and thus we excluded
Twitch from future experiments. Next, we detail how we chose the set keywords
for searching on YouTube. Then, we go over the characteristics of a subset of
the scam livestreams identified during the pilot study. We conclude with a
short description of four anti-bot mechanisms employed by scam sites and the
strategies we employed to counter them.

###  B.1. Twitch details

In this section, we discuss our methodology for identifying potential scam
livestreams on Twitch.

#### Identifying relevant streams

The Twitch API allows callers to retrieve all livestreams at a given point in
time. We retrieve the list of all livestreams every 30 minutes, which is
roughly the amount of time we need to record and process the streams in that
batch.

After retrieving all streams we identify the ones that are the most worth
recording; i.e., that seem most likely to be promoting giveaway scams. The
YouTube API already performs filtering, as it returns only streams related to
the provided keywords. To filter out irrelevant streams on Twitch, we use
their title, tags, and category. We start by marking a stream as relevant if
its tags or title contain any relevant keywords. We derived our set of
relevant keywords from the keywords used by CryptoScamTracker, but removed 16
non-crypto keywords (e.g., “event” and “give”) as they introduced too many
irrelevant results. After filtering out streams that are not relevant, we then
further remove any streams that are games using the category of the stream.
This step effectively limits the number of streams to around 250.

#### Recording duration in pilot study

Twitch randomly inserts a 15-second advertisement clip before the actual
stream content. To account for this issue, we record every Twitch stream for
20 seconds. For chat messages, Twitch’s API does not provide history for chat
messages. As a result, we record the Twitch chat for the entire duration of
the stream.

#### Stream and chat recording

We recorded streams and chats at our maximum capacity: sampling a stream for
20 seconds every 30 minutes (the minimum time needed to record and process a
batch of streams) and recording the chat messages of a stream while it is
live. We extracted any URL embedded in the stream as a QR code or in the chat
messages.

We ran this experiment for non-gaming streams that we identified in the
filtering module as well as for 250 gaming streams that contained the most
keywords. We did not find any scam livestreams during our 14-day pilot study,
and thus excluded Twitch from further analysis.

Figure 5. Effectiveness of keywords Category |  Keywords used
---|---
Coin names |  bitcoin btc ethereum eth tether usdt ripple xrp bnb “usd coin” usdc cardano ada coin ada dogecoin doge solana sol tron trx litecoin ltc polkadot dot polygon matic wrapped bitcoin wbtc bitcoin cash bch toncoin ton dai avalanche avax shiba inu shib binance usd busd algorand algo hex cryptocurrency crypto
Domain keywords |  kf event musk elon give coin shiba drop double get doge kefu vitalik claim binance hoskinson free charles star garling
HTML keywords |  giveaway participate send address rules crypto bonus immediately hurry
Table 3. Keywords used to find relevant streams.

###  B.2. Effectiveness of YouTube Search Keywords

Based on our pilot analysis, we opted for a set of keywords consisting of the
names and ticker symbols of the top 20 cryptocurrencies (as listed on
coinmarketcap.com on July 1, 2023),999In three cases (for ADA, SOL, and DOT),
we appended the word “coin” to the ticker symbol to avoid introducing
irrelevant streams. along with two catch-all terms: “cryptocurrency” and
“crypto.” We added other keywords used by the CryptoScamTracker tool
(li2023giveaway, , Table XI) to increase coverage. We found that many of the
additional keywords were not useful in expanding our list of relevant streams,
and thus stopped expanding our keyword set.

In more detail, we examine the incremental contribution of each search keyword
(Table 3) to the set of livestreams returned from the YouTube API.
Specifically, we compute the number of YouTube streams (over the 14-day pilot
study) that contain search keywords in their metadata (title and description).
If one stream contains multiple keywords, we give credit evenly among all
keywords.

Figure 5 shows the result. Overall, around 55% of the streams contained at
least one of the search keywords, and the top 20 keywords account for 90% of
those streams. Among the streams that do not contain any search keyword,
roughly 50% are not in English. Sampling English streams that do not contain
any keyword, we find that they are either on crypto-adjacent topics (e.g.,
trading) or completely irrelevant topics (e.g., live street camera).

###  B.3. Characteristics of scam livestreams

In our 14-day pilot study, we sought to understand the basic characteristics
of scam livestreams and websites, which in turn informed the configuration of
the parameters in our pipeline (e.g., for how long we record a scam
livestream). Specifically, we are interested in two questions: (1) how long a
QR code lasts once it appears and (2) how many chat messages are posted during
the lifespan of a scam livestream. We started by running our pipeline at
maximum capacity on YouTube, with the default setup of recording a stream for
two seconds as well as retrieving up to 70 historical chat messages every 7.5
minutes. In addition, for a subset of the livestreams, we kept recording the
stream upon detecting a QR code.

With this setup, we manually identified 276 streams that promoted 59 unique
giveaway websites. To answer the first question, we examined how long a QR
code lasted after we first detected it. For 41 scam livestreams, we continued
to record them upon detecting a QR code. We found that for the majority of the
livestreams, a QR code remained displayed continuously for a reasonable amount
of time (mean 7,200 seconds and median 3,140 seconds), except for one case
where the QR code was displayed periodically for around 15 seconds. This
finding suggests that two-second samples are enough to capture a QR code in
the livestream, if present. For the latter question, we examined scam streams
that had chat messages. We found that all scam streams had few chat messages
(less than 10) and no user interactions, and thus decided that 70 historical
chat messages every 7.5 minutes is sufficient.

###  B.4. Cloaking Behaviors of Scam Websites

As part of our pilot, we identified four types of cloaking behavior: (1) IP-
based cloaking, in which requests from an institutional network resulted in
403 responses but ones from a residential network did not; (2) user agent-
based cloaking, in which requests from browsers that were not running on
Windows or Mac resulted in 403 responses; (3) customized front pages, in which
human interaction was required to click on a button or select a desired
cryptocurrency; and (4) anti-bot detection through Cloudflare. As previously
mentioned, to counter these behaviors, we (1) used a commercial VPN to mask
our IP address; (2) spoofed our User-Agent string to appear as a popular
device and browser; (3) added a heuristic module to click through common front
pages; and (4) verified our crawler with Cloudflare to avoid triggering their
anti-bot detection.


# Finite distributive nearlattices

Luciano J. González  CIC and Departamento de Matemática, Facultad de Ciencias
Exactas, Universidad Nacional del Centro, Pinto 399, 7000 Tandil, Argentina.
<EMAIL_ADDRESS>and  Ismael Calomino

###### Abstract.

Our main goal is to develop a representation for finite distributive
nearlattices through certain ordered structures. This representation
generalizes the well-known representation given by Birkhoff for finite
distributive lattices through finite posets. We also study finite distributive
nearlattices through the concepts of dual atoms, boolean elements,
complemented elements and irreducible elements. We prove that the sets of
boolean elements and complemented elements form semi-boolean algebras. We show
that the set of boolean elements of a finite distributive lattice is a boolean
lattice.

###### Key words and phrases:

Nearlattice; Lattice; Representation theorem; Irreducible elements

###### 2010 Mathematics Subject Classification:

Primary 06D75; Secondary 06A12; 06D05

##  1\. Introduction and preliminaries

It is well known that the class of distributive lattices have many and
important applications, especially in logic and computer science. Thus, it is
also important to study some natural generalizations of distributive lattices
which may result interesting. An important class of distributive lattices for
computer science is the class of finite distributive lattices. This paper
deals with the concept of _finite distributive nearlattice_ , which is a
natural and nice generalization of finite distributive lattice.

Distributive nearlattices were study from different points of view: algebraic,
topological and logical [15, 21, 14, 2, 7, 11, 19, 8, 12, 6, 9, 10, 17, 18].

In Section 2 we consider the notions of dual atom and boolean element. We
prove that the boolean elements form a semi-boolean algebra ([1]). In Section
3 we show that the semi-boolean algebra of boolean elements of a finite
distributive nearlattice $\mathbf{A}$ is a homomorphic image of $\mathbf{A}$.
We generalize the concept of complement elements from distributive lattice to
distributive nearlattice and, we show that the set of complement elements of a
distributive nearlattice is a semi-boolean algebra. Section 4 is concerned
with the free distributive lattice extension of a distributive nearlattice
([11]). Finally, in Section 5 we develop a discrete representation for the
class of finite distributive nearlattice, which is a generalization of the
well-known representation for finite distributive lattice given by Birkhoff
[4].

We assume that the reader is familiar with the theory of ordered sets and
lattices. Our main references for Order and Lattice theory are [16, 20, 3].
Now we introduce the notational conventions that we use throughout the paper.

Let $P=\langle P,\leq\rangle$ be a poset. A subset $U\subseteq P$ is said to
be an _upset_ when for all $a,b\in P$, if $a\leq b$ and $a\in U$, then $b\in
U$. Dually we have the notion of _downset_. Given an element $a\in P$, the
_principal upset_ generated by $a$ is $[a):=\\{x\in P:a\leq x\\}$. Dually we
have the _principal downset_ $(a]$. Given a subset $Q$ of $P$ and $a\in P$,
$[a)_{Q}:=\\{x\in Q:a\leq x\\}$.

######  Definition 1.1 ([15]).

A _distributive nearlattice_ is a join-semilattice $\mathbf{A}=\langle
A,\vee,1\rangle$ with top element 1 such that for all $a\in A$, the principal
upset $[a)$ is a bounded distributive lattice (with respect to the induced
order).

Let $\mathbf{A}=\langle A,\vee,1\rangle$ be a distributive nearlattice. For
each $a\in A$, we denote the meet in $[a)$ by $\wedge_{a}$. It should be noted
that for all $x,y\in A$, the meet $x\wedge y$ exists in $A$ if and only if
$x,y$ have a common lower bound in $A$. Thus, for all $x,y\in[a)$, the meet of
$x,y$ in $[a)$ coincides with their meet in $A$, that is,
$x\wedge_{a}y=x\wedge y$. This should be kept in mind since we will use it
without mention. A subset $F\subseteq A$ is called _filter_ if (i) $1\in F$;
(ii) $F$ is an upset; and (iii) if $a,b\in F$ and $a\wedge b$ exists in $A$,
then $a\wedge b\in F$. Let us denote by $\mathrm{Fi}(\mathbf{A})$ the
collection of all filters of $\mathbf{A}$. It is straightforward that
$\mathrm{Fi}(\mathbf{A})$ is a closure system. For $X\subseteq A$, we denote
by $\mathrm{Fig}_{A}(X)$ the filter generated by $X$. Thus, we have that
$\langle\mathrm{Fi}(\mathbf{A}),\cap,\veebar,\\{1\\},A\rangle$ is a
distributive lattice where $F\veebar G=\mathrm{Fig}_{A}(F\cup G)$, for all
$F,G\in\mathrm{Fi}(\mathbf{A})$.

A non-empty subset $I\subseteq A$ is said to be an _ideal_ when (i) $I$ is a
downset; (ii) if $a,b\in I$, then $a\vee b\in I$. A proper ideal $I$ of $A$ is
called _prime_ when for all $a,b\in A$, if $a\wedge b$ exists and $a\wedge
b\in I$, then $a\in I$ or $b\in I$.

######  Theorem 1.2 ([21]).

Let $\mathbf{A}$ be a distributive nearlattice. If $I$ is an ideal and $F$ is
a filter of $\mathbf{A}$ such that $I\cap F=\emptyset$, then there exists a
prime filter $P$ of $A$ such that $I\subseteq P$ and $P\cap F=\emptyset$.

Given two distributive nearlattices $\mathbf{A}_{1}$ and $\mathbf{A}_{2}$, a
map $f\colon A_{1}\to A_{2}$ is said to be an _N-homomorphism_ if $f(1)=1$;
for all $a,b\in A_{1}$, $f(a\vee b)=f(a)\vee f(b)$, and $f(a\wedge
b)=f(a)\wedge f(b)$, whenever $a\wedge b$ exists in $A_{1}$.

The reader is referred to [13, 15, 5] for further information about
distributive nearlattices.

##  2\. Dual atoms and boolean elements

######  Definition 2.1.

Let $\mathbf{A}$ be a distributive nearlattice and $a\in A$. We will say that
$a$ is a dual atom if $a\neq 1$ and for each $x\in A$, $a\leq x\leq 1$ implies
$x=a$ or $x=1$. We say that $a$ is a boolean element if the principal filter
$[a)$ is a boolean lattice.

Let us denote by $\mathrm{At}_{\mathrm{d}}(\mathbf{A})$ and
$\mathrm{Bo}(\mathbf{A})$ the collections of all dual atoms of $\mathbf{A}$
and of all boolean elements of $\mathbf{A}$, respectively. It follows that
$\mathrm{At}_{\mathrm{d}}(\mathbf{A})\subseteq\mathrm{Bo}(\mathbf{A})$. If
$a\in\mathrm{Bo}(\mathbf{A})$ and $x\in[a)$, we shall denote the complement of
$x$ in $[a)$ by $\neg_{a}x$. The next properties will be used throughout the
paper without mention.

######  Remark 2.2.

Let $a\in\mathrm{At}_{\mathrm{d}}(\mathbf{A})$. Then, for all $x\in A$, we
have that $x\leq a$ or $x\vee a=1$. Moreover, if $\mathbf{A}$ is a finite
distributive nearlattice, then for every $x<1$, there exists
$a\in\mathrm{At}_{\mathrm{d}}(\mathbf{A})$ such that $x\leq a$.

######  Proposition 2.3.

Let $\mathbf{A}$ be a distributive nearlattice. Let
$a_{1},\dots,a_{n}\in\mathrm{At}_{\mathrm{d}}(\mathbf{A})$ be such that
$a_{1}\wedge\dots\wedge a_{n}$ exists in $A$. Then $a_{1}\wedge\dots\wedge
a_{n}\in\mathrm{Bo}(\mathbf{A})$.

###### Proof.

Let $a^{*}=a_{1}\wedge\dots\wedge a_{n}$. Since
$\langle[a^{*}),\wedge_{a^{*}},\vee,a^{*},1\rangle$ is a bounded distributive
lattice, it only remains to verify that each element of $[a^{*})$ has a
complement. For each $a_{i}$, let us denote the complement of an element
$x\in[a_{i})$ by $\neg_{i}x$. That is, for every $x\in[a_{i})$,
$\neg_{i}x\in[a_{i})$ such that $x\wedge_{a_{i}}\neg_{i}x=a_{i}$ and
$x\vee\neg_{i}x=1$.

Let $x\in[a^{*})$. Thus $x=(a_{1}\vee x)\wedge\dots\wedge(a_{n}\vee x)$. We
define the element $\neg x:=\neg_{1}(a_{1}\vee
x)\wedge\dots\wedge\neg_{n}(a_{n}\vee x)$ (notice that the last meet exists
because $a^{*}\leq a_{i}\leq\neg_{i}(a_{i}\vee x)$, for all $i$). It is clear
that $\neg x\in[a^{*})$. Now we show that $\neg x$ is the complement of $x$ in
$[a^{*})$. On the one hand, we have

$\displaystyle\neg x\wedge x$ $\displaystyle=\bigwedge_{1\leq i\leq
n}\neg_{i}(a_{i}\vee x)\wedge\bigwedge_{1\leq i\leq n}(a_{i}\vee
x)=\bigwedge_{1\leq i\leq n}[\neg_{i}(a_{i}\vee x)\wedge(a_{i}\vee x)]$
$\displaystyle=\bigwedge_{1\leq i\leq n}a_{i}=a^{*}.$

On the other hand, we have

$\neg x\vee x=\left(\bigwedge_{1\leq i\leq n}\neg_{i}(a_{i}\vee x)\right)\vee
x=\bigwedge_{1\leq i\leq n}\left(\neg_{i}(a_{i}\vee x)\vee x\right).$

Since every $a_{i}$ is a dual atom, it follows that

$\neg_{i}(a_{i}\vee x)=\begin{cases}1\ &\text{ if }\ x\leq a_{i}\\\ a_{i}\
&\text{ if }\ x\nleq a_{i}\end{cases}$

Hence, in any case, we obtain that $\neg_{i}(a_{i}\vee x)\vee x=1$. Then $\neg
x\vee x=1$. We have proved that $\neg x$ is the complement of $x$ in
$[a^{*})$. Therefore, $\langle[a^{*}),\wedge_{a^{*}},\vee,a^{*},1\rangle$ is a
boolean lattice, and thus $a^{*}=a_{1}\wedge\dots\wedge
a_{n}\in\mathrm{Bo}(\mathbf{A})$. ∎

Let $\mathbf{A}$ be a distributive nearlattice and $a\in A$. We define the
following set:

$X_{a}:=\\{x\in\mathrm{At}_{\mathrm{d}}(\mathbf{A}):a\leq x\\}.$

Thus,
$X_{a}=\mathrm{At}_{\mathrm{d}}(\mathbf{A})\cap[a)=\mathrm{At}_{\mathrm{d}}([a))$.
If $\mathbf{A}$ is finite and $a\in\mathrm{Bo}(\mathbf{A})$, then since $[a)$
is a finite boolean lattice it follows that

$a=\bigwedge X_{a}.$

######  Lemma 2.4.

Let $\mathbf{A}$ be a distributive nearlattice and $a,b\in A$. Then $X_{a\vee
b}=X_{a}\cap X_{b}$. If $a\wedge b$ exists, then $X_{a\wedge b}=X_{a}\cup
X_{b}$.

###### Proof.

It is straightforward to show directly that $X_{a\vee b}=X_{a}\cap X_{b}$.

It is clear that $X_{a}\cup X_{b}\subseteq X_{a\wedge b}$. Let now $x\in
X_{a\wedge b}$. Since $a\wedge b\leq x$, we have that $x=(a\vee x)\wedge(b\vee
x)$. Given that $x\in\mathrm{At}_{\mathrm{d}}(\mathbf{A})$, it follows that
$(a\leq x\text{ or }a\vee x=1)$ and $(b\leq x\text{ or }b\vee x=1)$. If $a\vee
x=1$ and $b\vee x=1$, then $x=1$, which is a contradiction. Hence, $a\leq x$
or $b\leq x$. That is, $x\in X_{a}\cup X_{b}$. Therefore $X_{a\wedge
b}\subseteq X_{a}\cup X_{b}$. ∎

######  Proposition 2.5.

Let $\mathbf{A}=\langle A,\vee,1\rangle$ be a finite distributive nearlattice.
Then $\langle\mathrm{Bo}(\mathbf{A}),\vee,1\rangle$ is a nearlattice
subalgebra of $\mathbf{A}$.

###### Proof.

We need to prove that $\mathrm{Bo}(\mathbf{A})$ is closed under $\vee$, and if
$a,b\in\mathrm{Bo}(\mathbf{A})$ are such that $a\wedge b$ exists in $A$, then
$a\wedge b\in\mathrm{Bo}(\mathbf{A})$.

Let $a,b\in\mathrm{Bo}(\mathbf{A})$. Notice that $a\vee b\leq\bigwedge
X_{a\vee b}$, and since $a,b\in\mathrm{Bo}(\mathbf{A})$, it follows that
$a=\bigwedge X_{a}$ and $b=\bigwedge X_{b}$. By Proposition 2.3, we have
$\bigwedge X_{a\vee b}\in\mathrm{Bo}(\mathbf{A})$. Let us show that $a\vee
b=\bigwedge X_{a\vee b}$. Suppose, towards a contradiction, that $a\vee
b\neq\bigwedge X_{a\vee b}$. So $\bigwedge X_{a\vee b}\nleq a\vee b$. Then,
there is a prime ideal $P$ such that $a\vee b\in P$ and $\bigwedge X_{a\vee
b}\notin P$. Thus, $a,b\in P$ and $P\cap X_{a\vee b}=\emptyset$. Since
$\bigwedge X_{a}\in P$, $\bigwedge X_{b}\in P$ and $P$ is prime, it follows
that there is $x_{a}\in X_{a}$ such that $x_{a}\in P$ and there is $x_{b}\in
X_{b}$ such that $x_{b}\in P$. Since $x_{a}$ and $x_{b}$ are dual atoms, we
have

$x_{a}\vee x_{b}=\begin{cases}1\ &\text{ if }\ x_{a}\neq x_{b}\\\ x_{a}=x_{b}\
&\text{ if }\ x_{a}=x_{b}\end{cases}$

Since $P$ is proper and $x_{a}\vee x_{b}\in P$, we have that $x_{a}=x_{b}$.
Let $x:=x_{a}=x_{b}$. Thus $a\vee b\leq x$. Then $x\in X_{a\vee b}\cap P$,
which is a contradiction. Hence $a\vee b=\bigwedge X_{a\vee
b}\in\mathrm{Bo}(\mathbf{A})$. Let now $a,b\in\mathrm{Bo}(\mathbf{A})$ be such
that $a\wedge b$ exists in $A$. By Lemma 2.4, we have

$\bigwedge X_{a\wedge b}=\bigwedge(X_{a}\cup X_{b})=\bigwedge
X_{a}\wedge\bigwedge X_{b}=a\wedge b.$

Then, it follows by Proposition 2.3 that $a\wedge b=\bigwedge X_{a\wedge
b}\in\mathrm{Bo}(\mathbf{A})$. Therefore, $\mathrm{Bo}(\mathbf{A})$ is a
nearlattice subalgebra of $\mathbf{A}$. ∎

######  Proposition 2.6.

Let $\mathbf{A}$ be a finite distributive nearlattice. Let
$a\in\mathrm{Bo}(\mathbf{A})$ and $b\in A$. If $a\leq b$, then
$b\in\mathrm{Bo}(\mathbf{A})$. That is,
$[a)_{\mathrm{Bo}(\mathbf{A})}=[a)_{A}$.

###### Proof.

Let $a\in\mathrm{Bo}(\mathbf{A})$ and $b\in A$ be such that $a\leq b$. Since
$a$ is a boolean element, we have $a=\bigwedge X_{a}$. We also have that
$b\leq\bigwedge X_{b}$. Suppose that $\bigwedge X_{b}\nleq b$. Then, there is
a prime ideal $P$ such that $b\in P$ and $\bigwedge X_{b}\notin P$. Thus,
$a\in P$ and $X_{b}\cap P=\emptyset$. Since $\bigwedge X_{a}\in P$ and $P$ is
prime, we obtain that there is $x\in X_{a}\cap P$. Then, $x\notin X_{b}$.
Since $x\in\mathrm{At}_{\mathrm{d}}(\mathbf{A})$ and $b\nleq x$, it follows
that $b\vee x=1$. This is a contradiction because $b,x\in P$ and $P$ is a
proper ideal. Hence $b=\bigwedge X_{b}\in\mathrm{Bo}(\mathbf{A})$. ∎

A _semi-boolean algebra_ ([1]) is a join-semilattice with a top element such
that every principal upset is a Boolean lattice. In [1] Abbot show that the
semi-boolean algebras are in a one-to-one correspondence with the implication
algebras (also known as Tarski algebras).

######  Theorem 2.7.

Let $\mathbf{A}=\langle A,\vee,1\rangle$ be a finite distributive nearlattice.
Then, the nearlattice subalgebra
$\langle\mathrm{Bo}(\mathbf{A}),\vee,1\rangle$ is a semi-boolean algebra.

###### Proof.

Let $a\in\mathrm{Bo}(\mathbf{A})$. By Proposition 2.6, we have
$[a)_{\mathrm{Bo}(\mathbf{A})}=[a)_{A}$, and since $\mathrm{Bo}(\mathbf{A})$
is a nearlattice subalgebra of $\mathbf{A}$, it follows that
$[a)_{\mathrm{Bo}(\mathbf{A})}$ is a boolean lattice. Hence
$\mathrm{Bo}(\mathbf{A})$ is a semi-boolean algebra. ∎

######  Proposition 2.8.

Let $A$ be a finite distributive nearlattice. The following are equivalent:

  1. (1)

$\bigwedge\mathrm{At}_{\mathrm{d}}(\mathbf{A})$ exists in $A$;

  2. (2)

There is $a\in A$ such that $\mathrm{Bo}(\mathbf{A})=[a)$;

  3. (3)

$\mathrm{Bo}(\mathbf{A})$ is a boolean lattice.

###### Proof.

$(1)\Rightarrow(2)$ Let $a:=\bigwedge\mathrm{At}_{\mathrm{d}}(\mathbf{A})$. It
follows by Proposition 2.3 that $a\in\mathrm{Bo}(\mathbf{A})$. Let
$b\in\mathrm{Bo}(\mathbf{A})$. Then,
$a=\bigwedge\mathrm{At}_{\mathrm{d}}(\mathbf{A})\leq\bigwedge X_{b}=b$. Thus
$b\in[a)$. Hence $\mathrm{Bo}(\mathbf{A})\subseteq[a)$. Now, by Proposition
2.6, we have
$[a)=[a)_{\mathrm{Bo}(\mathbf{A})}\subseteq\mathrm{Bo}(\mathbf{A})$.
Therefore, $\mathrm{Bo}(\mathbf{A})=[a)$.

$(2)\Rightarrow(3)$ It follows straightforward by Theorem 2.7.

$(3)\Rightarrow(1)$ Let $a\in A$ be the least element of
$\mathrm{Bo}(\mathbf{A})$. Thus
$[a)=[a)_{\mathrm{Bo}(\mathbf{A})}=\mathrm{Bo}(\mathbf{A})$. Since
$\mathrm{At}_{\mathrm{d}}(\mathbf{A})\subseteq\mathrm{Bo}(\mathbf{A})=[a)$, we
obtain that $\bigwedge\mathrm{At}_{\mathrm{d}}(\mathbf{A})$ exists in $A$. ∎

######  Corollary 2.9.

If $\mathbf{L}$ is a finite distributive lattice, then
$\mathrm{Bo}(\mathbf{L})$ is a boolean lattice. Moreover, for each
$a\in\mathrm{Bo}(\mathbf{L})$ and $x\in[a)$, we have $\neg_{a}x=\neg x\vee a$,
where $\neg x$ denotes the complement of $x$ in $\mathrm{Bo}(\mathbf{L})$.

###### Proof.

From the previous proposition, it is clear that $\mathrm{Bo}(\mathbf{L})$ is a
boolean lattice. Let $a\in\mathrm{Bo}(\mathbf{L})$ and $x\in[a)$. Since $[a)$
is a boolean lattice, there is unique $\neg_{a}x\in[a)$ such that
$x\vee\neg_{a}x=1$ and $x\wedge\neg_{a}x=a$. As $a\leq x$, by Proposition 2.6
we have $x\in\mathrm{Bo}(\mathbf{L})$ and, there is $\neg
x\in\mathrm{Bo}(\mathbf{L})$ such that $x\vee\neg x=1$ and $x\wedge\neg x=c$,
where $c\in A$ is the least element of $\mathrm{Bo}(\mathbf{L})$. Consider the
element $\neg x\vee a\in[a)$. So, $(\neg x\vee a)\vee x=(\neg x\vee x)\vee
a=1\vee a=1$ and $(\neg x\vee a)\wedge x=(\neg x\wedge x)\vee(a\wedge x)=c\vee
a=a$. Hence, we have $\neg_{a}x=\neg x\vee a$. ∎

##  3\. Annihilators

Let $\mathbf{A}=\langle A,\vee,1\rangle$ be a distributive nearlattice. For
every $a\in A$, let $a^{\top}=\\{x\in A:a\vee x=1\\}$ (see [7, 14]). It
follows that $a^{\top}\in\mathrm{Fi}(\mathbf{A})$, for all $a\in A$. We will
say that an element $a\in A$ is _dense_ if $a^{\top}=\\{1\\}$. Denote by
$\mathrm{D}(A)$ the set of all dense elements of $A$.

######  Proposition 3.1.

Let $\mathbf{A}$ be a finite distributive nearlattice. The following are
equivalent:

  1. (1)

$\bigwedge\mathrm{At}_{\mathrm{d}}(\mathbf{A})$ exists in $A$;

  2. (2)

There is $a\in A$ such that $\mathrm{D}(A)=(a]$;

  3. (3)

$\mathrm{D}(A)$ is not empty.

###### Proof.

$(1)\Rightarrow(2)$ Let $a:=\bigwedge\mathrm{At}_{\mathrm{d}}(\mathbf{A})$.
Let $x\in a^{\top}$. So $x\vee a=1$. Suppose that $x<1$. Then, there is
$b\in\mathrm{At}_{\mathrm{d}}(\mathbf{A})$ such that $x\leq b$. Thus $b=b\vee
a=1$, which is a contradiction. Hence $x=1$. It follows that
$a^{\top}=\\{1\\}$, and thus $a\in\mathrm{D}(A)$. We obtain that
$(a]\subseteq\mathrm{D}(A)$. Let now $b\in\mathrm{D}(A)$. So
$b^{\top}=\\{1\\}$. Let $x\in\mathrm{At}_{\mathrm{d}}(\mathbf{A})$. Since
$x\notin b^{\top}$, we have $b\vee x\neq 1$. Thus $b\leq x$. Then
$b\leq\bigwedge\mathrm{At}_{\mathrm{d}}(\mathbf{A})=a$. Hence $b\in(a]$.
Therefore, $\mathrm{D}(A)\subseteq(a]$.

$(2)\Rightarrow(3)$ It is immediate.

$(3)\Rightarrow(1)$ Let us show that $\mathrm{At}_{\mathrm{d}}(\mathbf{A})$
has a lower bound. As $\mathrm{D}(A)\neq\emptyset$, there is $a\in A$ such
that $a^{\top}=\\{1\\}$. Let $x\in\mathrm{At}_{\mathrm{d}}(\mathbf{A})$ and
suppose that $a\nleq x$. Thus, $x\vee a=1$. Then $x\in a^{\top}=\\{1\\}$,
which is a contradiction because $x\in\mathrm{At}_{\mathrm{d}}(\mathbf{A})$.
Hence $a\leq x$, for all $x\in\mathrm{At}_{\mathrm{d}}(\mathbf{A})$.
Therefore, $\bigwedge\mathrm{At}_{\mathrm{d}}(\mathbf{A})$ exists in $A$. ∎

######  Proposition 3.2.

Let $\mathbf{A}$ be a finite distributive nearlattice. The following
properties are satisfied:

  1. (1)

Let $a,b\in\mathrm{Bo}(\mathbf{A})$. Then, $a\leq b$ if and only if
$a^{\top}\subseteq b^{\top}$.

  2. (2)

For each $a\in A$, we have $a^{\top}=\left(\bigwedge X_{a}\right)^{\top}$.

###### Proof.

$(1)$ It is easy to show that $a\leq b$ implies $a^{\top}\subseteq b^{\top}$.
Conversely, assume that $a^{\top}\subseteq b^{\top}$. Let $x\in X_{b}$. Then
$x\in\mathrm{At}_{\mathrm{d}}(\mathbf{A})$ and $b\vee x=x\neq 1$, i.e.,
$x\notin b^{\top}$. So, $x\notin a^{\top}$. Since $a\vee x\neq 1$ and $x$ is
dual atom, we have $a\leq x$. Thus $x\in X_{a}$. Thus $X_{b}\subseteq X_{a}$.
Therefore, $a=\bigwedge X_{a}\leq\bigwedge X_{b}=b$.

$(2)$ Let $a\in A$ and $b:=\bigwedge X_{a}$. By Proposition 2.3,
$b\in\mathrm{Bo}(\mathbf{A})$. Since $a\leq b$ we have $a^{\top}\subseteq
b^{\top}$. Let $x\in b^{\top}$. Thus,

$1=b\vee x=\left(\bigwedge X_{a}\right)\vee x=\bigwedge\\{y\vee x\colon y\in
X_{a}\\}.$

Then $y\vee x=1$, for all $y\in X_{a}$. Suppose that $a\vee x<1$. So, there
exists $y\in\mathrm{At}_{\mathrm{d}}(\mathbf{A})$ such that $a\vee x\leq y<1$.
Thus $a\leq y$ and $x\leq y$. It follows that $y\in X_{a}$. Then $y=y\vee
x=1$, which is a contradiction. Hence, $a\vee x=1$. Thus $x\in a^{\top}$.
Then, we obtain that $b^{\top}\subseteq a^{\top}$. Therefore,
$a^{\top}=b^{\top}=\left(\bigwedge X_{a}\right)^{\top}$. ∎

Given a finite distributive nearlattice $\mathbf{A}$ we define the map
$\pi_{A}\colon A\to\mathrm{Bo}(\mathbf{A})$ as follows: for every $a\in A$,

$\pi_{A}(a)=\bigwedge X_{a}.$

From Proposition 2.3, it follows that $\pi_{A}$ is well defined.

######  Remark 3.3.

Notice that $a\leq\pi_{A}(a)$, for all $a\in A$. Moreover, for each $a\in A$,
$\pi_{A}(a)$ is the least boolean element $b$ such that $a\leq b$. Thus, we
also have that $\pi_{A}(a)=a$, for all $a\in\mathrm{Bo}(\mathbf{A})$.

######  Proposition 3.4.

Let $\mathbf{A}$ be a finite distributive nearlattice. Then, the map
$\pi_{A}\colon A\to\mathrm{Bo}(\mathbf{A})$ is an onto N-homomorphism.

###### Proof.

Let $a,b\in A$. By definition and Lemma 2.4, we have $\pi_{A}(a\vee
b)=\bigwedge X_{a\vee b}=\bigwedge\left(X_{a}\cap X_{b}\right)$. Since
$X_{a}\cap X_{b}\subseteq X_{a},X_{b}$, it follows that
$\pi_{A}(a)\vee\pi_{A}(b)\leq\pi_{A}(a\vee b)$. By Remark 3.3, we have
$a\leq\pi_{A}(a)$ and $b\leq\pi_{A}(b)$. Thus $a\vee
b\leq\pi_{A}(a)\vee\pi_{A}(b)$. Then, since $\pi_{A}(a)\vee\pi_{A}(b)$ is a
boolean element, it follows by Remark 3.3 that $\pi_{A}(a\vee
b)\leq\pi_{A}(a)\vee\pi_{A}(b)$. Therefore, $\pi_{A}(a\vee
b)=\pi_{A}(a)\vee\pi_{A}(b)$.

Now assume that $a\wedge b$ exists in $\mathbf{A}$. Then,

$\pi_{A}(a\wedge b)=\bigwedge X_{a\wedge b}=\bigwedge\left(X_{a}\cup
X_{b}\right)=\bigwedge X_{a}\wedge\bigwedge X_{b}=\pi_{A}(a)\wedge\pi_{A}(b).$

Finally, $\pi_{\mathbf{A}}(1)=\bigwedge X_{1}=\bigwedge\emptyset=1$. ∎

It is well known that if $\mathbf{L}$ is a bounded distributive lattice, then
the subset $\mathrm{Comp}(\mathbf{L})$ of all complemented elements of $L$
form a boolean algebra. Given that in a distributive nearlattice may not
exists the least element, we generalize the concept of complemented element as
follows. Recall that $\veebar$ denotes the supremum in
$\mathrm{Fi}(\mathbf{A})$ and $a^{\top}\in\mathrm{Fi}(\mathbf{A})$, for all
$a\in A$.

######  Definition 3.5.

Let $\mathbf{A}$ be a distributive nearlattice. An element $a\in A$ is said to
be _complemented_ if $[a)\veebar a^{\top}=A$.

We denote by $\mathcal{C}(\mathbf{A})$ the set of all complemented elements of
$\mathbf{A}$.

######  Proposition 3.6.

If $\mathbf{A}$ is a bounded distributive lattice, then
$\mathrm{Comp}(\mathbf{A})=\mathcal{C}(\mathbf{A})$.

###### Proof.

Let $a\in\mathcal{C}(\mathbf{A})$. So $[a)\veebar a^{\top}=A$. Let $0$ be the
least element of $A$. Since $0\in[a)\veebar a^{\top}$, there is $b\in
a^{\top}$ such that $a\wedge b=0$. Thus, we have $a\vee b=1$ and $a\wedge
b=0$. Then, $b$ is the complement of $a$ in $A$. Hence $a\in\mathrm{Comp}(A)$.

Let now $a\in\mathrm{Comp}(A)$. Let $a^{*}\in A$ be the complement of $a$.
Thus $a\wedge a^{*}=0$ and $a\vee a^{*}=1$. Let $x\in A$. Then $(a\vee
x)\wedge(a^{*}\vee x)=(a\wedge a^{*})\vee x=x$. Since $x\vee a\in[a)$ and
$a^{*}\vee x\in a^{\top}$, it follows that $x\in[a)\veebar a^{\top}$. Hence
$[a)\veebar a^{\top}=A$. Therefore, $a\in\mathcal{C}(\mathbf{A})$. ∎

######  Proposition 3.7.

Let $\mathbf{A}=\langle A,\vee,1\rangle$ be a distributive nearlattice. Then
$\langle\mathcal{C}(\mathbf{A}),\vee,1\rangle$ is a nearlattice subalgebra of
$\mathbf{A}$.

###### Proof.

Let $a,b\in\mathcal{C}(\mathbf{A})$. Thus $[a)\veebar a^{\top}=[b)\veebar
b^{\top}=A$. Then,

$[a\vee b)\veebar(a\vee b)^{\top}=\left([a)\cap[b)\right)\veebar(a\vee
b)^{\top}=\left([a)\veebar(a\vee b)^{\top}\right)\cap\left([b)\veebar(a\vee
b)^{\top}\right).$

Since $a\leq a\vee b$, we have $a^{\top}\subseteq(a\vee b)^{\top}$. Thus
$A=[a)\veebar a^{\top}\subseteq[a)\veebar(a\vee b)^{\top}$. Then
$A=[a)\veebar(a\vee b)^{\top}$. Analogously, we have $A=[b)\veebar(a\vee
b)^{\top}$. It follows $[a\vee b)\veebar(a\vee b)^{\top}=A$. Hence, $a\vee
b\in\mathcal{C}(\mathbf{A})$.

Assume now that $a\wedge b$ exists in $A$. Then,

$\displaystyle[a\wedge b)\veebar(a\wedge b)^{\top}$
$\displaystyle=([a)\veebar[b))\veebar\left(a^{\top}\cap b^{\top}\right)$
$\displaystyle=\left([a)\veebar[b)\veebar
a^{\top}\right)\cap\left([a)\veebar[b)\veebar b^{\top}\right)=A.$

Hence $a\wedge b\in\mathcal{C}(\mathbf{A})$. Therefore,
$\langle\mathcal{C}(\mathbf{A}),\vee,1\rangle$ is a nearlattice subalgebra of
$\mathbf{A}$. ∎

######  Proposition 3.8.

Let $\mathbf{A}=\langle A,\vee,1\rangle$ be a distributive nearlattice. Then
$\langle\mathcal{C}(\mathbf{A}),\vee,1\rangle$ is a semi-boolean algebra.

###### Proof.

By the previous proposition we know that
$\langle\mathcal{C}(\mathbf{A}),\vee,1\rangle$ is a distributive nearlattice.
Thus, $[a)_{\mathcal{C}(\mathbf{A})}$ is a bounded distributive lattice, for
all $a\in\mathcal{C}(\mathbf{A})$. Let $a\in\mathcal{C}(\mathbf{A})$. Let us
show that each element of the lattice $[a)_{\mathcal{C}(\mathbf{A})}$ has a
complement. Let $b\in[a)_{\mathcal{C}(\mathbf{A})}$. So $a\leq b$ and
$b\in\mathcal{C}(\mathbf{A})$. Then $[b)\veebar b^{\top}=A$. Given that
$a\in[b)\veebar b^{\top}$, there exist $x\in[b)$ and $y\in b^{\top}$ such that
$a=x\wedge y$. Thus $a\leq y$ and $y\vee b=1$. Notice that $a\leq b,y$. Then
$a\leq b\wedge y$. Since $b\leq x$, it follows that $b\wedge y\leq x\wedge
y=a$. Hence $a=b\wedge y$. Now we show that $y\in\mathcal{C}(\mathbf{A})$.
Since $a=b\wedge y$, we obtain that $[a)=[b)\veebar[y)$ and
$a^{\top}=b^{\top}\cap y^{\top}$. Moreover, since $b\in y^{\top}$, it follows
that $[b)\subseteq y^{\top}$. Then, we have

$\displaystyle A$ $\displaystyle=[a)\veebar
a^{\top}=\left([b)\veebar[y)\right)\veebar\left(b^{\top}\cap
y^{\top}\right)=\left([b)\veebar[y)\veebar
b^{\top}\right)\cap\left([b)\veebar[y)\veebar y^{\top}\right)$
$\displaystyle=[y)\veebar y^{\top}.$

Hence, we obtain that $y\in[a)_{\mathcal{C}(\mathbf{A})}$, $b\vee y=1$ and
$b\wedge y=a$. That is, $y$ is the complement of $b$ in
$[a)_{\mathcal{C}(\mathbf{A})}$. Then $[a)_{\mathcal{C}(\mathbf{A})}$ is a
boolean algebra. Therefore, $\mathcal{C}(\mathbf{A})$ is a semi-boolean
algebra. ∎

######  Proposition 3.9 ([6]).

Let $\mathbf{A}$ be a distributive nearlattice. Then, $\mathbf{A}$ is a semi-
boolean algebra if and only if $A=\mathcal{C}(\mathbf{A})$.

######  Proposition 3.10.

Let $\mathbf{A}$ be a finite distributive nearlattice. Then, $\mathbf{A}$ is a
semi-boolean algebra if and only if
$\mathrm{Bo}(\mathbf{A})=\mathcal{C}(\mathbf{A})$.

###### Proof.

Assume that $\mathbf{A}$ is a semi-boolean algebra. Then,
$A=\mathrm{Bo}(\mathbf{A})$ and $A=\mathcal{C}(\mathbf{A})$. Hence
$\mathrm{Bo}(\mathbf{A})=\mathcal{C}(\mathbf{A})$. Conversely, assume that
$\mathrm{Bo}(\mathbf{A})=\mathcal{C}(\mathbf{A})$. Let $a\in A$. Then, by
Proposition 3.2, $a^{\top}=b^{\top}$, where $b=\bigwedge X_{a}$. Thus,
$b\in\mathrm{Bo}(\mathbf{A})=\mathcal{C}(\mathbf{A})$ and $a\leq b$. Then
$A=[b)\veebar b^{\top}\subseteq[a)\veebar b^{\top}=[a)\veebar a^{\top}$. Thus
$a\in\mathcal{C}(\mathbf{A})$. Hence $A=\mathcal{C}(\mathbf{A})$. Therefore,
$\mathbf{A}$ is a semi-boolean algebra. ∎

##  4\. Connection with the free distributive lattice extension

######  Definition 4.1.

Let $\mathbf{A}$ be a distributive nearlattice. A pair $\langle
L(\mathbf{A}),e\rangle$, where $L(\mathbf{A})$ is a bounded distributive
lattice and $e\colon A\to L(\mathbf{A})$ is an N-embedding, is said to be a
_free distributive lattice extension_ of $\mathbf{A}$ if $e[A]$ is finitely
meet-dense in $L(\mathbf{A})$ and the following universal property holds: for
every bounded distributive lattice $\mathbf{M}$ and every N-homomorphism
$h\colon A\to M$, there exists a unique lattice homomorphism
$\widehat{h}\colon L(\mathbf{A})\to M$ such that $h=\widehat{h}\circ e$.

######  Lemma 4.2.

Let $\mathbf{A}$ be a distributive nearlattice and let $\langle
L(\mathbf{A}),e\rangle$ be the free distributive lattice extension of
$\mathbf{A}$. The following properties are satisfied:

  1. (1)

$e\left[[a)\right]=\left[e(a)\right)$.

  2. (2)

$\mathrm{Fig}_{L(\mathbf{A})}\left(e\left[a^{\top}\right]\right)=e(a)^{\top}$.

###### Proof.

$(1)$ It is clear that $e\left[[a)\right]\subseteq[e(a))$. Conversely, if
$u\in\left[e(a)\right)$, then $e(a)\leq u$. Since $e[A]$ is finitely meet-
dense in $L(\mathbf{A})$, there exist $x_{1},\ldots,x_{n}\in A$ such that
$u=e(x_{1})\wedge\ldots\wedge e(x_{n})$. Then $e(a)\leq e(x_{i})$, for all
$i\in\\{1,\ldots,n\\}$. Since $e$ is injective, we have $a\leq x_{i}$,
$\forall i\in\\{1,\ldots,n\\}$. Then, there exists $y:=x_{1}\wedge\ldots\wedge
x_{n}\in[a)$ such that $e(y)=u$. Hence $u\in e\left[[a)\right]$. Therefore,
$e\left[[a)\right]=\left[e(a)\right)$.

$(2)$ We know that $e(a)^{\top}$ is a filter of $L(\mathbf{A})$, and it is
clear that $e\left[a^{\top}\right]\subseteq e(a)^{\top}$. Then, we have
$\mathrm{Fig}_{L(\mathbf{A})}\left(e\left[a^{\top}\right]\right)\subseteq
e(a)^{\top}$. Let now $u\in e(a)^{\top}$. So $u\vee e(a)=1_{L(\mathbf{A})}$.
Let $a_{1},\dots,a_{n}\in A$ be such that $u=e(a_{1})\wedge\dots\wedge
e(a_{n})$. Thus $1_{L(\mathbf{A})}=u\vee e(a)=(e(a_{1})\vee
e(a))\wedge\dots\wedge(e(a_{n})\vee e(a))$. Then
$1_{L(\mathbf{A})}=e(a_{i}\vee a)$ for all $i$. Hence $a_{i}\vee a=1$ for all
$i$. That is, $a_{1},\dots,a_{n}\in a^{\top}$. Thus,
$e(a_{1}),\dots,e(a_{n})\in e\left[a^{\top}\right]$. Then
$u=e(a_{1})\wedge\dots\wedge
e(a_{n})\in\mathrm{Fig}_{L(\mathbf{A})}\left(e\left[a^{\top}\right]\right)$.
Hence,
$\mathrm{Fig}_{L(\mathbf{A})}\left(e\left[a^{\top}\right]\right)=e(a)^{\top}$.
∎

######  Proposition 4.3.

Let $\mathbf{A}$ be a finite distributive nearlattice and $\langle
L(\mathbf{A}),e\rangle$ its free distributive nearlattice extension. Then,

  1. (1)

$e\left[\mathrm{At}_{\mathrm{d}}(\mathbf{A})\right]=\mathrm{At}_{\mathrm{d}}(L(\mathbf{A}))$;

  2. (2)

$e\left[\mathrm{Bo}(\mathbf{A})\right]\subseteq\mathrm{Bo}(L(\mathbf{A}))$;

  3. (3)

$e\left[\mathcal{C}(\mathbf{A})\right]\subseteq\mathrm{Comp}(L(\mathbf{A}))$.

###### Proof.

(1) Let $a\in\mathrm{At}_{\mathrm{d}}(\mathbf{A})$. Let $u\in L(\mathbf{A})$
be such that $e(a)\leq u\leq 1_{L(\mathbf{A})}$ (recall that
$1_{L(\mathbf{A})}=e(1)$). Thus, there are $a_{1},\dots,a_{n}\in A$ such that
$u=e(a_{1})\wedge\dots\wedge e(a_{n})$. Since $e(a)\leq u\leq e(a_{i})$ for
all $i$, we obtain that $a\leq a_{i}\leq 1$ for all $i$. Then, for every $i$,
$a=a_{i}$ or $a_{i}=1$. If $a_{i}=1$ for all $i$, then $u=1_{L(\mathbf{A})}$.
Otherwise, there is $i\in\\{1,\dots,n\\}$ such that $a_{i}=a$. Thus, $e(a)\leq
u\leq e(a_{i})=e(a)$. Then $u=e(a)$. Hence
$e(a)\in\mathrm{At}_{\mathrm{d}}(L(\mathbf{A}))$. Let now
$u\in\mathrm{At}_{\mathrm{d}}(L(\mathbf{A}))$. There are $a_{1},\dots,a_{n}\in
A$ such that $u=e(a_{1})\wedge\dots\wedge e(a_{n})$. Since $u\neq
1_{L(\mathbf{A})}$, it follows that there is $i$, such that $a_{i}\neq 1$.
Then, since $u\leq e(a_{i})<1_{L(\mathbf{A})}$, we have $u=e(a_{i})$.
Moreover, it is straightforward to show directly that
$a_{i}\in\mathrm{At}_{\mathrm{d}}(\mathbf{A})$. Hence $u\in
e\left[\mathrm{At}_{\mathrm{d}}(\mathbf{A})\right]$.

(2) Let $a\in\mathrm{Bo}(\mathbf{A})$. It is clear that
$\langle[e(a)),\wedge,\vee,e(a),1_{L(\mathbf{A})}\rangle$ is a bounded
distributive sublattice of $L(\mathbf{A})$. We need only to prove that every
element of $[e(a))$ is complemented. Recall that $[e(a))=e\left[[a)\right]$.
Thus, let $e(x)\in[e(a))$ with $x\in[a)$. Since $[a)$ is a boolean lattice,
there is $\neg_{a}x\in[a)$ such that $x\vee\neg_{a}x=1$ and
$x\wedge_{a}\neg_{a}x=a$. Then
$1_{L(\mathbf{A})}=e\left(x\vee\neg_{a}x\right)=e(x)\vee
e\left(\neg_{a}x\right)$ and
$e(a)=e\left(x\wedge_{a}\neg_{a}x\right)=e(x)\wedge e\left(\neg_{a}x\right)$.
Thus, $e\left(\neg_{a}x\right)$ is the complement of $e(x)$ in
$\left[e(a)\right)$, i.e., $e\left(\neg_{a}x\right)=\neg_{e(a)}e(x)$.
Therefore, $e(a)\in\mathrm{Bo}(L(\mathbf{A}))$.

(3) Let $a\in\mathcal{C}(\mathbf{A})$. So $[a)\veebar a^{\top}=A$. We need to
prove that $e(a)\in\mathrm{Comp}(L(\mathbf{A}))$. From Proposition 3.6, it is
equivalent to show that $[e(a))\veebar e(a)^{\top}=L(\mathbf{A})$ (here
$\veebar$ is the join of $\mathrm{Fi}(L(\mathbf{A}))$). Since $[a)\veebar
a^{\top}=A$, it follows that

$\mathrm{Fig}_{L(\mathbf{A})}\left(e\left[[a)\veebar
a^{\top}\right]\right)=\mathrm{Fig}_{L(\mathbf{A})}\left(e[A]\right)=L(\mathbf{A}).$

By [11, Thm. 3.3], we know that the map
$\Phi\colon\mathrm{Fi}(\mathbf{A})\to\mathrm{Fi}(L(\mathbf{A}))$ defined by
$\Phi(F)=\mathrm{Fig}_{L(\mathbf{A})}\left(e[F]\right)$ for every
$F\in\mathrm{Fi}(\mathbf{A})$, is a lattice isomorphism. Then,

$\begin{split}\mathrm{Fig}_{L(\mathbf{A})}\left(e\left[[a)\veebar
a^{\top}\right]\right)&=\Phi\left([a)\veebar
a^{\top}\right)=\Phi\left([a)\right)\veebar\Phi\left(a^{\top}\right)\\\
&=\mathrm{Fig}_{L(\mathbf{A})}\left(e\left[[a)\right]\right)\veebar\mathrm{Fig}_{L(\mathbf{A})}\left(e\left[a^{\top}\right]\right)\\\
&=[e(a))\veebar\mathrm{Fig}_{L(\mathbf{A})}\left(e\left[a^{\top}\right]\right).\end{split}$

Hence, by Lemma 4.2, we have

$L(\mathbf{A})=\mathrm{Fig}_{L(\mathbf{A})}\left(e\left[[a)\veebar
a^{\top}\right]\right)=[e(a))\veebar e(a)^{\top}.$

Therefore, we have shown that $e(a)\in\mathcal{C}(L(\mathbf{A}))$. ∎

######  Remark 4.4.

If $\mathbf{A}$ is a finite distributive nearlattice, then $L(\mathbf{A})$ is
a finite distributive lattice. Hence, by Corolary 2.9, it follows that
$\mathrm{Bo}\left(L(\mathbf{A})\right)$ is a boolean lattice and for all
$a\in\mathrm{Bo}(\mathbf{A})$, $e\left(\neg_{a}x\right)=\neg e(x)\vee e(a)$
for all $x\in[a)$.

######  Theorem 4.5.

Let $\mathbf{A}$ be a finite distributive nearlattice and $\langle
L(\mathbf{A}),e\rangle$ its free distributive lattice extension. Then
$\pi_{L(\mathbf{A})}\circ e=e\circ\pi_{A}$, i.e., the following diagram
commutes:

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{A}}$$\scriptstyle{e}$$\textstyle{\mathrm{Bo}({\bf{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\textstyle{L({\bf{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{L({\bf{A}})}}$$\textstyle{\mathrm{Bo}\left(L({\bf{A}})\right)}$
---

###### Proof.

Let $a\in A$. By Proposition 4.3, we obtain that

$\displaystyle\pi_{L(\mathbf{A})}\left(e(a)\right)$
$\displaystyle=\bigwedge\\{z\in\mathrm{At}_{\mathrm{d}}(L(\mathbf{A})):e(a)\leq
z\\}$
$\displaystyle=\bigwedge\\{e(x):x\in\mathrm{At}_{\mathrm{d}}(\mathbf{A}),\,e(a)\leq
e(x)\\}$
$\displaystyle=e\left[\bigwedge\\{x\in\mathrm{At}_{\mathrm{d}}(\mathbf{A}):a\leq
x\\}\right]=e\left(\pi_{\mathbf{A}}(a)\right).\qed$

##  5\. A discrete representation

In this section we develop a representation for the class of finite
distributive nearlattices through certain ordered structures. This
representation is a nice generalization of that given by Birkhoff for finite
distributive lattice through finite posets ([4]).

######  Definition 5.1.

Let $\mathbf{A}$ be a distributive nearlattice. An element $a\in A$ is said to
be _meet-irreducible_ (or simply _irreducible_) if for all $x,y\in A$ such
that $x\wedge y$ exists in $A$, $a=x\wedge y$ implies $a=x$ or $a=y$.

Let us denote by $\mathrm{Irr}(\mathbf{A})$ the set of all irreducible
elements of $\mathbf{A}$. Notice that
$\mathrm{At}_{\mathrm{d}}(\mathbf{A})\subseteq\mathrm{Irr}(\mathbf{A})$.

######  Proposition 5.2.

Let $\mathbf{A}$ be a distributive nearlattice and $a\in A$. Then, $a$ is
irreducible if and only if for all $x,y\in A$ such that $x\wedge y$ exists,
$x\wedge y\leq a$ implies $x\leq a$ or $y\leq a$.

######  Theorem 5.3.

Let $\mathbf{A}$ be a finite distributive nearlattice. Then, for every $a\in
A$,

$a=\bigwedge\\{x\in\mathrm{Irr}(\mathbf{A}):a\leq x\\}.$

###### Proof.

Let $S=\\{a\in A:a\neq\bigwedge\\{x\in\mathrm{Irr}(\mathbf{A}):a\leq x\\}\\}$.
We suppose by contradiction that $S\neq\emptyset$. Then, since $A$ is finite,
$S$ has a maximal element $m$. Thus $m\notin\mathrm{Irr}(\mathbf{A})$. Then,
there exist $a,b\in A$ such that $m=a\wedge b$, $m<a$ and $m<b$. Since $m$ is
a maximal element of $S$, it follows that $a,b\notin S$. Thus
$a=\bigwedge\\{x\in\mathrm{Irr}(\mathbf{A}):a\leq x\\}$ and
$b=\bigwedge\\{\mathrm{Irr}(\mathbf{A}):b\leq x\\}$. Hence, by Proposition
5.2, we have

$\displaystyle m$
$\displaystyle=\left(\bigwedge\\{x\in\mathrm{Irr}(\mathbf{A}):a\leq
x\\}\right)\wedge\left(\bigwedge\\{x\in\mathrm{Irr}(\mathbf{A}):b\leq
x\\}\right)$ $\displaystyle=\bigwedge\\{x\in\mathrm{Irr}(\mathbf{A}):m\leq
x\\}.$

Thus, we obtain that $m\notin S$, which is a contradiction. Therefore,
$S=\emptyset$. ∎

######  Proposition 5.4.

Let $\mathbf{A}$ be a finite distributive nearlattice and $\langle
L(\mathbf{A}),e\rangle$ its free distributive lattice extension. Then,
$e\left[\mathrm{Irr}(\mathbf{A})\right]=\mathrm{Irr}(L(\mathbf{A}))$.

###### Proof.

Let $x\in\mathrm{Irr}(\mathbf{A})$. Let $u,v\in L(\mathbf{A})$ be such that
$e(x)=u\wedge v$. There are $a_{1},\dots,a_{n},b_{1},\dots,b_{m}\in A$ such
that $u=e(a_{1})\wedge\dots\wedge e(a_{n})$ and $v=e(b_{1})\wedge\dots\wedge
e(b_{m})$. Thus, we have that $e(x)\leq e(a_{i}),e(b_{j})$, for all
$i=1,\dots,n$ and $j=1,\dots,m$. Then $x\leq a_{i},b_{j}$, for all $i,j$,
which implies that there exists $a_{1}\wedge\dots\wedge a_{n}\wedge
b_{1}\wedge\dots\wedge b_{m}$. Then $e(x)=e(a_{1}\wedge\dots\wedge a_{n}\wedge
b_{1}\wedge\dots\wedge b_{m})$. So $x=a_{1}\wedge\dots\wedge a_{n}\wedge
b_{1}\wedge\dots\wedge b_{m}$. Since $x$ is irreducible, it follows that
$x=a_{1}\wedge\dots\wedge a_{n}$ or $x=b_{1}\wedge\dots\wedge b_{m}$. Then
$e(x)=u$ or $e(x)=v$. Hence $e(x)\in\mathrm{Irr}(L(\mathbf{A}))$. Therefore
$e\left[\mathrm{Irr}(\mathbf{A})\right]\subseteq\mathrm{Irr}(L(\mathbf{A}))$.
Now it is straightforward to show the inclusion
$\mathrm{Irr}(L(\mathbf{A}))\subseteq e\left[\mathrm{Irr}(\mathbf{A})\right]$.
∎

From now on, given a poset $\langle X,\leq\rangle$, $\mathcal{D}(X)$ will
denote the collection of all downsets of $X$ and let us consider the bounded
distributive lattice $\langle\mathcal{D}(X),\cap,\cup,\emptyset,X\rangle$.

Let $\mathbf{A}$ be a finite distributive nearlattice and $\langle
L(\mathbf{A}),e\rangle$ its free distributive lattice extension. Let us
consider $\mathrm{Irr}(\mathbf{A})$ ($\mathrm{Irr}(L(\mathbf{A}))$) as a sub-
poset of $A$ ($L(\mathbf{A})$), that is, $x\leq y$ if and only if $x\vee y=y$,
for all $x,y\in\mathrm{Irr}(\mathbf{A})$. Since $L(\mathbf{A})$ is a finite
distributive lattice, it follows that the map $\alpha\colon
L(\mathbf{A})\to\mathcal{D}(\mathrm{Irr}(L(\mathbf{A})))$ defined by
$\alpha(u)=\\{z\in\mathrm{Irr}(L(\mathbf{A})):u\nleq z\\}$ is an isomorphism.
Hence, by Proposition 5.4, we have that the map $\widehat{\alpha}\colon
L(\mathbf{A})\to\mathcal{D}(\mathrm{Irr}(\mathbf{A}))$ given by
$\widehat{\alpha}(u)=\\{x\in\mathrm{Irr}(\mathbf{A}):u\nleq e(x)\\}$ is an
isomorphism. Therefore, we have the following.

######  Proposition 5.5.

For every finite distributive nearlattice $\mathbf{A}$, we have that
$\langle\mathcal{D}(\mathrm{Irr}(\mathbf{A})),\widehat{e}\rangle$ is the free
distributive lattice extension of $\mathbf{A}$, where $\widehat{e}\colon
A\to\mathcal{D}(\mathrm{Irr}(\mathbf{A}))$ is given by
$\widehat{e}(a)=(\widehat{\alpha}\circ
e)(a)=\\{x\in\mathrm{Irr}(\mathbf{A}):a\nleq x\\}$.

######  Definition 5.6.

A _DN-structure_ is a pair $\langle X,\gamma\rangle$ such that $X$ is a poset
and $\gamma\colon\mathcal{D}(X)\to\\{0,1\\}$ is a map satisfying the
following:

  1. (S1)

$\gamma(X)=1$;

  2. (S2)

$\gamma([x)^{c})=1$, for all $x\in X$;

  3. (S3)

for all $U,V\in\mathcal{D}(X)$, $U\subseteq V$ implies
$\gamma(U)\leq\gamma(V)$.

We say that a DN-structure $\langle X,\gamma\rangle$ is _finite_ if the poset
$X$ is finite. Let $\langle X,\gamma\rangle$ be a DN-structure. We define

$\mathbf{N}(X):=\\{U\in\mathcal{D}(X):\gamma(U)=1\\}.$

######  Proposition 5.7.

Let $\langle X,\gamma\rangle$ be a finite DN-structure. Then
$\langle\mathbf{N}(X),\cup,X\rangle$ is a distributive nearlattice and
$\langle
X,\leq\rangle\cong\langle\mathrm{Irr}(\mathbf{N}(X)),\subseteq\rangle$.

###### Proof.

First let us show that $\mathbf{N}(X)$ is closed under $\cup$. Let
$U_{1},U_{2}\in\mathbf{N}(X)$. Thus $\gamma(U_{1})=\gamma(U_{2})=1$. Then,
since $U_{1}\subseteq U_{1}\cup U_{2}$, it follows by (S3) that
$\gamma(U_{1}\cup U_{2})=1$. Hence $U_{1}\sqcup U_{2}\in\mathbf{N}(X)$

Now let $U_{1},U_{2},V\in\mathbf{N}(X)$ be such that $V\subseteq U_{1},U_{2}$.
Thus $V\subseteq U_{1}\cap U_{2}$. By (S3), we have $U_{1}\cap
U_{2}\in\mathbf{N}(X)$. Then, $U_{1}\cap U_{2}$ is the meet of $U_{1}$ and
$U_{2}$ in $[V)_{\mathbf{N}(X)}$. Hence
$\langle[V)_{\mathbf{N}(X)},\cap,\cup,U,X\rangle$ is a bounded distributive
lattice. Therefore, $\langle\mathbf{N}(X),\cup,X\rangle$ is a distributive
nearlattice.

It is well-known that $\mathrm{Irr}(\mathcal{D}(X))=\\{[x)^{c}:x\in X\\}$, and
thus $\langle\mathrm{Irr}(\mathcal{D}(X)),\subseteq\rangle\cong\langle
X,\leq\rangle$. Let us prove that
$\mathrm{Irr}(\mathbf{N}(X))=\mathrm{Irr}(\mathcal{D}(X))$. By (S2), we have
$\mathrm{Irr}(\mathcal{D}(X))\subseteq\mathrm{Irr}(\mathbf{N}(X))$. Let now
$U\in\mathrm{Irr}(\mathbf{N}(X))$. Since $U^{c}$ is a finite upset of $X$, we
have that $U^{c}=[x_{1})\cup\dots\cup[x_{n})$, for some $x_{1},\dots,x_{n}\in
U^{c}$. Thus $U=[x_{1})^{c}\cap\dots\cap[x_{n})^{c}$. Then, since
$U\in\mathrm{Irr}(\mathbf{N}(X)$, we obtain that $U=[x_{i})^{c}$, for some
$i=1,\dots,n$. Hence
$\mathrm{Irr}(\mathbf{N}(X))\subseteq\mathrm{Irr}(\mathcal{D}(X))$. Therefore,
we have that $\langle\mathrm{Irr}(\mathbf{N}(X)),\subseteq\rangle\cong\langle
X,\leq\rangle$. ∎

Let $\langle A,\vee,1\rangle$ be a finite distributive nearlattice. Let
$\mathbf{S}(A)=\langle\mathrm{Irr}(\mathbf{A}),\leq\rangle$.

######  Proposition 5.8.

Let $\langle A,\vee,1\rangle$ be a finite distributive nearlattice. Then, the
pair $\langle\mathbf{S}(A),\gamma_{A}\rangle$ is a DN-structure, where
$\gamma_{A}\colon\mathcal{D}(\mathbf{S}(A))\to\\{0,1\\}$ is the map defined by
$\gamma_{A}(U)=1$ if and only if $\bigwedge(\mathbf{S}(A)\setminus U)$ exists
in $A$.

###### Proof.

It is straightforward to show directly that the map $\gamma_{A}$ satisfies
conditions (S1)–(S3). ∎

Given a finite distributive nearlattice $\mathbf{A}$, we have that
$\langle\mathbf{N}(\mathbf{S}(A)),\cup,\mathbf{S}(A)\rangle$ is a finite
distributive nearlattice, where
$\mathbf{N}(\mathbf{S}(A))=\\{U\in\mathcal{D}(\mathbf{S}(A)):\gamma_{A}(U)=1\\}$.

######  Theorem 5.9 (Discrete representation).

Let $\mathbf{A}=\langle A,\vee,1\rangle$ be a finite distributive nearlattice.
Then, $\mathbf{A}\cong\mathbf{N}(\mathbf{S}(A))$.

###### Proof.

From Proposition 5.5, we have that the map $\widehat{e}\colon
A\to\mathcal{D}(\mathbf{S}(A))$ is an N-embedding, where
$\widehat{e}(a)=\\{x\in\mathrm{Irr}(\mathbf{A}):a\nleq x\\}$, for every $a\in
A$. Thus, $\mathbf{A}\cong\widehat{e}[A]$. Let us show that
$\widehat{e}[A]=\mathbf{N}(\mathbf{S}(A))$. Then,

$\displaystyle U\in\mathbf{N}(\mathbf{S}(A))$ $\displaystyle\iff
U\in\mathcal{D}(\mathbf{S}(A))\text{ and }\gamma_{A}(U)=1$ $\displaystyle\iff
U\in\mathcal{D}(\mathbf{S}(A))\text{ and }\bigwedge\mathbf{S}(A)\setminus
U\text{ exists in }A$ $\displaystyle\iff\text{there exists }a\in A\text{ such
that }U=\\{x\in\mathrm{Irr}(\mathbf{A}):a\nleq x\\}$
$\displaystyle\iff\text{there exists }a\in A\text{ such that
}U=\widehat{e}(a)$ $\displaystyle\iff U\in\widehat{e}[A].$

Hence, $\mathbf{A}\cong\widehat{e}[A]=\mathbf{N}(\mathbf{S}(A))$. ∎

######  Remark 5.10.

Let $\mathbf{A}$ be a finite distributive nearlattice. If $\mathbf{A}$ is in
fact a lattice, then it follows that
$\mathbf{N}(\mathbf{S}(\mathbf{A}))=\mathcal{D}(\mathbf{S}(\mathbf{A}))$. Thus
$\mathbf{A}\cong\mathcal{D}(\mathbf{S}(\mathbf{A}))$. Moreover, it is clear
that the finite posets are in one-to-one correspondence with the DN-structures
$\langle X,\gamma\rangle$ such that $\gamma(U)=1$, for all
$U\in\mathcal{D}(X)$. Therefore, from the representation above established we
can obtain the representation given by Birkhoff for finite distributive
lattices.

## Acknowledgments

This work was partially supported by ANPCyT (Argentina) under the Grant
PICT-2019-00882 and by CONICET (Argentina) under the Grant PIP
112-201501-00412. The first author was also partially supported by ANPCyT
under the Grant PICT-2019-00674 and by Universidad Nacional de La Pampa under
the Grant P.I. No 78M, Res. 523/19.

## References

  * [1] J. Abbott.  Semi-boolean algebra.  Matematički Vesnik, 4(19):177–198, 1967.
  * [2] J. Araújo and M. Kinyon.  Independent axiom systems for nearlattices.  Czech. Math. J., 61(4):975–992, 2011.
  * [3] R. Balbes and P. Dwinger.  Distributive lattices.  University of Missouri Press, 1974.
  * [4] G. Birkhoff.  Rings of sets.  Duke Math. J., 3(3):443–454, 1937.
  * [5] I. Calomino.  Supremo álgebra distributivas: una generalización de las álgebra de Tarski.  PhD thesis, Universidad Nacional del Sur, 2015.
  * [6] I. Calomino.  Note on $\alpha$-filters in distributive nearlattices.  Mathematica Bohemica, 144(3):241–250, 2019.
  * [7] I. Calomino and S. Celani.  A note on annihilators in distributive nearlattices.  Miskolc Math. Notes, 16(1):65–78, 2015.
  * [8] I. Calomino, S. Celnai, and L. J. González.  Quasi-modal operators on distributive nearlattices.  Rev. Un. Mat. Argentina, In Press, 2019.
  * [9] I. Calomino and L. J. González.  Remarks on normal distributive nearlattices.  Quaest. Math., pages 1–12, 2020.
  * [10] S. Celani and I. Calomino.  Stone style duality for distributive nearlattices.  Algebra universalis, 71(2):127–153, 2014.
  * [11] S. Celani and I. Calomino.  On homomorphic images and the free distributive lattice extension of a distributive nearlattice.  Rep. Math. Logic, 51:57–73, 2016.
  * [12] S. Celani and I. Calomino.  Distributive nearlattices with a necessity modal operator.  Math. Slovaca, 69:35–52, 2019.
  * [13] I. Chajda, R. Halaš, and J. Kühr.  Semilattice structures, volume 30 of Research and Exposition in Mathematics.  Heldermann Verlag, Lemgo, 2007.
  * [14] I. Chajda and M. Kolařík.  Ideals, congruences and annihilators on nearlattices.  Acta Univ. Palacki. Olomuc. Fac. rer. nat. Mathematica, 46(1):25–33, 2007.
  * [15] I. Chajda and M. Kolařík.  Nearlattices.  Discrete Math., 308(21):4906–4913, 2008.
  * [16] B. Davey and H. Priestley.  Introduction to lattices and order.  Cambridge University Press, 2002.
  * [17] L. J. González.  The logic of distributive nearlattices.  Soft Computing, 22(9):2797–2807, 2018.
  * [18] L. J. González.  Selfextensional logics with a distributive nearlattice term.  Arch. Math. Logic, 58:219–243, 2019.
  * [19] L. J. González and I. Calomino.  A completion for distributive nearlattices.  Algebra Universalis, 80: 48, 2019.
  * [20] G. Grätzer.  Lattice theory: foundation.  Springer Science & Business Media, 2011.
  * [21] R. Halaš.  Subdirectly irreducible distributive nearlattices.  Miskolc Math. Notes, 7:141–146, 2006.