# Understanding Digital Government Transformation

Mamdouh Alenezi
College of Computer and Information Sciences
Prince Sultan University
Riyadh, Saudi Arabia
<EMAIL_ADDRESS>

###### Abstract

In today’s era of innovation of technological progression, digitalisation has
not only transformed individual lives but also has a prominent influence on
business activities. The world is surviving in a global yet complex
technological progression that not only changes the lives of civilians but is
also transforming the public, private, and academic spheres of life. This
research focuses on the digitalisation of governments, their challenges, and
success factors. It is found that government faces difficulties in formulating
strategies, proper planning, execution strategies, and a lack of organised
information and expertise. However, success can be achieved by working on
capabilities of the future workforce, creating leaders for tomorrow,
generating digitalisation capabilities, and bringing a purpose-driven
digitalisation before digital government transformation. Overall, the study’s
findings suggest that digital government transformation creates value,
enhances relations, improves service delivery, grows economy, pushes economic
activities, enhances citizen engagement, increases the policy implementation
and their efficiency, and affects business growth positively.

_K_ eywords Digital government $\cdot$ Government transformation $\cdot$
eGovernment

##  1 Introduction

The world today is striving towards a global yet complex technological
progression that has not only changed the lives of civilians but is also
transforming the public, private, and academic spheres of life. In addition,
this technology-driven change extends opportunity for nations to converge
advancements with an aim to create a more human-centric and inclusive future
[1]. It is very important to understand the reason prior to discussing the
significance of digital transformation. The rationale to shift towards digital
transformation is linked with changing needs of consumers and market dynamics,
existing threats with the traditional and conventional methods, and need of
innovation to sustain the competitiveness [2]. One reason could be to leverage
emerging technologies. With the emergence and prioritization of the digital
economies facilitating a user-centric and seamless experiences, citizens now
anticipate public sector to be highly accessible, flexible, and efficient.
Governments across the globe are transitioning towards digital transformation
(DT) as a strategic motive to enhance the service performance, improve
consumer experience, streamline the activities and operations, and develop new
business models [3]. This paper aims to provide an overview for the government
institutions regarding the impact of digital transformation through the
analysis of statistics and information, highlighting the reliance on
digitisation.

The adoption of DT in governmental institutions is continuously changing the
citizen’s expectations [3]. E-government is the latest incorporation of
digital government transformation. E-government is a digital application
transformation that facilitates country services and products for its industry
and civilians by the implementation of information technologies and electronic
tools [4]. These expectations include real-time and high-quality digital
services. Similarly, European Commission [5] stated that the objective of
digital transformation in the public sector tends to focus on creating new
forms of relationships, building new frameworks of services delivery, and
generating innovative ways of working with stakeholders. The governments need
to address such decisions very critically. Furthermore, the strong and fast
adoption of digital government transformation can support governments heavily
while working on these objectives. Similarly, the demands of politicians,
businesses, and civilians tend to be extremely important as they determine the
efficiency of governments. Technological changes are heavily experienced by
these individuals in their work, life, and environment [6]. Hence, one of the
aims of this study is to identify how governments can address these
expectations. Specifically, the paper will evaluate the digital government
transformation. In addition, the study will evaluate the influence of digital
transformation on the decision making of governments, policymakers, and
researchers by analysing the success factors and challenges of the DT.

##  2 Digital Transformation

Digital Transformation refers to the leveraging of business capabilities and
digital technologies for the sake of enhancing customer experience,
operational processes, and business models which create customer value [7]. In
the current era, organisations adopt emerging technologies to achieve rapid
development, which in return changes their business competition positively
whereby, such organisations create new ways of operations [8]. Moreover,
according to Fitzgerald [9], all the organisations and across the industries
are affected by digital transformation; such effect pushes organisations to
adopt digital transformation to remain competitive in their market or segment.

Precisely, digital transformation enhances the value chain, stakeholders’
relationships, and efficient deliveries [10]. Moreover, digital transformation
provides an edge to enterprises by increasing the ease of monitoring,
controlling, and executing the strategies while reducing barriers of
connectivity and burdens of responsibilities. Furthermore, DT is regarded as
the backbone of sustainability, survivability, and competitive edge of the
business through efficient utilisation, innovation, and operational
integration [11]. In simple words, all the activities, operations, policy
execution are synchronised by interconnectivity which gives access to each
other. As a result, businesses can effectively monitor, control, and design
strategies accordingly. Hence, it has now become the need of time for the
organisations to integrate digitalisation to sustain its competitive edge.

##  3 Elements of Digital Transformation

The term digital transformation, which is mostly used in the private
industrial segment, forces the organisations to adopt new and advanced
technologies so remain differentiated in the world of the internet due to the
provision of services offline and online [12]. The elements of digital
transformation include transforming value creation, relations with citizens
and organisational culture, and service delivery.

###  3.1 Transforming to Create Value

The value chain could lead to a positive shift by acquiring digital
transformation. Berman [12] determined that the activities such as co-funding,
co-distribution, co-marketing, co-production, co-creation, and co-design are
heavily improved by the incorporation of digital transformation. This is
because it increases the connectivity and reliability between these
departments, which then result in higher performance [13]. Thus, integration
of DT leads to overall efficiency in the operations through higher
interconnectivity that adds value to the firm.

###  3.2 Transforming to Enhance Relations

Digital transformation is considered a technological revolution that increases
the consumer’s expectations, whereby businesses can fulfil them effectively by
the use of digital transformations. This indicates that DT can lead to the
accomplishment of high expectations, that is regarded as a cultural
modification. Similarly, government businesses inject consumer engagement,
integrity, and transparency by implementing digital solutions.

###  3.3 Transforming to Enhance Service Delivery

The digital transformation provides new ways of service delivery, customer
interactions, and delivery modes [12]. It supports the value chain actors,
including connections between consumers, external producers, and suppliers.
These benefits pinch enterprises to adopt digital transformation for the sake
of efficiency. Hence, it can be said that digital transformation is basically
upgrading the value chain to achieve the enterprise’s objectives.

These benefits can also be utilised in government institutions by adopting
digital transformation. It is now clear that there is a huge need for digital
government transformations. This study emphasises transformational government,
digital government, e-governance, and e-government.

##  4 Digital Government Transformation

###  4.1 E-government and Technology

Technology is an essential element of public infrastructure. It supports
creating value in the society at large, private, and public sector through
sustainable innovation. In addition, it is adopted to treat economic
disruptions, environmental disruptions, and transferable diseases [14].
However, if the technology is used in the public sector, different things are
at stake that includes trust in institutions and government, risk adversity,
inter-governmental models, governance, legal traditions, organisation and
operations of government, socio-economic factors, culture, and technology
itself [15].

In the last two decades, researchers have intensively studied the digital
government transformation giving many definitions of e-government. Rooks,
Matzat, and Sadowski [16] defined that e-government uses Information
communication technologies (ICTs) to increase facilities for its residents
along with providing government information to its residents by using
information communication technologies and internet. The literature on this
topic considers transformational government, digital government, and
e-government similar to digital government transformation.

In simple words, these concepts are interrelated, sharing a similar nature of
the ground [3]. In addition, they evaluate the impact of using information
communication technologies on culture, organisational process, and service
delivery along with the influence on value creation. At first, it is necessary
for the government to break the traditional process to bring transformation in
the operations in the form of digitalisation. Primarily, it includes the re-
engineering of process and structure known as Business Process Engineering
(BPR) [17]. Therefore, incorporation of ICT facilitates in seamless and
efficient government operations to improve citizen experience and
satisfaction.

The main idea of incorporating electronic government is to improve and
facilitate governmental services with the use of communication and information
technology. In simple words, the traditional governments are transformed into
e-governments to integrate the administration and enhance connectivity among
the policymakers [18]. Although the idea looks very simple, yet its
installation and implementation tend to be very complex [19]. The complexity
is related to the availability of various factors, including economic, legal,
political, organisational, social, and technical. Hence, the success of
transforming becomes subjective to how efficient transformation is carried
out.

###  4.2 E-government Characteristics

The comprehensive use of technology has the ability to offer several social
and economic benefits. The digital transformation enables inclusion allowing
small and large organisations to sell their product and service offerings
online while competing on a global scale. Similarly, e-government facilitates
country services and products for its industry and civilians by the
implementation of information technologies and electronic tools [20].
Moreover, digitalisation contributes to increasing efficiency. Digital
technologies allow companies as well as government entities to make better use
of resources, labour while leveraging the intelligence, speed, and reliability
of digital tools. According to Amayah [4], DT refers to organisational changes
which use new business models and new digital technologies to enhance consumer
experience and organisational performance.

It is essential to digitise public services as it helps in economic growth,
pushes economic activities, enhances citizen engagement, and positively
affects business growth. In the last decade, the use of digital transformation
has trended in developed as well as the developing countries [21]. Most
importantly, there is increasing problem of unskilled personnel in
governmental organisations and a lack of knowledge in the staff [21].
Weerakkody et al. [22] explained that these problems could be solved as the
adoption of digital transformation leads to innovative solutions in political,
social, and economic areas whereby the strong decision-making process is
followed. In addition, these changes are necessary to be understood by all the
individuals who prepare and implement policies or make a decision regarding
government practices [23]. These individuals include researchers, government
executives, and policymakers. Hence, considering these benefits, it is very
crucial for the government to adopt DT for the purpose of predicting and
understanding the ongoing changes.

Digital Government Transformation can support the Sustainable Development
Goals (SDGs) in multiple directions. The development of digital inclusion in
the government administration supports the fast and efficient execution of
operations. According to Nielsen [14], digital government transformation
reduces the administration burden and increases productivity, acquires higher
productivity, and facilitate in the adherence to Sustainable Development Goals
(SDGs). Moreover, effective use of technology with efficient re-engineering
and organisational processes and regulations also bring employment,
sustainable economic growth, and inclusive and transparent society. Thus, DT
in the government promotes sustainable measures for the betterment of the
environment.

##  5 Digital Government Transformation Challenges

With the emerging era of engaging the public sectors with digital
transformation, this journey combats a number of challenges. To discuss most
of the prominent ones, the digital government transformation challenges can be
generally categorised into two common categories; internal and external
challenges. According to Lindgren and van Veenstra [24], internal challenges
refer to the difficulties that are raised within the public sector, whereas
the external challenges are the ones that arise from the bodies that are not
the immediate organs of the government but hold external effects on it.

As to identify the internal challenges, studies reveal that to attain a goal
or objective, an organisation must have a mutual vision and mission for it
[25]. Similarly, the initial issue a government organisation faces when it
comes to digital transformation is the management of formulating an
appropriate digital transformation strategy. The public sectors must be aware
of their digital/IT needs, so an appropriate strategy must be designed.
Research by Jonathan et al. [6] described that none of the interviewees – who
were involved in the research – were aware of any kind of digitalising
strategy. Some of the interviewees had heard about it, but they find it
problematic. Also, proper planning for sustaining the strategy is clearly
associated with this challenge.

Organisational structure and its culture are also considered a significant
challenge that fosters the digitalisation transformation process in
organisations. Public sectors are the most vulnerable organisations that
commonly face these challenges [20]. The employees themselves show a lack of
motivation to adopt the digital changes and willingness towards continuing the
existing system. According to studies that conducted interviews for analysing
the prominent challenges, most of the respondents were found to be happier and
satisfied with the existing typical analogue culture they use in their
organisation [6]. Therefore, in the way to bring up a digitalised
transformation in a public sector, organisational culture becomes resistant.

Last but not least, information security is one of the radical challenges
among all. According to Mamonov and Benbunan-Fich [26], digital services can
never be categorised as services to the public if the sensitive information in
them is not secure. When anything is brought up to a digital platform, the
level of information security threats gets increased. Both risks for
information security threats as well as the lack of organised information
secured tactics go in parallel to each other. Digital tactics for information
security is highly necessary for any public sector; otherwise, the system will
no longer be beneficial in terms of providing services to the public.

In terms of external challenges, there exist two major challenges that are
responsible for fostering the emergence of digital growth in the public
sector. The first is lack of expertise in the market, and the other is
stakeholder relationships. Evidence suggested that the availability of skilled
and trained experts who have solid grips on digital technologies is considered
to be the foremost requirement to digitally transform any sector, whether a
private or public sector firm [27]. It is also evident that skilled technical
individuals can easily be found, but the main problem lies within the lack of
people who have firm experience in recent forms of digitalisation and training
the relevant staff accordingly [6]. One of the most affected countries who
possess lack of trained and expertise in the market are the developing
countries. Nevertheless, strict public sector regulations and protocols also
affect the expert individuals to get attracted towards the public sector
digitalisation processes.

On the other hand, Nambisan, Wright, and Feldman [28] explained that the
stakeholder relationship can also play a challenging role for the public
sectors to incorporate the digital transformation if not maintained
accordingly. Primarily, these stakeholder relationships are associated with
the private organisations, but they can be associated with the government
departments as well, depending on the interests of the public sector
organisations. Even though the main objective of digitally transforming the
public sector is to enhance transparency and accountability, but multiple
engaging stakeholders who can assist in different dimensions of decision
making is one of the significant purposes [6]. However, the management of the
stakeholder relationship, including their participation, is a potential
challenge for public sector organisations.

##  6 Digital Government Transformation Success Factors

Organisations certainly experience challenges and hurdles to incorporate
digital transformation, but implementing appropriate strategies and including
some of the prominent success factors may help gain the objective. Evidence
suggested that one of the prior success factors is to enhance the capabilities
of the future workforce [29]. Success can only be achieved if an effective
environment is provided to the employees where they can work and represent
their talents as well as achieve new capabilities. With the initiation of
digital transformation, it is immensely important to establish digital
capabilities for the working and future workforce. Once the employees get
trained over the digital transformation, they start to work as prime
integrators of the system, where the integrators refer to the ones who adopt
the latest changes of the digital frameworks and integrate them into the
existing organisational culture.

The second success factor is to create leaders who are digitally well-aware
and demonstrated. According to a research survey conducted by Liferay [30], it
has been examined that more than 30% of the respondents involved in the survey
used to believe that organisational change was the biggest barrier towards
digitalising their public sector. The challenging aspect is to break and
transform the traditional analogue system of the organisation and bring a
different way for the public sector. Therefore, it is not only important to
bring transformation across the board, but it is also potentially needed to
create a team of leaders who play a key role in the success of digital
transformation [20]. It is because a digital transformation in the public
sector can only be seen if the leaders are strongly willing to implement it.
In this regard, they must be aware of all the pros and cons of the
digitalisation process, as well as the reason behind integrating it into the
public sector organisations.

Another significant success factor is related to the intercommunication
between the digital and non-digital approaches. When a public sector
incorporates digital transformation, it is crucially important to formally
communicate the actual narrative for bringing up the change. According to
Morakanyane et al. [2], the doer must help understand the workers
comprehensively why this digitalising is being transformed. This helps in
building a concept for the employees and also creates a passion for working
for them. The narrative to demonstrate the reasons behind digital
transformation should be rich, covering every aspect of the latest changes to
excite the employees to work accordingly. This would also cover describing the
short and long term goals as well as the key performance indicators (KPIs).

Another factor that determines the success of digital transformation in the
public sector is to enable accessibility to everyone in the country [2].
Different citizens may have different needs, but their expectations from the
government remain the same. The transformation must ensure that the public
sectors are easily accessible to every citizen by using any device. Currently,
only 76 out of 193 United Nations (UN) member states are completely
accommodating web access nationwide, which is relatively very low. Therefore,
it is high time for the government to change the typical procedure of
engagement and enable everyone to participate digitally in multiple domains of
the public sector.

A far better success factor is to propose a purpose-driven and seamless
experience into the digital transformation of public sectors. It is predefined
that the target audience might have different needs and behaviours depending
on the nature of service they require from the government. Therefore,
incorporating a purpose-driven digital transformation into the government
agencies can bring positive outcomes in terms of successful implementation of
the transformation. Every government agent must have a relevant platform that
is desired by the audience. For example, audiences applying for a visa to one
country is different from the audience applying for a vehicle license. So,
there is a need to integrate certain relevant and appropriate digital corners
in the transformation that assist the incoming audience to their best
services. It is evident because the audience who experience satisfactorily
through any website is more than 105% likely to use such websites as their
primary source to have services [29]. A more proportion of this audience is
also prevalent to recommend these sites to friends and their community, which
further enhances the application and services of the digital transformation.

##  7 Discussion

Digital transformation in the government or public sector refers to different
and innovative ways of engaging and working with stakeholders, developing
frameworks for efficient service delivery mechanisms and forming new
relationships. Nevertheless, ahead of the availability of reports of leading
corporations, i.e. Deloitte, there is limited systematic empirical evidence
found regarding the way public administrations define digital transformation
in their daily practices, what the anticipated outcomes are, and how they
approach digital transformation projects. The terms such as digital
transformation, digitalisation or digitisation are used interchangeably in the
studies [3]. Considering the role of digital transformation in government,
studies have observed that digital transformation are the changes in service
mode delivery, and are also new types of direct engagement with clients, for
instance, via social media sites to adapt services and products as per
changing consumer demands or needs [3]. The conception of e-government over
the past two decades has been studied extensively, and researchers like Rooks,
Matzat and Sadowski [16] differentiated narrow and broad explanations of
e-governments. In terms of narrow definition, the emphasis is on the use of
information communication technologies (ICTs) to deliver services to people,
while broad definition includes the use of ICTs and the internet to provide
government and administrative information to the people. Other definitions, as
per Ma and Zheng [19], of digital transformation focus on the engagement with
people via ICTs. Other than the definitions, the role or the effectiveness of
e-government on companies is still a disputed issue. Nevertheless, the
advantage of e-government emphasises largely on the improvement of service
delivery, leading to increase efficiency in government performance.

Digitalisation has been adopted in developed countries, and Denmark is
considered one of the leading countries that have established e-government
programs. As per the UN e-government survey, Denmark was rated to be one of
the mature countries regarding e-government [15]. The collaborative,
consensus-seeking, and coordinated approach that has taken to the digital
transformation of the Danish government sector are a few of the key elements
behind the success of Danish authorities in e-government. Additionally, the
formal and informal discussions are the norm of e-government strategies in
Denmark, and it is unluckily the end-user group, academia, and private sector
that are not part of the governance model, as they could assist in ensuring a
holistic approach to the use of technology in government sector.

E-government strategies in Denmark are facilitated by strategic initiatives
associated with the necessary business case model and IT-project, which assist
in reducing the risk of failure and further support active benefit of
understanding at both strategy and project levels [15]. Similar findings have
been observed in the study of Janowski [31] who argued that the concept of
digital government has become widespread, and the evolution model of digital
government can now be categorised into technology or digitisation in
government, electronic or engagement governance, policy-driven or
contextualisation electronic governance and electronic or transformation
government. Gong, Yang, and Shi [32] discussed the understanding of digital
transformation in government by asserting that it is a slow process with
adaptations in numerous structural elements, affecting the entire
administrative system from local to the federal level, including both
incremental and radical alterations. Flexibility increases together with the
development of digital transformation. However, the flexibility more relies on
bureaucratic levels and organisational elements.

Other than the adoption of digitalisation in the public sector, there are
certain challenges, which are faced when implementing it on a wider level. For
instance, the study of Jonathan et al. [6] pointed out the management of
developing a relevant and suitable digital transformation strategy is one of
the key issues faced by the government at initial levels. The public sector
should be conscious of their IT/digital requirements, and hence appropriate
strategy should be crafted accordingly. Additionally, proper planning and
effective mechanism are important for sustaining the strategy and addressing
challenges. The culture and structure of the organisation, as per Jonathan et
al. [6], is a considerable challenge that increases the digital transformation
procedure in the organisation. The vulnerability is high among public sector
organisations as they commonly encounter such challenges. The workforce
themselves exhibit low motivation and enthusiasm to adopt digital alteration
and show eagerness to continue with the existing system.

Considering the external problems, there are two main challenges that restrict
the increase of digital growth in the governmental sector. Firstly, lack of
experience and stakeholder relationship are the two challenges found in the
public sector, which affects the growth of digitalisation. Evidence from the
research of Gil-Garcia et al. [27] showed that the availability of trained and
skilled experts who possess better understanding and knowledge regarding
digital technologies are viewed to be the primary need for digital
transformation in both the public and private sector. It is also noted that
technically skilled people are available, yet the issue is with people who
lack experience with reference to recent digitalisation and training the
workforce accordingly [6]. The developing countries are viewed to be highly
affected in this regard, besides strict protocols, and public sector
regulations are other challenges that affect public sector organisations to
attract experts in increasing their digitisation process.

In contrary to the issues, there exist success factors as well, which can be
accomplished if employees are provided with an effective environment where
they can represent and work their talents and attain new capabilities. It is
highly important with the start of digital transformation to create digital
capabilities for the working and for the future workforce. The digital
training of employees is important because they will be the key integrators of
the tech-based system, where they will have the key role in integrating
digital framework and developing a new culture by replacing or reshaping the
existing one. Secondly, the role of leadership is also important; who must be
knowledgeable and have experience in addressing digital changes. Morakanyane
et al. [2] add that accessibility in the public sector is the factor that
determines the successful integration of digital transformation. A different
citizen may have a varying requirement, and therefore their expectation from
the authorities remains the same. The digital transformation should be
developed according to the public needs and needs to be accessible for a
citizen with multiple different options. Therefore, it is pivotal for the
government to adapt to changes and work on the environment that facilitates
technological changes in the organisation.

##  8 Conclusion

This research emphasised digital transformation in government and examined how
the digital transformation over the past few decades and have transformed and
what challenges/issues the public sector faces in integrating such
technologies. It is observed that Denmark is one of the leading developed
countries to adopt digital transformation due to effective e-government
strategies and a suitable environment. However, developing countries are found
to be lacking in adopting digital transformation due to the lack of tech
culture in their public and private organisation. Additionally, the lack of
trained and skilled employees is also one of the key challenges that restrict
digitisation in the government sector. The research highlights the effective
and favourable environment, skilled workforce, leadership, and governmental
policies and regulations are the key success factor that may encourage and
facilitate the quick adaption of digital transformation in public sector
organisations.

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# Notes on a short-cut to the proof of the $\mathbf{M}_{3}$-$\mathbf{N}_{5}$
Theorem

M. R. Emamy-K  Gustavo A. Meléndez Ríos
Department of Mathematics, UPR Río Piedras

(February 22, 2024)

###### Abstract

This paper presents two shortcuts to a classical proof of the
$\mathbf{M}_{3}$-$\mathbf{N}_{5}$ Theorem, which can be found in [2] and [1].
To be precise, the shortcuts pertain a particular step of the proof that
requires showing an algebraic equality. In addition, we briefly discuss how to
compare the lengths of the three proofs (the original and our two proposed
shortcuts). To do so, we introduce two methods to compare the lengths of
proofs based on algebraic lattice expressions. We call them the proof count
method and the proof poset method. Both methods indicate that our proofs are
shorter but the difference is more pronounced in the former.

Keywords: lattices, posets, proof length

##  1 Introduction

The $\mathbf{M}_{3}$-$\mathbf{N}_{5}$ Theorem is a basic theorem in lattice
theory that characterizes all distributive and modular lattices. A classical
proof of this theorem can be found in [2] (also in [1]). Part of this proof
consists of showing that a non-distributive modular lattice must have a
sublattice isomorphic to $\mathbf{M}_{3}$. Showing this, in turn, requires
proving an algebraic identity satisfied by three elements of the lattice. This
paper proposes two shortcuts to the proof of this identity in the
aforementioned references. They both consist of a reduction in the number of
algebraic steps needed to show said identity.

We then use these shortcuts to discuss how to compare the lengths of proofs
that consist of algebraic lattice expressions. After all, the length of a
proof depends on how it is presented. To this end, we give an overview of
results in [3] where we introduce two methods to compare the lengths of
proofs: the proof count method and the proof poset method. After applying them
to the three proofs of the equation (the original in [2], [1] and our two
shortcuts), we get that our proofs are shorter but that the difference is more
pronounced in the former method.

The rest of this paper is organized as follows. Section 2 provides some
background material that includes a review of the
$\mathbf{M}_{3}$-$\mathbf{N}_{5}$ Theorem with a sketch of its proof that
shows where the algebraic equality of interest fits in it. We then present the
original proof of the equality with our shortcuts in Section 3. Afterwards, we
summarize our analysis of the lengths of the three proofs in Section 4.
Finally, Section 5 concludes with a summary of our results and ideas for
continuing this work.

##  2 Background

We set the stage for our proposed proof shortcuts in two steps. First, we
briefly review some lattice theory basics and establish the notation we use in
Subsection 2.1. Afterwards, Subsection 2.2 reviews the
$\mathbf{M}_{3}$-$\mathbf{N}_{5}$ Theorem and gives a sketch of the proof in
[2] and [1]. For more information regarding lattices and posets, we refer the
reader to [2].

###  2.1 Preliminaries

We begin with some notation. Given a lattice $L$, we denote its meet operation
with $\land$ and its join operation with $\lor$. In addition, we denote its
induced order with $\leq$, using $<$ for strict order. If two lattices $L$ and
$K$ are isomorphic, we write $L\cong K$. Finally, recall that $\mathbf{M}_{3}$
and $\mathbf{N}_{5}$ denote the diamond and pentagon lattices, respectively.
These are shown in Figures 2 and 2.

$1$ $a$ $b$ $c$ $0$ Figure 1: The diamond $\mathbf{M}_{3}$

$1$ $u$ $v$ $w$ $0$ Figure 2: The pentagon $\mathbf{N}_{5}$

We also recall some basic properties and results about lattices. First, the
lattice operations $\land$ and $\lor$ satisfy four axioms: associativity,
commutativity, idempotency, and absorption. Second, all distributive lattices
are modular. Last, we formally state two results that we will make reference
to.

######  Lemma 1.

(Connecting Lemma) Let $L$ be a lattice with $a,b\in L$ and induced order
$\leq$. Then the following are equivalent:

  1. 1.

$a\leq b$;

  2. 2.

$a\lor b=b$;

  3. 3.

$a\land b=a$.

######  Lemma 2.

(half-way lemma) Let $L$ be a lattice with $a,b,c\in L$. Then

  1. 1.

$a\land(b\lor c)\geq(a\land b)\lor(a\land c);$

  2. 2.

$a\geq c\implies a\land(b\lor c)\geq(a\land b)\lor c.$

###  2.2 $\mathbf{M}_{3}$-$\mathbf{N}_{5}$ Theorem

We now review the $\mathbf{M}_{3}$-$\mathbf{N}_{5}$ Theorem and sketch the
proof in [2] and [1]. This theorem characterizes distributive and modular
lattices. The idea is that $\mathbf{N}_{5}$ is a forbidden sublattice of
modular lattices in the sense that all lattices that contain it are non-
modular. Similarly, $\mathbf{M}_{3}$ and $\mathbf{N}_{5}$ are forbidden
sublattices of distributive lattices.

######  Theorem 3.

($\mathbf{M}_{3}$-$\mathbf{N}_{5}$ Theorem) Let $L$ be a lattice.

  1. 1.

$L$ is non-modular if and only if $\mathbf{N}_{5}$ is a sublattice of $L$.

  2. 2.

$L$ is non-distributive if and only if $\mathbf{N}_{5}$ or $\mathbf{M}_{3}$ is
a sublattice of $L$.

Proof Sketch We provide a sketch of the proof in Davey and Priestley (page 89
of [2]; also in [1]). Given that sublattices of modular (distributive)
lattices are modular (distributive) and that $\mathbf{N}_{5}$ and
$\mathbf{M}_{3}$ are non-modular and modular but not distributive
respectively, it suffices to show that (i) a non-modular lattice has
$\mathbf{N}_{5}$ as a sublattice and (ii) that a non-distributive modular
lattice has $\mathbf{M}_{3}$ as a sublattice.

Part (i): Suppose that L is a non-modular lattice. Then there exist $d,e,f$ in
$L$ with $d>f$ such that $d\land(e\lor f)>(d\land e)\lor f$. This is because
by Lemma 2,

$d\land(e\lor f)\neq(d\land e)\lor f\implies d\land(e\lor f)>(d\land e)\lor
f.$ (1)

Define $a=(d\land e)\lor f$ and $b=d\land(e\lor f)$. It can be shown that
$a\land e=b\land e$ and $a\lor e=b\lor e$. Let $p=a\land e$ and $q=a\lor e$.
Then $K=\\{p,q,a,b,e\\}$ is a sublattice of $L$ that satisfies the necessary
joins and meets to be isomorphic to $\mathbf{N}_{5}$. It can be verified that
all 5 elements of $K$ are distinct. Therefore, $K\cong\mathbf{N}_{5}$ and $L$
has a sublattice isomorphic to $\mathbf{N}_{5}$.

Part (ii): Suppose that $L$ is a modular but non-distributive lattice. Then
there exist $d,e,f\in L$ such that $d\land(e\lor f)>(d\land e)\lor(d\land f)$.
This is because by Lemma 2,

$d\land(e\lor f)\neq(d\land e)\lor(d\land f)\implies d\land(e\lor f)>(d\land
e)\lor(d\land f).$ (2)

Define the following five elements in $L$:

$\displaystyle p$ $\displaystyle=(d\land e)\lor(e\land f)\lor(f\land d),$ (3)
$\displaystyle q$ $\displaystyle=(d\lor e)\land(e\lor f)\land(f\lor d),$ (4)
$\displaystyle u$ $\displaystyle=(d\land q)\lor p,$ (5) $\displaystyle v$
$\displaystyle=(e\land q)\lor p,$ (6) $\displaystyle w$ $\displaystyle=(f\land
q)\lor p.$ (7)

It can be shown that $K=\\{p,q,u,v,w\\}$ is a sublattice of $L$ isomorphic to
$\mathbf{M}_{3}$. This requires proving that

  1. 1.

$p<q;$

  2. 2.

$p\leq u,v,w\leq q;$

  3. 3.

$u\land v=v\land w=w\land u=p;$

  4. 4.

$u\lor v=v\lor w=w\lor u=q;$

  5. 5.

All five elements of K are distinct.

This concludes our proof sketch. ∎

We have finally arrived at our point of interest. Note that item 3 in the list
above requires showing three identities: $u\land v=p$, $v\land w=p$, and
$w\land u=p$. We remark that the proofs of all three of them are basically the
same. Therefore, we focus on the proof of the identity $u\land v=p$ for which
we propose the two shortcuts mentioned at the beginning of this paper. Thus,
we have established the context of our shortcuts within the proof of the
$\mathbf{M}_{3}$-$\mathbf{N}_{5}$ Theorem and are ready to present them.

##  3 3 Proofs of $u\land v=p$

We now discuss the three proofs of $u\land v=p$, the original in [2] and [1]
plus our two shortcuts. All three have the same general strategy: algebraic
manipulation of lattice expressions starting from $u\land v$ and ending with
$p$. In particular, they involve substitution of $u$, $v$, $p$, and $q$ for
their definitions and applications of lattice axioms, modular law, and the
Connecting Lemma.

Each proof is followed by a list of justifications for each algebraic step. In
these justifications we make several references to the modular law, that is;
for all $a,b,c\in L$ with $a\geq c,$ we have $a\land(b\lor c)=(a\land b)\lor
c$ or without distinction to its dual, that is; for all $a\leq c,$ we have
$a\lor(b\land c)=(a\lor b)\land c$ (we also underline $a$ and $c$ just before
each step when modularity is applied to facilitate the reading). Not all uses
of commutativity and associativity in the proofs are mentioned. The reader is
advised to consult the proof sketch of Theorem 3 to recall the definitions of
and the relations among the elements treated in the proofs.

We begin with the classical proof given in page 91 of [2] (book by B. A. Davey
and H. A. Priestley). It is also in [1]. We will call it Proof 1.

######  Proof 1.

(Classical proof in [2] and [1])

$\displaystyle u\land v$ $\displaystyle=((d\land
q)\lor\underline{p})\land(\underline{(e\land q)\lor p})$ (8a)
$\displaystyle=(((e\land\underline{q})\lor\underline{p})\land(d\land q))\lor
p$ (8b) $\displaystyle=((q\land(e\lor p))\land(d\land q))\lor p$ (8c)
$\displaystyle=((e\lor p)\land(d\land q))\lor p$ (8d)
$\displaystyle=((d\land(e\lor f))\land(e\lor(f\land d)))\lor p$ (8e)
$\displaystyle=(d\land((\underline{e\lor f})\land(\underline{e}\lor(f\land
d))))\lor p$ (8f) $\displaystyle=(d\land(((e\lor f)\land(f\land d))\lor
e))\lor p$ (8g) $\displaystyle=(\underline{d}\land((\underline{f\land d})\lor
e))\lor p$ (8h) $\displaystyle=((d\land e)\lor(f\land d))\lor p=p$ (8i)

Justification for each step of Proof 1

  * •

(8a) By definition of $u$ and $v$.

  * •

(8b) Applying modular law with $a=(e\land q)\lor p$, $b=d\land q$, and $c=p$.

  * •

(8c) Applying modular law with $a=q$, $b=e$, and $c=p$.

  * •

(8d) By associativity, commutativity, and idempotency on $q$.

  * •

(8e) By definitions of $p$ and $q$, associativity, commutativity, and
absorption.

  * •

(8f) By associativity.

  * •

(8g) Applying modular law with $a=e\lor f$, $b=f\land d$, and $c=e$.

  * •

(8h) Since $f\land d\leq f\leq e\lor f$.

  * •

(8i) Applying modular law with $a=d$, $b=e$, and $c=f\land d$ and using the
fact that $(d\land e)\lor(f\land d)\leq p$ by definition of $p$. ∎

We now present our two new proofs of the same result. These will be Proofs 2
and 3 respectively. We remark that the versions of both presented here are an
improvement from those in [3].

  1. 1.

Proof 2: It takes a short-cut in Proof 1.

  2. 2.

Proof 3: It results from trying to re-create either of the first two proofs
from memory but taking a different route to the same result instead.

######  Proof 2.

$\displaystyle u\land v$ $\displaystyle=((d\land
q)\lor\underline{p})\land(\underline{(e\land q)\lor p})$ (9a)
$\displaystyle=(((e\land\underline{q})\lor\underline{p})\land(d\land q))\lor
p$ (9b) $\displaystyle=((q\land(e\lor p))\land(d\land q))\lor p$ (9c)
$\displaystyle=((e\lor p)\land(d\land q))\lor p$ (9d)
$\displaystyle=((\underline{d}\land(e\lor f))\land(e\lor(\underline{f\land
d})))\lor p$ (9e) $\displaystyle=((e\lor f)\land((d\land e)\lor(f\land
d)))\lor p=p$ (9f)

Justification for each step of Proof 2

  * •

(9a)-(9e) Identical to (8a)-(8e) in Proof 1.

  * •

(9f) By commutativity, associativity, applying modular law with $a=d$, $b=e$,
and $c=f\land d$, and the fact that $(e\lor f)\land((d\land e)\lor(f\land
d))\leq p$ by definition of $p$. ∎

######  Proof 3.

$\displaystyle u\land v$
$\displaystyle=[(d\land\underline{q})\lor\underline{p}]\land[(e\land\underline{q})\lor\underline{p}]$
(10a) $\displaystyle=[q\land(d\lor p)]\land[q\land(e\lor p)]$ (10b)
$\displaystyle=q\land\\{(d\lor p)\land(e\lor p)\\}$ (10c)
$\displaystyle=q\land\\{[\underline{d\lor(e\land
f)}]\land[e\lor(\underline{d\land f})]\\}$ (10d)
$\displaystyle=q\land\\{[(d\lor(\underline{e\land
f}))\land\underline{e}]\lor(d\land f)\\}$ (10e)
$\displaystyle=q\land\\{[(e\land d)\lor(e\land f)]\lor(d\land f)\\}$ (10f)
$\displaystyle=q\land\\{(e\land d)\lor(e\land f)\lor(d\land f)\\}=p$ (10g)

Justification for each step of Proof 3

  * •

(10a) By definition of $u$ and $v$.

  * •

(10b) Applying modular law twice:

    1. 1.

With $a=q$, $b=d$, and $c=p$.

    2. 2.

With $a=q$, $b=e$, and $c=p$.

  * •

(10c) By commutativity, associativity, and idempotency.

  * •

(10d) By definition of $p$ and absorption.

  * •

(10e) Applying modular law with $a=d\lor(e\land f)$, $b=e$, and $c=d\land f$.
Modular law can be applied because $a=d\lor(e\land f)\geq d\geq d\land f=c$.

  * •

(10f) Applying modular law with $a=e$, $b=d$, and $c=e\land f$.

  * •

(10g) By associativity, the definition of $p$, and $p<q$. ∎

##  4 Comparing Proof Lengths

Having shown the three proofs of the identity $u\land v=p$, we turn our
attention to comparing their length. It can readily be seen that Proofs 1, 2,
3 respectively have 9, 6, and 7 lines. However, we know that the length of a
proof is not an immutable property since it depends on the level of detail
given (and hence on presentation). Thus, we study this matter further by
coming up with two methods to compare the length of the three proofs. We will
call them proof count and proof poset respectively. The first will be naïve
while the second will be more rigorous. Both methods are of our own design.
Here, we limit ourselves to an overview of the results of our analysis. The
details can be found in [3].

###  4.1 Proof Count Method

The proof count method is a straightforward brute-force approach. We measure
the length of each proof by counting the lines and characters it uses. To be
more precise, we count the following:

  1. 1.

lines,

  2. 2.

variables,

  3. 3.

operation symbols,

  4. 4.

grouping symbols,

  5. 5.

equal signs,

  6. 6.

all symbols.

Table 1 shows the results of comparing the three proofs with the proof count
method. The main observation we make is that Proofs 2 and 3 use significantly
less symbols than Proof 1 (129 and 155 vs. 185). There are two discrepancies
between the data here and that in [3] that must be explained. The first is
that we moved the final $=p$ of each proof to the previous line for the sake
of efficiency. The second is that we made improvements to Proofs 2 and 3, as
mentioned in Section 3.

Proof | lines | variables | operations | grouping | equal | symbol total
---|---|---|---|---|---|---
Proof 1 | 9 | 57 | 46 | 72 | 10 | 185
Proof 2 | 6 | 40 | 32 | 50 | 7 | 129
Proof 3 | 7 | 48 | 39 | 60 | 8 | 155
Table 1: Results of Proof Count

###  4.2 Proof Poset Method

The second method we use is the proof poset method. In it, we build a poset
(specifically a chain) to represent each of the proofs as follows:

  1. 1.

Top element: $u\land v=[(d\land q)\lor p]\land[(e\land q)\lor p]$;

  2. 2.

Bottom element: $p$;

  3. 3.

Each vertex is a statement in proof;

  4. 4.

Vertex $i$ covers vertex $j$ if we can go from statement $i$ to statement $j$
by applying only one rule from a small list of basic rules (to be given in
Definition 4);

  5. 5.

Must add statements to proof if going from $i$ to $i+1$ requires more than one
rule.

The idea is to examine in detail the logical structure of each of the proofs
by decomposing them into their building blocks. We admit the following list of
basic rules for the covering relation of the proof posets.

######  Definition 4.

(basic rules for proof poset)

  * •

Lattice axioms

    1. 1.

L1: associative laws

    2. 2.

L2: commutative laws

    3. 3.

L3: idempotency laws

    4. 4.

L4: absorption laws

  * •

Modular law

  * •

Other rules

    1. 1.

Def: Substitute an element for its definition or vice-versa.

    2. 2.

Applying Connecting Lemma or some other order property of lattices.

In order to shorten the posets, we will also allow the following combinations
of the modular law with commutativity as a single step (covering relation of
poset).

######  Definition 5.

(list of combinations of modular law with commutativity)

  1. 1.

M: Modular law

$a\geq c\implies a\land(b\lor c)=(a\land b)\lor c$ (11)

  2. 2.

M1: Modular law with inner commute

$a\geq c\implies a\land(c\lor b)=(a\land b)\lor c$ (12)

  3. 3.

M2: Modular law with outer commute

$a\geq c\implies(b\lor c)\land a=(a\land b)\lor c$ (13)

  4. 4.

M3: Modular law with reverse inner commute

$a\geq c\implies(b\land a)\lor c=a\land(b\lor c)$ (14)

  5. 5.

M4: Modular law with reverse outer commute

$a\geq c\implies c\lor(a\land b)=a\land(b\lor c)$ (15)

Needless to say, none of the proofs given above is fully decomposed based on
the lists of Definitions 4 and 5. Hence, they require adding intermediate
statements. Doing so results in having three fully decomposed proofs from
which we can construct three proof posets as indicated above. The fully
decomposed proofs and the proof posets can be found in [3].

We now give an overview of the results from the proof poset method. We compute
the lengths of the decomposed proofs (number of vertices of poset) in Table 2
and count how many times each proof uses each basic rule in Table 3. From
Table 2, we get that Proofs 2 and 3 are only slightly shorter than Proof 1
based on our notion of “basic rule” in Definitions 4 and 5. The main
observation from Table 3 is that Proof 2 uses the modular law one time less
than the other proofs (3 vs. 4). Noting that the modular law is less basic
than the axioms, this could be interpreted as saying that Proof 2 is somehow
logically simpler than the other two.

Therefore, we conclude the following from our proof comparison exercise:

  1. 1.

Proof Count Method: Proofs 2 and 3 are shorter than Proof 1.

  2. 2.

Proof Poset Method: Proofs 2 and 3 are slightly shorter than Proof 1.

  3. 3.

Proof 2 is the shortest proof either way.

Proof 1 | 32
---|---
Proof 2 | 29
Proof 3 | 30
Table 2: Length of proof posets Proof | L1 | L2 | L3 | L4 | M | M1 | M2 | M3 | M4 | Def | Other
---|---|---|---|---|---|---|---|---|---|---|---
1 | 9 | 9 | 1 | 4 | 0 | 2 | 1 | 1 | 0 | 2 | 2
2 | 9 | 8 | 1 | 4 | 1 | 0 | 1 | 1 | 0 | 2 | 1
3 | 10 | 6 | 1 | 4 | 1 | 0 | 1 | 2 | 0 | 3 | 1
Table 3: Count of basic rules invoked by Proofs 1-3

##  5 Conclusion

We bring this paper to a close with a summary of our results and some
suggestions for continuing this work. We presented two shortcuts to a part of
a classical proof of the $\mathbf{M}_{3}$-$\mathbf{N}_{5}$ Theorem found in
[2, 1]. These were the result of “algebraic efficiencies” in proving the
equation $u\land v=p$ in said proof. We also gave an overview of our work
comparing the length of the three proofs in [3] where we proposed and applied
two methods for doing so: the proof count method and the proof poset method.
Both demonstrated that the new proofs were shorter than the proof in [2, 1]
but the difference was more marked in the proof count method.

Needless to say, it is possible to keep pushing the boundaries of this work.
In particular, we can develop further both proof length comparison methods and
apply them to other proofs in lattice theory. This could lead to a better-
defined notion of “proof length” and, thus, a potential classification of
lattice theory proofs based on it. We finish with some ideas for the
development of both methods.

######  Problem 1.

(Proof count method ideas)

  1. 1.

Consider other aspects that can be counted.

  2. 2.

Add different weights to different symbols.

######  Problem 2.

(Proof poset method ideas)

  1. 1.

Refine better basic rules. For instance, break the combined modular law rules
(M1-M4) to see the impact on the final results.

  2. 2.

Add different weights to different rules based on how basic they are.

## References

  * [1] Stanley Burris and Hanamantagouda P. Sankappanavar.  A Course in Universal Algebra.  Springer-Verlag, 1st edition, 1981.  Retrieved from the Millenium Edition, an online reprint from 2009 (updated in 2012) of the original 1981 Springer-Verlag edition.
  * [2] Brian Davey and Hilary Priestley.  Introduction to Lattices and Order.  Cambridge University Press., Cambridge, UK, 2nd edition, 2002.
  * [3] Gustavo A. Meléndez Ríos.  $ST$-distributive and $ST$-modular lattices.  Master’s thesis, Universidad de Puerto Rico, Recinto de Río Piedras, April 2022.