# SMSMix: Sense-Maintained Sentence Mixup for Word Sense Disambiguation

Hee Suk Yoon1, Eunseop Yoon111footnotemark: 1, John Harvill2, Sunjae Yoon1,
Mark Hasegawa-Johnson2, and Chang D. Yoo1
1Korea Advanced Institute of Science and Technology (KAIST)
2University of Illinois at Urbana-Champaign (UIUC)
{hskyoon, esyoon97, sunjae.yoon<EMAIL_ADDRESS>
{harvill2<EMAIL_ADDRESS>Equal contribution Corresponding author

###### Abstract

Word Sense Disambiguation (WSD) is an NLP task aimed at determining the
correct sense of a word in a sentence from discrete sense choices. Although
current systems have attained unprecedented performances for such tasks, the
nonuniform distribution of word senses during training generally results in
systems performing poorly on rare senses. To this end, we consider data
augmentation to increase the frequency of these least frequent senses (LFS) to
reduce the distributional bias of senses during training. We propose Sense-
Maintained Sentence Mixup (SMSMix), a novel word-level mixup method that
maintains the sense of a target word. SMSMix smoothly blends two sentences
using mask prediction while preserving the relevant span determined by
saliency scores to maintain a specific word’s sense. To the best of our
knowledge, this is the first attempt to apply mixup in NLP while preserving
the meaning of a specific word. With extensive experiments, we validate that
our augmentation method can effectively give more information about rare
senses during training with maintained target sense label.

##  1 Introduction

Determining the meaning of a word in a particular sentence is a fundamental
problem in Natural Language Processing (NLP), as it can enable a better
understanding of natural languages and help solve various NLP problems. Word
Sense Disambiguation (WSD) is a crucial task towards accomplishing this goal,
where it involves choosing the most relevant meaning of a target word in
context from predefined sense labels. Like most other NLP tasks, the
advancement of Deep Learning has led supervised learning of neural models to
be the primary method of WSD Huang et al. (2019); Blevins and Zettlemoyer
(2020); Barba et al. (2021). However, one of the biggest challenges of WSD is
overcoming the data bias that naturally stems from the distributional bias of
senses in language Kilgarriff (2004).

Figure 1: Schematic illustration of SMSMix filling underpopulated areas in the
training data space ($\mathcal{D}_{1}$) - linked to least frequent senses
(LFS) - with synthetic examples created through mixup with external corpus
($\mathcal{D}_{2}$) sentences. Figure 2: Overview of SMSMix. (a) $x^{A}$ is a
sentence containing target word $x_{t}^{A}$ with sense label $y^{A}$. The
saliency score of each token is visualized by the concentration of the color.
The sense-maintained span $x_{S}^{A}$ is determined by setting the highest
saliency token from the front and back of the target word $x_{t}^{A}$ as the
start and end of the span, respectively. (b) A random sentence $x^{B}$ with
random span $x_{S}^{B}$ is sampled from an external corpus. (c) $x_{S}^{A}$ is
replaced into $x_{S}^{B}$ with mask tokens. (d) Mask token prediction with T5
smoothly blends the two sentences to minimize the strong-edge problem. The
output results in $\tilde{x}$ containing the target word $x_{t}^{A}$ with
label $\tilde{y}$ set to $y^{A}$.

Because the dataset tends to have this bias toward certain senses, the WSD
system shows high accuracy on the most frequent sense (MFS) and low accuracy
on the least frequent sense (LFS) of a word.

A common solution in machine learning to address data imbalance involves
oversampling under-represented categories, although merely doing so often
leads to overfitting of minority classes Chawla et al. (2002). Another
approach uses data augmentation to drive the learning process toward a more
suitable solution. Due to the expensive cost of data collection, data
augmentation has become one of the essential tools in modern deep learning,
especially when dealing with low-resource tasks. In image classification, some
approaches Chou et al. (2020); Kabra et al. (2020); Galdran et al. (2021) have
been recently explored, involving the well-known data augmentation method
Mixup Zhang et al. (2017) to alleviate data imbalance. This motivation
encourages us to fill underpopulated areas in the training data of WSD -
linked to least frequent senses (LFS) - with synthetic examples created
through mixup (Figure 1).

In this work, we propose Sense-Maintained Sentence Mixup (SMSMix), a novel
mixup data augmentation method where the meaning of a specific word in a
sentence is unaffected as much as possible while exposing it to various
contexts. Inspired by the recently proposed word-level mixup in NLP for text
classification tasks Yoon et al. (2021), we first determine a span of text in
a sentence containing a target word that is most relevant to maintaining the
target word sense using gradient-based saliency scores. Second, because Wu et
al. (2020) mentions that mixup within training data cannot give more
information to the model and acts merely as a regularizer, we inject the
aforementioned span of text into a random sentence from a Wikipedia corpus.
Lastly, when carrying out this injection, we consider smoothly blending the
span to the Wikipedia corpus sentence using mask prediction, inspired by
SmoothMix Lee et al. (2020), where they consider smoothly mixing two images to
solve the strong-edge problem.

SMSMix has empirically shown to be an effective augmentation method that can
give more information about rare senses during training a WSD model in a
supervised setting. We especially show that performing mixup with an external
sentence (i.e., a sentence that is not from the training data) can outperform
internal mixup within training data. Also, by visualizing the latent vector of
the target words in the augmented sentences, we show that SMSMix effectively
maintains the sense of the target word.

##  2 SMSMix

SMSMix generates a new sentence $\tilde{x}$ by injecting span $x_{S}^{A}$,
which contains a target word $x_{t}^{A}$ with sense label $y^{A}$, from
sentence $x^{A}$ into another span $x_{S}^{B}$ from sentence $x^{B}$. Prior to
injection, we concatenate mask tokens to the front and back of span
$x_{S}^{A}$ and perform mask prediction to smoothly blend the two sentences.
Because we use saliency score to preserve the sense label when determining
span $x_{S}^{A}$, we set the label of $x_{t}^{A}$ in the new sentence
$\tilde{x}$ to $y^{A}$ (Figure 2).

###  2.1 Saliency and Sense-Maintained Span

Saliency, which shows how each input fraction affects the final prediction, is
usually measured using gradient-based methods. Yoon et al. (2021) recently
proposed a mixup strategy in NLP using a gradient-based saliency score to
preserve the locality of the two sentences performing mixup. Similarly, we
compute the gradient of classification loss $L$ with respect to the input
token embedding $e$, and apply the L2 norm to get the saliency for each input
token: i.e., $s=||\frac{\partial L}{\partial e}||_{2}$. The saliency of each
input token signifies how influential each input token is for the meaning of
the target word. The most salient span for maintaining the target word sense
is determined by taking the token with the highest saliency score at the front
and back of the target word and setting it to the beginning and the end of the
span, respectively.

###  2.2 Mixing Sentence

We consider two scenarios when injecting the sense-maintained span. First, we
consider injecting the span into a random MFS sentence from the training set
for the corresponding sense (internal). In this case, $x_{S}^{B}$ is
determined in the same way as $x_{S}^{A}$. However, Wu et al. (2020) mentions
that mixup inside the training data has limitations in giving new information
to the model and is effective only as a regularization role. Thus, we also
consider injecting the span into a random sentence from the Wikipedia corpus
(external).

###  2.3 Smoothing

SmoothMix Lee et al. (2020) has been shown to be an effective mixup strategy
in image classification for minimizing the problem of unnatural strong-edge of
the boundary between two images performing the mixup. Motivated by SmoothMix,
we propose to minimize the strong-edge problem in the boundary between the two
contributing sentences. Prior to injecting the sense-maintained span onto the
other sentence, we put a mask token at the front and back of the span. We then
use T5 111We use T5 Version 1.1-large for all the mask predictions.
https://huggingface.co/google/t5-v1_1-large Raffel et al. (2019) for mask
prediction to generate a smoothly transitioning mask and use this to blend two
different texts to form an augmented sample.

##  3 Experimental Setup

###  3.1 Dataset and Model

#### Dataset

We use the SemCor Miller et al. (1993) for training, the largest dataset
manually annotated with sense from WordNet that contains 226,036 annotated
examples covering 33,362 unique senses. As standard procedure, we use
SemEval-2007 (SE07, Pradhan et al. (2007)) as the validation set, while
performing testing on Senseval-2 (SE2; Palmer et al. (2001)), Senseval-3 (SE3,
Snyder and Palmer (2004)), SemEval-2013 (SE13, Navigli et al. (2013)), and
SemEval-2015 (SE15, Moro and Navigli (2015)). Additionally, to measure the
degree to which the system generalizes to LFS, unseen words, and definitions
(zero-shot settings), we consider five subsets of the data (MFS, LFS, 0-lex,
0-lex-def, 0-def) proposed in Barba et al. (2021).

#### Model

We evaluate our augmentation scheme on the BEM Blevins and Zettlemoyer (2020)
system, which employs a bi-encoder to represent the target word and its sense
definitions within the same space.

###  3.2 Training

We adopt a two-stage training strategy as Yoon et al. (2021) and Liu et al.
(2021). After fully training the WSD system, we train one additional epoch
with data augmentation with a learning rate of 5e-7. All other hyperparameters
are set equal to the original BEM training
222https://github.com/facebookresearch/wsd-biencoders. We consider three types
of data augmentation: (1) simple oversampling of LFS (Oversample), (2) SMSMix
with internal training data sentence (SMSMixint), (3) SMSMix with random
external Wikipedia corpus sentence (SMSMixext). Additionally, for both
settings of SMSMix, we automatically filter out those augmented sentences that
are not grammatically acceptable, which could happen when the T5 could not
smoothly blend the span into a sentence 333We use the sentence acceptability
judgment capability of T5-large using the ”cola sentence:” prompt.. For all
three data augmentations, we add three data samples for half of the LFS in the
SemCor training data. When training with data augmentation, we concatenate the
original data with the augmented data to prevent the training data
distribution from getting too far from the original data distribution, as
mentioned in He et al. (2019). All training was done on NVIDIA Quadro RTX
8000.

###  3.3 Evaluation

Performance of the WSD task has generally been reported so far in terms of
micro-average F1 scores. However, doing so gives more weight to frequent
senses simply because they occur more often 444Out of 6,368 test data in the
aggregated ALL evaluation set, 4949 are MFS and 1419 are LFS., thus resulting
in an underrepresentation of the low performances of the least frequent
senses. Therefore, in addition to the micro-averaged F1 scores, we also choose
to report the macro-averaged F1 scores as Maru et al. (2022).

| Dev Set | Test Sets
---|---|---
Methods | SE07 | SE2 | SE3 | SE13 | SE15 | ALL
| m-F1 | M-F1 | m-F1 | M-F1 | m-F1 | M-F1 | m-F1 | M-F1 | m-F1 | M-F1 | m-F1 | M-F1
BEM | 74.5 | 72.4 | 79.4 | 75.5 | 77.4 | 73.0 | 79.7 | 77.0 | 81.7 | 79.1 | 79.0 | 73.9
+Oversample | 74.1 | 72.0 | 79.6 | 75.8 | 77.7 | 73.5 | 79.7 | 77.7 | 81.8 | 79.4 | 79.1 | 74.6
+SMSMixint | 74.3 | 72.0 | 79.4 | 75.7 | 77.9 | 73.8 | 79.3 | 77.5 | 81.9 | 79.1 | 79.1 | 74.4
+SMSMixext | 74.5 | 72.4 | 79.9 | 76.4 | 77.8 | 73.8 | 79.8 | 77.7 | 82.2 | 79.6 | 79.3 | 74.8
|  |  |  |  |  |  |  |  |  |  |  |
Table 1: Comparison of both micro (m-F1) and macro (M-F1) F1 scores on the all-words WSD task against its baseline (BEM). The reported values are an average over five runs with different seeds (The standard deviations are reported in Appendix 5). Best macro and micro F1 scores for each test set are in bold. | MFS | LFS | 0-lex | 0-lex-def | 0-def
---|---|---|---|---|---
| m-F1 | M-F1 | m-F1 | M-F1 | m-F1 | M-F1 | m-F1 | M-F1 | m-F1 | M-F1
BEM | 89.3 | 79.0 | 53.3 | 57.7 | 90.2 | 90.3 | 68.2 | 65.7 | 69.1 | 67.1
\+ Oversample | 89.0 | 79.3 | 54.7 | 59.0 | 90.7 | 90.9 | 68.8 | 66.7 | 69.7 | 68.0
\+ SMSMixint | 89.2 | 79.2 | 54.6 | 59.3 | 90.5 | 90.9 | 68 | 66.3 | 68.2 | 67.2
\+ SMSMixext | 89.3 | 79.7 | 55.0 | 59.7 | 91 | 91.6 | 69.0 | 66.9 | 69.7 | 68.2
|  |  |  |  |  |  |  |  |  |
Table 2: Comparison of both micro (m-F1) and macro (M-F1) F1 scores on MFS,
LFS, and zero-shot datasets. The reported values are an average over five runs
with different seeds (The standard deviations are reported in Appendix 6).
Best macro and micro F1 scores for each dataset are in bold.

##  4 Results and Discussion

###  4.1 Overall Results

Table 1 shows the overall micro (m-F1) and macro F1 (M-F1) results on the
English all-words WSD task. We find that oversampling and SMSMix within
training data (SMSMixint) have a similar slight increase over the original
BEM. Performing SMSMix with external Wikipedia corpus (SMSMixext) shows the
highest increase in performance, obtaining 79.3 m-F1 and 74.8 M-F1 on the
aggregated ALL evaluation set, outperforming the original BEM by 0.3 m-F1 and
0.9 M-F1 points, respectively.

###  4.2 Results on Sense Frequency

In Table 2 we report the results on the five subsets of the data (MFS, LFS,
0-lex, 0-lex-def, 0-def). Compared with BEM without data augmentation, the
performance on LFS of oversampling, SMSMixint, and SMSMixext improved by 1.3,
1.6, and 2.0 M-F1, respectively. As the performance is improved in all three
cases, it can be seen that the approach to mitigate sense imbalance through
data augmentation is effective. In addition, the results have shown that
performing mixup augmentation through external information obtained more
performance gain than the internal mixup.

Figure 3: Latent space visualization of the embeddings of target word _plant
(noun)_ in the labeled OMSTI dataset sentences (original) and augmented
sentences using SMSMix on SemCor training dataset. The embeddings of SMSMix
target words closely overlap with those of original target words, suggesting
that SMSMix maintains the true sense label of the target word.

###  4.3 Sense-Maintained Augmentation

We visually analyze whether SMSMix changes the meaning of the target word. We
first take the context encoder of a BEM system fully trained on the SemCor
dataset without augmentation. Then, we apply SMSMix to the SemCor training set
to generate 50 augmented sentences per sense. Moreover, we obtain 50 labeled
sentences per sense from the OMSTI dataset Taghipour and Ng (2015). These are
then fed into the pre-trained BEM system, and we extract the embedding of the
target words. We plot the 2-D representation of these embeddings using t-SNE
Van Der Maaten (2014). We find that the resulting latent space visualization
of the target words in augmented sentences closely overlaps with those in
labeled sentences (Figure 3), which suggests that SMSMix effectively preserves
the meaning of the target word while mixing it into various contexts.

##  5 Conclusion

This paper introduced a novel input-level mixup data augmentation scheme
SMSMix for improving the Least Frequent Sense (LFS) data imbalance in the Word
Sense Disambiguation task. SMSMix maintains the meaning of a specific word in
a sentence by keeping the sense-maintained span using saliency score and
smoothly injects the span into a different context using mask prediction.
Throughout the experiment, we show that instead of injecting the sense-
maintained span with an internal training data sentence, injecting it into a
random external corpus sentence allows the model to better improve the
performance on LFS.

## Limitations

In this paper, we considered word-level mixup data augmentation to create new
synthetic sentences containing a specific word with preserved meaning. We show
in Appendix A.2 that when the two sentences performing the mixup have a high
contextual difference, the T5 model fails to smoothly blend the two sentences
during mask prediction, resulting in a sentence that is grammatically
incorrect or does not make sense. For future work, we plan on considering
sentence similarity to choose sentences for mixup instead of random selection,
as in the paper.

## Ethics Statement

This paper does not violate the use of others’ work without reference.
Furthermore, the paper does not involve introducing new datasets and the
experiments conducted do not utilize demographic or identity characteristics.

## Acknowledgements

This work was supported by Institute of Information & communications
Technology Planning & Evaluation (IITP) grant funded by the Korea government
(MSIT) (No. 2022-0-00951, Development of Uncertainty-Aware Agents Learning by
Asking Questions), and was partly supported by the National Research
Foundation of Korea (NRF) grant funded by the Korea government(MSIT) (No.
2022R1A2C201270611).

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##  Appendix A Appendix

###  A.1 t-SNE Plots

We present additional t-SNE plots in Figure 4 to show that the augmentation
data generated by our method retain the label of the original data. The method
of obtaining the t-SNE plots is the same as in section 4.3.

###  A.2 Augmentation Examples

We show examples of augmented sentences for several senses in Tables 3 and 4.
Table 3 shows some examples of well-augmented results using our proposed
SMSMix. The chosen span with a target word, determined by the saliency score
from the original sentence, smoothly blends in with a randomly sampled
Wikipedia sentence. On the other hand, the examples shown in Table 4 are
cherry-picked failure examples. There are a few failure cases where there are
incorrect grammar uses or the sentence does not make sense due to the high
discrepancy of context between the two sentences.

###  A.3 Standard Deviation of Results

In Table 5 and Table 6, we report the standard deviation of the results
obtained in Table 1 and Table 2, respectively. The values were obtained by
running the same experiments with five different random seeds.

###  A.4 Location for Span Injection

We randomly choose where to inject the sense-maintained span in a sentence, as
we use T5’s span prediction capability to complete the sentence. To
demonstrate the effectiveness of T5’s span prediction for SMSMix, we use the
example in Figure 2 to create various augmented sentences by injecting the
sense-maintained span into different random locations (Figure 5).

(a) earth (noun)

(b) heart (noun)

(c) school (noun)

Figure 4: Latent space visualization showing that SMSMix maintains the sense
lablel of the target word. Examples
---
target sense |  output%1:10:02:: (signal that comes out of an electronic system)
original |  Outputs of the two systems are measured by a pulse timing circuit and a resistance bridge, followed by a simple analogue computer which feeds a multichannel recorder.
Wikipedia |  The book tells the story of Hendrix and his life through reproductions of rare material such as letters, drawings, postcards and posters.
SMSMix |  The book tells the story of how the inputs and outputs of the two systems are measured through rare material such as letters, drawings, postcards and posters.
target sense |  work%1:04:01:: (the occupation for which you are paid)
original |  In a few places cooperative programs between schools and employers in clerical work have shown the same possibilities for allowing the student, while still in school, to develop skills which are immediately marketable upon graduation.
Wikipedia |  Throughout his career, he was the recipient of more than 30 awards and honors related to engineering, manufacturing, and the development of heavy equipment.
SMSMix |  Throughout his career, his engineering skills and clerical work have been the recipient of more than 30 awards and honors related to engineering, manufacturing ,and the development of heavy equipment.
target sense |  work%2:38:00:: (proceed along a path)
original |  Several photographs and charts of galaxies help the non-scientist keep up with the discussion, and the smooth language indicates the contributors were determined to avoid the jargon that seems to work its way into almost every field.
Wikipedia |  Barbour gave his time free for the next 25 summers to manage field parties throughout the state, surveying the geological and paleontological resources of the State of Nebraska.
SMSMix |  Barbour gave his time free for the next 25 summers to deal with all the jargon that seems to work its way into the process of surveying the geological and paleontological resources of the State of Nebraska.
target sense |  condition%1:10:01:: (an assumption on which rests the validity or effect of something else)
original |  This is what we mean when we say this demand must be accepted without condition.
Wikipedia |  The Society was wound up in the year 2001 when no ordinary members wanted to be nominated as new committee members.
SMSMix |  The Society is open to all, without condition , except in the year 2001 when no ordinary members wanted to be nominated as new committee members.
target sense |  lighting%1:06:00:: (apparatus for supplying artificial light effects for the stage or a film)
original |  When improvements are recommended in working conditions - such as lighting , rest rooms , eating facilities , air-conditioning - do you try to set a measure of their effectiveness on productivity?
Wikipedia |  However, when de Gaulle first introduced the Fouchet Plan in 1961, it faced opposition from many of the member states.
SMSMix |  However, when de Gaulle first introduced the idea, improvements are recommended in working conditions - such as lighting, rest rooms and transport between member states.
|
Table 3: Examples of well-augmented sentences, generated by SMSMix. Red
represents the sense-maintained span in the original sentence, blue represents
a random span in an external sentence that is going to be replaced, and green
represents the span prediction made by T5.  Examples
---
target sense |  make%2:40:02:: (achieve a point or goal)
original |  It is interesting to note that medium compulsives in the unstructured schools made lowest achievement scores ( although not significantly lower ) .
Wikipedia |  Josh tries to distance himself as much as possible from her, for fear of what might happen if she finds out what he is.
SMSMix |  Josh tries to distance himself as much as possible from her, for fear of losing her, but the compulsives in the unstructured schools made the lowest achievement scores in the.
target sense |  employment%1:04:01:: (the act of using)
original |  Another case may be given in illustration of a successful use of analysis, and also of the employment of a procedure for intensive analysis.
Wikipedia |  Ahh!, which was released on May 21, 1988 and would ultimately go on to sell 8 million copies worldwide.
SMSMix |  Ahh! of analysis, and also of the employment of a procedure that would ultimately go on to sell 8 million copies worldwide.
target sense |  replace%2:41:00:: (take the place or move into the position of)
original |  This and raw sugar replace ordinary refined sugar on the tables and very little sugar is used in cooking.
Wikipedia |  These are public housing units and estates aimed at Singaporeans who do not want a HDB flat but might find private property too expensive.
SMSMix |  These are public housing units and estates aimed at Singaporeans who want to sugar replace ordinary sugar a HDB flat but might find private property too expensive.
target sense |  people%1:14:03:: (the common people generally)
original |  Linguists have not always been more enlightened than “practical people” and sometimes have insisted on incredibly trivial points while neglecting things of much greater significance.
Wikipedia |  According to the law of marginal utility, the value of each good in a stock of identical goods is utility of the last and most easily dispensable unit.
SMSMix |  According to the law of marginal utility , the value of each good in a stock of identical goods is “the most practical people” and sometimes have insisted on incredibly trivial details.
target sense |  shift%2:38:02:: (move around)
original |  Important as these differences are, they should not obscure the basic fact that by shifting the hypothalamic balance sufficiently to the parasympathetic side, we produce depressions, whereas a shift in the opposite direction causes excitatory effects and, eventually, maniclike changes.
Wikipedia |  The district is the mining and forestry centre of Suriname, with many large bauxite mining operations operating.
SMSMix |  The district is the mining and forestry centre of Suriname, with many mines by shifting the hypothalamic balance of.
|
Table 4: Failure examples of augmentation by SMSMix. Red represents the sense-maintained span in the original sentence, blue represents a random span in an external sentence that is going to be replaced, and green represents the span prediction made by T5.  Figure 5: Examples of SMSMix using various different injection locations. Red represents the sense-maintained span in the original sentence, blue represents a random span in an external sentence that is going to be replaced, and green represents the span prediction made by T5. | Dev Set | Test Sets
---|---|---
Methods | SE07 | SE2 | SE3 | SE13 | SE15 | ALL
| m-F1 | M-F1 | m-F1 | M-F1 | m-F1 | M-F1 | m-F1 | M-F1 | m-F1 | M-F1 | m-F1 | M-F1
+Oversample | 0.38 | 0.41 | 0.10 | 0.10 | 0.11 | 0.18 | 0.10 | 0.20 | 0.06 | 0.10 | 0.49 | 0.10
+SMSMixint | 0.15 | 0.18 | 0.09 | 0.11 | 0.18 | 0.20 | 0.23 | 0.30 | 0.10 | 0.06 | 0.05 | 0.05
+SMSMixext | 0.19 | 0.22 | 0.04 | 0.09 | 0.08 | 0.06 | 0.09 | 0.15 | 0.08 | 0.09 | 0.05 | 0.05
|  |  |  |  |  |  |  |  |  |  |  |
Table 5: Standard deviation values for the experimental results in Table 1. The values were obtained by running the same experiments with five different random seeds. | MFS | LFS | 0-lex | 0-lex-def | 0-def
---|---|---|---|---|---
| m-F1 | M-F1 | m-F1 | M-F1 | m-F1 | M-F1 | m-F1 | M-F1 | m-F1 | M-F1
\+ Oversample | 0.11 | 0.17 | 0.14 | 0.15 | 0.12 | 0.15 | 0.15 | 0.13 | 0.18 | 0.25
\+ SMSMixint | 0.07 | 0.06 | 0.19 | 0.10 | 0.04 | 0.10 | 0.10 | 0.13 | 0.10 | 0.13
\+ SMSMixext | 0.08 | 0.04 | 0.12 | 0.12 | 0.04 | 0.08 | 0.10 | 0.07 | 0.08 | 0.07
|  |  |  |  |  |  |  |  |  |
Table 6: Standard deviation values for the experimental results in Table 2.
The values were obtained by running the same experiments with five different
random seeds.


# Pointless proofs of the Menger and Rothberger games

Renan M. Mezabarba  Centro de Ciências Exatas, Universidade Federal do
Espírito Santo, Vitória, ES, 29075-910, Brazil<EMAIL_ADDRESS>
###### Abstract.

We present proofs of Hurewicz’s and Pawlikowski’s Theorems concerning
topological games, but in the context of lattices, in which one cannot argue
with points.

###### Key words and phrases:

Menger property, Rothberger property, selection principles, lattices,
pointless topology

###### 2020 Mathematics Subject Classification:

54C65, 91A44, 54H12.

## Introduction

The framework of topological games and selection principles, the latter
introduced by Scheepers [9], have been a fruitful source for both questions
and answers in the field, as can be seen in the surveys [10, 12]. We refer the
reader to the article [1], by Aurichi and Dias, which presents an introduction
to the subject, as well as the main notations we follow along this work.

In an attempt to extend some results regarding selective variations of
tightness in the context of $C_{p}$-theory [2] to the realm of topological
groups, the following two important theorems about open covers appeared to be
relevant:

  * •

Hurewicz’s Theorem [4]. A topological space satisfies
${\mathsf{S}_{\textnormal{fin}}}({\mathcal{O}},{\mathcal{O}})$ if, and only
if, $\mathsf{Player~{}I}$ does not have a winning strategy in the game
$\mathsf{G}_{\operatorname{fin}}({\mathcal{O}},{\mathcal{O}})$;

  * •

Pawlikowski’s Theorem [7]. A topological space satisfies
${\mathsf{S}_{\textnormal{1}}}({\mathcal{O}},{\mathcal{O}})$ if, and only if,
$\mathsf{Player~{}I}$ does not have a winning strategy in the game
$\mathsf{G}_{\operatorname{1}}({\mathcal{O}},{\mathcal{O}})$.

Roughly speaking, our original goal was to settle the same equivalences stated
above but replacing the family ${\mathcal{O}}$, of open covers of a
topological space, with the set $\Omega_{x}$ whose elements are those subsets
of a topological group containing the point $x$ in their closures. Although
Hurewicz’s and Pawlikowski’s Theorems are not directly related to this
problem, their proofs could provide useful insights. This ended up taking us
to the work of Szewczak and Tsaban [11], in which the authors present
conceptual proofs for both the results in such a way that the role of each
hypothesis becomes very clear.

Even though the analysis made so far about their proofs has not helped in our
original problem, we have come to an unwittingly disclosure: both proofs can
be carried out in the topology of the space rather than in the space itself,
meaning that the results are of an order-theoretic nature. In other words,
Hurewicz’s and Pawlikowski’s Theorems can be stated and proved in the context
of _Pointless Topology_ , in the sense of Johnstone [6, 5]. The purpose of
this work is to present this generalization, based on the proofs in [11] and
in a few basic concepts of pointless topology, which we shall recall along the
next sections111For a more systematic treatment of pointless topology, we
refer the reader to [8, 5]..

##  1\. The Menger game played on nice lattices

Let $(\mathbb{P},\leq)$ be a _lattice_ , i.e., a poset such that for every
$a,b\in\mathbb{P}$ there are $\sup\\{a,b\\}$ and $\inf\\{a,b\\}$, usually
denoted by $a\vee b$ and $a\wedge b$, respectively the _join_ and the _meet_
of $a$ and $b$. For a fixed element $p\in\mathbb{P}$, let us denote by
$\mathsf{V}_{p}$ the family of all subsets $A$ of $\mathbb{P}$ such that $\sup
A=p$.

The family $\mathsf{V}_{p}$ may be thought as a generalization of the family
of open covers of a topological space. Indeed, if $(X,\tau)$ is a topological
space, then $(\tau,\subseteq)$ is a lattice222Actually, it is more than a
lattice: it is a frame, as we shall recall soon. such that
${\mathcal{U}}\subseteq\tau$ is an open cover for $X$ if, and only if,
$\sup{\mathcal{U}}=X$. Thus, with the notation of the previous paragraph, the
family $\mathcal{O}_{X}$ of the open covers of $X$ coincides with the family
$\mathsf{V}_{X}$. In this sense, the main task of this section is to discuss
the proof of the following theorem, which is a generalization of Hurewicz’s
Theorem.

######  Theorem 1.1.

Let $\mathbb{P}$ be a lattice with enough prime elements. For every element
$p\in\mathbb{P}$,
${\mathsf{S}_{\textnormal{fin}}}(\mathsf{V}_{p},\mathsf{V}_{p})$ holds if, and
only if,
${\mathsf{I}}\\!\not{\\!\uparrow}\,\mathsf{G}_{\operatorname{fin}}(\mathsf{V}_{p},\mathsf{V}_{p})$
holds.

A prime element of the lattice $\mathbb{P}$ is an element $q\in\mathbb{P}$
(not equal to $1$ if $\mathbb{P}$ is bounded), such that $a\wedge b\leq q$
implies $a\leq q$ or $b\leq q$ for every $a,b\in\mathbb{P}$. The _enoughness_
means that if $a,b\in\mathbb{P}$ satisfy $a\not\leq b$, then there exists a
prime element $q\in\mathbb{P}$ such that $b\leq q$ and $a\not\leq q$. These
elements shall play the role of the (complement of) points in our adaptation
of the arguments of Szewczak and Tsaban.

Let us begin by noticing that if
${\mathsf{S}_{\textnormal{fin}}}(\mathsf{V}_{p},\mathsf{V}_{p})$ holds, then
for every $A\in\mathsf{V}_{p}$ there exists a countable subset ${B\subseteq
A}$ such that $\sup B=p$, i.e., $\mathbb{P}$ is _Lindelöf_ in an order-
theoretic sense. Thus we may restrict $\mathsf{Player~{}II}$’s moves to
countable subsets of $\mathbb{P}$. On the other hand, since suprema are
associative, we may assume $\mathsf{Player~{}I}$’s moves are countable and
increasing, in such a way that $\mathsf{Player~{}II}$ may select a single
element per inning.

Indeed, if $\mathsf{Player~{}I}$ plays a subset $A_{n}\in\mathsf{V}_{p}$ in
the inning $n$, then there exists a countable subset
$\\{a^{n}_{m}:m\in\omega\\}\subseteq A_{n}$ such that
$\sup_{m\in\omega}a_{m}^{n}=p$, implying that
$B_{n}=\left\\{b_{m}:m\in\omega\right\\}\in\mathsf{V}_{p}$ is countable and
increasing, where $b_{m}=a_{0}^{n}\vee\dotso\vee a_{m}^{n}$ for each $m$. Now,
if $\mathsf{Player~{}II}$ selects $b_{m_{n}}\in B_{n}$ for each $n$ such that
$\sup_{n\in\omega}b_{m_{n}}=p$, the associativity of joins yields

$p=\sup_{n\in\omega}b_{m_{n}}=\sup_{n\in\omega}\bigcup\left\\{a_{0}^{n},\dotso,a^{n}_{m_{n}}\right\\},$

showing that there is no loss of generality in our assumption. As a last
remark, notice that we may suppose that an answer
$A=\\{a_{n}:n\in\omega\\}\in\mathsf{V}_{p}$ of $\mathsf{Player~{}I}$ to some
element $a\in\mathbb{P}$ selected by $\mathsf{Player~{}II}$ is such that
$a_{0}=a$, since we can replace $A$ with $\\{a\vee
a_{n}:n\in\omega\\}\in\mathsf{V}_{p}$.

With these assumptions, a strategy for $\mathsf{Player~{}I}$ in the game
$\mathsf{G}_{\operatorname{fin}}(\mathsf{V}_{p},\mathsf{V}_{p})$ may be
identified with a family $\sigma=\\{A_{s}:s\in\omega^{<\omega}\\}$ of subsets
of $\mathbb{P}$ such that $A_{s}\in\mathsf{V}_{p}$ for every finite sequence
$s\in\omega^{<\omega}$, where $A_{\langle\,\rangle}=\\{a_{j}:j\in\omega\\}$ is
$\mathsf{Player~{}I}$’s first move, $A_{\langle
m\rangle}=\\{a_{m,j}:j\in\omega\\}$ is $\mathsf{Player~{}I}$’s answer to
$a_{m}$, $A_{\langle m,n\rangle}=\\{a_{m,n,j}:j\in\omega\\}$ is
$\mathsf{Player~{}I}$’s answer to $a_{m,n}$ and so on. Let us say that such a
strategy $\sigma$ is a nice strategy for $\mathsf{Player~{}I}$ in the game
$\mathsf{G}_{\operatorname{fin}}(\mathsf{V}_{p},\mathsf{V}_{p})$.

All the previous simplifications were made by Szewczak and Tsaban [11] in the
context of open covers in order to reduce the proof of Hurewicz’s Theorem to a
tricky lemma about tail covers. Recall that a countable open cover
${\mathcal{U}}$ of a topological space $X$ is called a _tail cover_ [11] if
the set of intersections of cofinite subsets of ${\mathcal{U}}$ is an open
cover for $X$. Here, the translation is straightforward: a subset
$A\subseteq\mathbb{P}$ is a tail set (with respect to $p\in\mathbb{P}$) if
$\sup A=p$ and for every cofinite subset $B$ of $A$ there exists $\inf B$ such
that $\sup\\{\inf B:B$ is a cofinite subset of $A\\}=p$. The order-theoretic
version of their lemma is the following.

######  Lemma 1.2.

Let $\sigma=\\{A_{s}:s\in\omega^{<\omega}\\}$ be a nice strategy for
$\mathsf{Player~{}I}$ and for $n\in\omega$ let
$B_{n+1}=\bigcup_{s\in\omega^{n}}A_{s}$. Then $B_{n+1}$ is a tail set with
respect to $p$.

Note that $B_{n+1}$ is the union of all possible answers of
$\mathsf{Player~{}I}$ in the $n$-th inning.

###### Proof.

We proceed by induction.

Since $B_{1}=A_{\langle\,\rangle}$ is countable and increasing, it follows
easily that

$\sup\left\\{\inf{B}:B\text{ is a cofinite subset of
$A_{\langle\,\rangle}$}\right\\}=p,$

showing that $B_{1}$ is a tail set. Now, we assume $B_{n}$ is a tail set for
$n>1$. Since every $A_{s}$ is countable, we may write
$B_{n}=\\{b_{n}:n\in\omega\\}$ and then we put

$B_{n+1}=\bigcup_{j\in\omega}\left\\{b^{j}_{n}:n\in\omega\right\\},$

where $\\{b_{m}=b_{0}^{m},b_{1}^{m},b_{2}^{m},\dotso\\}$ is the (increasing)
answer of $\mathsf{Player~{}I}$ to $b_{m}\in B_{n}$. We shall show that
$B_{n+1}$ is a tail set with respect to $p$.

Let $B\subseteq B_{n+1}$ be a cofinite subset of $B_{n+1}$ and for every
$k\in\omega$ consider $m_{k}=\min\\{n:b_{n}^{k}\in B\\}$. Note that since $B$
is cofinite, $m_{k}=0$ for all but finitely many $k$, hence
$C=\\{k:m_{k}=0\\}$ is cofinite. Now, the niceness of strategy $\sigma$
implies that $D=\\{b_{m_{k}}^{k}:k\in C\\}$ is also a cofinite subset of
$B_{n}$, while the induction hypothesis guarantees the existence of $\inf D$.
Finally, since $\inf B\cap\\{b_{n}^{k}:n\in\omega\\}=b^{k}_{m_{k}}$ for every
$k\in\omega$, it follows that

$\inf
B=\inf\bigcup_{k\in\omega}B\cap\left\\{b_{n}^{k}:n\in\omega\right\\}=\inf_{k\in\omega}b_{m_{k}}^{k}=\inf
D\wedge\bigwedge_{k\not\in C}b_{m_{k}}^{k},$

which existence follows because $\mathbb{P}$ is a lattice. Now, we show that
the supremum of those cofinal subsets is $p$.

Clearly we have $\inf B\leq p$ for all cofinite subsets $B\subseteq B_{n+1}$.
So, we just need to show that if $q\in\mathbb{P}$ is such that $\inf B\leq q$
for all cofinite subsets $B$ of $B_{n+1}$, then $p\leq q$. Suppose, contrary
to our claim, that $p\not\leq q$. Since $\mathbb{P}$ has enough prime
elements, there exists a $\tilde{q}\in\mathbb{P}$ such that $q\leq\tilde{q}$
and $p\not\leq\tilde{q}$. Now, the inductive hypothesis yields a cofinite
subset $C\subseteq B_{n}$ such that $\inf C\not\leq\tilde{q}$, say
$C=\\{b_{n}:n\in J\\}$ for a cofinite subset $J\subseteq\omega$. Since
$\sup\\{b_{k}^{m}:k\in\omega\\}=p$ for all $m\in\omega$, for each $m\not\in J$
there exists an $n_{m}\in\omega$ such that $b_{n_{m}}^{m}\not\leq\tilde{q}$,
from which it follows that $\bigwedge_{m\in\omega\setminus
J}b_{n_{m}}^{m}\not\leq\tilde{q}$. Since the family

$E:=\left\\{b_{m}^{n}:n\in
J,m\in\omega\right\\}\cup\left\\{b_{m}^{k}:m\in\omega\setminus J\text{ and
}k\geq n_{m}\right\\}$

is a cofinite subset of $B_{n+1}$ such that $\inf E=\inf
C\wedge\bigwedge_{m\in\omega\setminus J}b^{m}_{n_{m}}$, the primeness of
$\tilde{q}$ gives $\inf E\not\leq\tilde{q}$. Therefore, $\inf E\not\leq q$. ∎

###### Proof of Theorem 1.1.

The simplifications made so far allows us to assume $\mathsf{Player~{}I}$ has
a nice strategy $\sigma$ such that, with the same notations of the previous
lemma, the family $B_{n+1}$ is a tail set for every $n\in\omega$.
Consequently,

$V_{n+1}:=\\{\inf B:B\subseteq B_{n+1}\text{ is
cofinite}\\}\in\mathsf{V}_{p}.$

Now, ${\mathsf{S}_{\textnormal{fin}}}(\mathsf{V}_{p},\mathsf{V}_{p})$ applies
to the sequence $(V_{1},V_{2},\dotso)$, giving a sequence
$(F_{1},F_{2},\dotso)$ of finite subsets $F_{n}\subseteq V_{n}$ such that
$\sup\bigcup_{n\in\omega}F_{n}=p$. Notice that for each $f\in F_{n}$ there
exists a cofinite subset $B_{f}^{n}\subseteq B_{n+1}$ such that $f=\inf
B_{f}^{n}$, and $C^{n}=\bigcap_{f\in F_{n}}B_{f}^{n}$ is a cofinite subset of
$B_{n+1}$.

Finally, let us see how $\mathsf{Player~{}II}$ can defeat the nice strategy
$\sigma$: in the $n$-th inning, $\mathsf{Player~{}I}$ selects an infinite
subset $A$ of $B_{n+1}$, which must intersect $C^{n}$, so
$\mathsf{Player~{}II}$ may choose $f_{n}\in C^{n}\cap A$. Then

$p=\sup\\{\inf B^{n}_{f}:f\in
F_{n},n\in\omega\setminus\\{0\\}\\}\leq\sup_{n\in\omega}f_{n}\leq p,$

showing that $\mathsf{Player~{}II}$ wins, as desired. ∎

##  2\. The Rothberger game played on nicer lattices

We now proceed to discuss the adaptation of Pawlikowski’s Theorem, which shall
require stronger conditions on the lattice $\mathbb{P}$. For our preliminary
considerations, let us suppose that $\mathbb{P}$ is a bounded lattice with
enough prime elements. For simplicity, such lattices will be called pre-
Pawlikowski.

As it happens in the original proof of Pawlikowski, we will need to use
Hurewicz’s theorem to provide nice plays in auxiliary games. At the beginning
of the proof, one uses the simple exercise that a countable union of Menger
spaces is again a Menger space. This still happens with pre-Pawlikowski
lattices when we consider the right generalization of union.

######  Lemma 2.1.

If for every $n\in\omega$ there is a pre-Pawlikowski lattice $\mathbb{P}_{n}$
such that
${\mathsf{S}_{\textnormal{fin}}}(\mathsf{V}_{1_{\mathbb{P}_{n}}},\mathsf{V}_{1_{\mathbb{P}_{n}}})$
holds, then $\mathbb{P}:=\prod_{n\in\omega}\mathbb{P}_{n}$ is a pre-
Pawlikowski lattice satisfying
${\mathsf{S}_{\textnormal{fin}}}(\mathsf{V}_{1_{\mathbb{P}}},\mathsf{V}_{1_{\mathbb{P}}})$,
where $\mathbb{P}$ is endowed with the pointwise ordering and
$1_{\mathbb{P}}:=(1_{\mathbb{P}_{n}})_{n\in\omega}$.

###### Proof.

The reader can easily show that $\mathbb{P}$ is a pre-Pawlikowski lattice such
that $1_{\mathbb{P}}$ is its maximum element. Now, for a sequence
$(A_{n})_{n\in\omega}$ of subsets of $\mathbb{P}$ such that $\sup
A_{n}=1_{\mathbb{P}}$ for each $n\in\omega$, we shall select a sequence
$(F_{n})_{n\in\omega}$ of subsets such that $F_{n}\in[A_{n}]^{<\omega}$ for
every $n$ and $\sup\bigcup_{n\in\omega}F_{n}=1_{\mathbb{P}}$.

Let $\\{Q_{n}:n\in\omega\\}$ be an infinite partition of $\omega$ such that
each $Q_{n}$ is infinite. For a fixed $n\in\omega$ and $j\in Q_{n}$ we have
$\sup A_{j}=1_{\mathbb{P}}$, implying that
$\sup\pi_{n}[A_{j}]=1_{\mathbb{P}_{n}}$, where
$\pi_{n}\colon\mathbb{P}\to\mathbb{P}_{n}$ is the obvious projection. Then,
for each $j\in Q_{n}$ there exists a finite subset $F_{j}\subseteq A_{j}$ such
that $\sup\bigcup_{j\in\omega}\pi_{n}[A_{j}]=1_{\mathbb{P}_{n}}$, from which
it easily follows that $\sup\bigcup_{n\in\omega}F_{n}=1_{\mathbb{P}}$. ∎

With this lemma, we can show that if $\mathbb{P}$ is a pre-Pawlikowski lattice
satisfying ${\mathsf{S}_{\textnormal{fin}}}(\mathsf{V}_{1},\mathsf{V}_{1})$,
then every strategy of $\mathsf{Player~{}I}$ in the game
$\mathsf{G}_{\operatorname{fin}}(\mathsf{V}_{1},\mathsf{V}_{1})$ can be
severely defeated, in the following sense.

######  Proposition 2.2.

Let $\mathbb{P}$ be a pre-Pawlikowski lattice and let $\sigma$ be a strategy
for $\mathsf{Player~{}I}$ in the game
$\mathsf{G}_{\operatorname{fin}}(\mathsf{V}_{1},\mathsf{V}_{1})$. If
${\mathsf{S}_{\textnormal{fin}}}(\mathsf{V}_{1},\mathsf{V}_{1})$ holds, then
there exists a play $(A_{0},F_{0},A_{1},F_{1},\dotso)$ according to $\sigma$
such that for every $p\in\mathbb{P}\setminus\\{1\\}$ we have $\sup
F_{n}\not\leq p$ for infinitely many $n$.

###### Proof.

The previous lemma guarantees that $\mathbb{P}^{\omega}$ is a pre-Pawlikowski
lattice satisfying
${\mathsf{S}_{\textnormal{fin}}}(\mathsf{V}_{1_{\omega}},\mathsf{V}_{1_{\omega}})$,
where $1_{\omega}:=(1)_{n\in\omega}$. Now, we use $\sigma$ to cook up a
strategy $\tilde{\sigma}$ for $\mathsf{Player~{}I}$ in the game
$\mathsf{G}_{\operatorname{fin}}(\mathsf{V}_{1_{\omega}},\mathsf{V}_{1_{\omega}})$
played on $\mathbb{P}^{\omega}$. For a subset $A\subseteq\mathbb{P}$, let
$\tilde{A}:=\\{\delta_{n}(a):a\in A$ and $n\in\omega\\}$, where
$\delta_{n}(a)\in\mathbb{P}^{\omega}$ is such that $\delta_{n}(a)(m):=0$ for
$m\neq n$ and $\delta_{n}(a)(n)=a$. It is easy to see that if $\sup A=1$, then
$\sup\tilde{A}=1_{\omega}$.

We define $\tilde{\sigma}(\emptyset):=\tilde{A_{0}}$, where
$\sigma(\emptyset):=A_{0}$. If $\mathsf{Player~{}II}$ selects a finite subset
$\tilde{F_{0}}\subseteq\tilde{A_{0}}$, let $F_{0}:=\\{a\in
A_{0}:\delta_{n}(a)\in\tilde{F_{0}}$ for some $n\in\omega\\}$, a finite subset
of $A_{0}$, and then define $\tilde{\sigma}(\tilde{F_{0}}):=\tilde{A_{1}}$,
where $A_{1}:=\sigma(F_{0})$. Proceeding like this, we obtain a strategy
$\tilde{\sigma}$ for $\mathsf{Player~{}I}$ in the game
$\mathsf{G}_{\operatorname{fin}}(\mathsf{V}_{1_{\omega}},\mathsf{V}_{1_{\omega}})$
played on $\mathbb{P}^{\omega}$.

The lattice version of Hurewicz’s theorem yields a play
$(\tilde{A_{0}},\tilde{F_{0}},\tilde{A_{1}},\tilde{F_{1}},\dotso)$ according
to $\tilde{\sigma}$ such that
$\sup\bigcup_{n\in\omega}\tilde{F_{n}}=1_{\omega}$, and by the way we defined
$\tilde{\sigma}$, this play corresponds to a play
$(A_{0},F_{0},A_{1},F_{1},\dotso)$ according to $\sigma$. We claim this latter
play has the desired property. Indeed, for a fixed
$p\in\mathbb{P}\setminus\\{1\\}$, we have
$\tilde{p}_{0}:=(p,1,1,\dotso){<}\,1_{\omega}$, from which it follows that
there exists $n_{0}\in\omega$ and
$f:=(f_{n})_{n\in\omega}\in\tilde{F}_{n_{0}}$ such that
$f\not\leq\tilde{p}_{0}$, implying $f_{0}\in F_{n_{0}}$ is such that
$f_{0}\not\leq p$, hence $\sup F_{n_{0}}\not\leq p$. Since $F:=\bigcup_{j\leq
n_{0}}F_{j}$ is finite, the set $N:=\\{n:\delta_{n}(a)\in F$ for some $a\\}$
is also finite, which allow us to take
$\tilde{p}_{m}:=(1,\dotso,1,p,1,\dotso)$, with $p$ in the $m$-th position, for
$m\not\in N$. Once again, there exists $n_{1}\geq m>n_{0}$ such that $\sup
F_{n_{1}}\not\leq p$. Continuing like this, we see that $\sup F_{n}\not\leq p$
for infinitely many $n$. ∎

In the argument presented by Szewczak and Tsaban [11], the topological
counterpart of Proposition 2.2 is used to obtain a play
$({\mathcal{U}}_{0},F_{0},{\mathcal{U}}_{1},F_{1},\dotso)$ of an auxiliary
instance of the Menger game such that for every $x$ one has $x\in\bigcup
F_{n}$ for infinitely many $n$. This allows one to select $U_{m}\in F_{m}$ for
each $m\in\omega$ such that the family $\\{U_{m}:m\in\omega\\}$ is an open
cover for $X$. As it happens with the previous propositions, this argument
also has a lattice counterpart.

######  Lemma 2.3.

Let $\mathbb{P}$ be a pre-Pawlikowski lattice satisfying
${\mathsf{S}_{\textnormal{1}}}(\mathsf{V}_{1},\mathsf{V}_{1})$. Let
$(F_{n})_{n\in\omega}$ be a sequence of finite subsets of $\mathbb{P}$ such
that for every $p\in\mathbb{P}\setminus\\{1\\}$ there are infinitely many $n$
such that $\sup F_{n}\not\leq p$. Then there are elements $f_{n}\in F_{n}$ for
every $n$ such that $\sup_{n\in\omega}f_{n}=1$.

###### Proof.

The proof is the same as in [11], up to terminology. For each $n\in\omega$ let

${\mathcal{V}}_{n}:=\left\\{\bigwedge G:\exists J\in[\omega]^{n+1}\,\forall
j\in J\,|G\cap F_{j}|=1\text{ and }|G|\leq n+1\right\\},$

i.e., the elements of ${\mathcal{V}}_{n}$ are the meets of $n+1$ elements
belonging to $n+1$ pairwise distinct sets of the sequence
$(F_{j})_{j\in\omega}$. The hypothesis about the sequence
$(F_{n})_{n\in\omega}$ implies that $\sup{\mathcal{V}}_{n}=1$ for every
$n\in\omega$: for if $p<1$, there exists a prime element $\tilde{p}<1$ such
that $p\leq\tilde{p}$, while the hypothesis gives infinitely many $m$ such
that $\sup F_{m}\not\leq\tilde{p}$; then we may take $J\in[\omega]^{n+1}$ and
$f_{j}\in F_{j}$ for each $j\in J$ such that $f_{j}\not\leq\tilde{p}$; since
$\tilde{p}$ is prime, it follows that $\bigwedge_{j\in
J}f_{j}\not\leq\tilde{p}$, hence $\bigwedge_{j\in J}f_{j}\not\leq p$.
Therefore, $1\in\mathbb{P}$ is the only upper bound of ${\mathcal{V}}_{n}$.

Now, since ${\mathsf{S}_{\textnormal{1}}}(\mathsf{V}_{1},\mathsf{V}_{1})$
holds, there exists a sequence $(v_{n})_{n\in\omega}$ with
$v_{n}\in{\mathcal{V}}_{n}$ for each $n$, such that
$\sup_{n\in\omega}v_{n}=1$. Notice that $v_{0}\leq f_{n_{0}}$ for some
$f_{n_{0}}\in F_{n_{0}}$, while $v_{1}\leq f_{n_{1}}$ for some $f_{n_{1}}\in
F_{n_{1}}$ such that $n_{1}\neq n_{2}$, and so on. Proceeding like this, we
obtain $f_{n_{k}}\in F_{n_{k}}$ such that $\sup_{k\in\omega}f_{n_{k}}=1$, so
we can pick arbitrary $f_{n}\in F_{n}$ for the remaining $n$’s and still get
$\sup_{n\in\omega}f_{n}=1$. ∎

We finally proceed to the order-theoretic version of Pawlikowski’s theorem.
However, we must impose a last condition over our lattices. Let us say that
$\mathbb{P}$ is a Pawlikowski lattice if $(\sup A)\wedge b=\sup\\{a\wedge
b:a\in A\\}$ holds for every $b\in\mathbb{P}$ and every subset
$A\subseteq\mathbb{P}$ having a supreme. This is the order-theoretic version
of the identity

$\left(\bigcup{\mathcal{U}}\right)\cap B=\bigcup_{U\in{\mathcal{U}}}U\cap B,$

holding for (open) sets of a topological space $X$. The distributivity over
arbitrary suprema is needed to guarantee that for every
$A,B\in\mathsf{V}_{1}$, the family $A\wedge B:=\\{a\wedge b:a\in A\text{ and
}b\in B\\}$ is still a member of $\mathsf{V}_{1}$, i.e., $\sup A\wedge B=1$.

######  Theorem 2.4.

Let $\mathbb{P}$ be a Pawlikowski lattice satisfying
${\mathsf{S}_{\textnormal{1}}}(\mathsf{V}_{1},\mathsf{V}_{1})$. Then
$\mathsf{Player~{}I}$ does not have a winning strategy in the game
$\mathsf{G}_{\operatorname{1}}(\mathsf{V}_{1},\mathsf{V}_{1})$.

###### Proof.

The hypothesis over $\mathbb{P}$ allows one to follow the same arguments of
Szewczak and Tsaban [11] by simply applying the correspondent order-theoretic
versions of the necessary lemmas. The details are left to the reader. ∎

##  3\. Further comments

Although the adaptations of Szewczak and Tsaban simplifications do not require
explicit use of points, one could ask how the further assumptions on
Pawlikowski lattices affect their proximity to topological spaces. Recall that
a lattice $\mathbb{P}$ is _spatial_ if there exists a topological space
$(X,\tau)$ such that $\mathbb{P}$ is a frame isomorphic to $\tau$ in the frame
category333The arrows are the increasing maps preserving arbitrary joins and
finite meets, thus including $0$ and $1$., where by a frame we mean a complete
lattice satisfying the distributive condition of Pawlikowski lattices. Thus,
the question rephrases as: is there a non-spatial Pawlikowski lattice? The
answer is yes.

Indeed, as implicitly444It was _pointed_ out by Eric Wofsey in this answer:
https://math.stackexchange.com/a/2606878/128988. showed by Johnstone in his
classic book _Stone Spaces_ [5], a lattice $\mathbb{P}$ is spatial if, and
only if, $\mathbb{P}$ is a _complete_ lattice with enough prime elements. On
the other hand, our assumption regarding distributivity and prime elements do
not yield completeness, as the following example of Eric Worsey shows555See:
https://math.stackexchange.com/a/4019499/128988..

######  Example 3.1.

For an infinite set $X$, let $L:=\\{A\subseteq X:A$ is finite or $A$ is
cofinite$\\}$. One can easily see that $L$ is a non-complete lattice with
respect to inclusion, with enough prime elements (the complementary of
singletons) and such that

$(\sup\mathcal{A})\cap B=\sup_{A\in\mathcal{A}}(A\cap B)$

for every $B\in L$ and $\mathcal{A}\subseteq L$ with
$\sup\mathcal{A}=\bigcup\mathcal{A}\in L$. In other words, $L$ is a non-
complete Pawlikowski lattice.

In this sense, the results obtained along this work show that, although a few
aspects of points (disguised as prime elements) seem to be necessary to prove
Hurewicz’s and Pawlikowski’s theorems, one does not need the full power of a
topology to do so. Therefore, assuming other problems regarding topological
games can be rephrased in the lattice language, it is natural to ask the
following.

######  Question 3.2.

Do Theorems 1.1 and 2.4 hold for larger classes of lattices?

While the last question implicitly aims for possible applications, another
direction could be to search for generalizations of known results in the
field. In this case, we mention the following theorem, which extends Lemma
3.12 from García-Ferreira and Tamariz-Mascarúa [3].

######  Theorem 3.3.

For a Pawlikowski lattice $\mathbb{P}$ and a function
$f\colon\omega\to\omega$, the following are equivalent:

  1. (1)

${\mathsf{S}_{\textnormal{1}}}(\mathsf{V}_{1},\mathsf{V}_{1})$ holds;

  2. (2)

for each sequence $(A_{n})_{n\in\omega}$ where $A_{n}\in\mathsf{V}_{1}$ for
every $n$, there are finite subsets $B_{n}\in[A_{n}]^{<f(n)+1}$ such that
$\bigcup_{n\in\omega}B_{n}\in\mathsf{V}_{1}$.

As a final remark, we notice that Hurewicz property
$\mathsf{U}_{\operatorname{fin}}({\mathcal{O}},\Gamma)$ can also be stated for
lattices: we may say that a bounded lattice $\mathbb{P}$ is _Hurewicz_ if for
every sequence $(A_{n})_{n\in\omega}\in(\mathsf{V}_{1})^{\omega}$ there are
finite subsets $F_{n}\subseteq A_{n}$ for each $n$ such that

$\bigvee\left\\{\bigvee F_{n}:n\in\omega\right\\}=1$

with $\left\\{n:\bigvee F_{n}\not\leq p\right\\}$ cofinite for every prime
element $p\in\mathbb{P}$. The translation of this property to the lattice
context suggests even further investigations.

## Acknowledgements

I wish to express my gratitude to Dione Andrade Lara, Rodrigo “Rockdays” Dias
and Vinicius Rodrigues for the fruitful discussions about the results
presented here.

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