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Open Access Theses & Dissertations

2012-01-01 Inverse and Inverse Categories Alexandra Macedo University of Texas at El Paso, [email protected]

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This is brought to you for free and open access by DigitalCommons@UTEP. It has been accepted for inclusion in Open Access Theses & Dissertations by an authorized administrator of DigitalCommons@UTEP. For more information, please contact [email protected]. INVERSE SEMIGROUPS AND INVERSE CATEGORIES

ALEXANDRA MACEDO

Department of Mathematical Sciences

APPROVED:

Emil Daniel Schwab, Ph.D., Chair

Helmut Knaust, Ph.D.

Cristian Botez, PhD.

Benjamin C. Flores, Ph.D. Interim Dean of the Graduate School

Copyright by ALEXANDRA MACEDO 2012

Dedicated to my parents Alexandru and Marta Bogdan, to my husband Oscar, and to my daughter Alexa. INVERSE SEMIGROUPS AND INVERSE CATEGORIES

by

ALEXANDRA MACEDO, M.S.

THESIS

Presented to the Faculty of the Graduate School of The University of Texas at El Paso in Partial Fulfillment of the Requirements for the Degree of

MASTER OF SCIENCE

Department of Mathematical Sciences THE UNIVERSITY OF TEXAS AT EL PASO August 2012

Acknowledgements

I hereby would like to express my deepest gratitude to the honorable members of my thesis committee for their guidance and suggestions. I am most grateful to my advisor and committee chair Dr. Emil Schwab for all his patience, motivation, help, time, and dedication.

Without his vast knowledge, superb work ethic, and guidance this Thesis Dissertation would have not been possible. I could not have asked for a better advisor and mentor.

I would like to give special thank to Dr. Helmut Knaust. With his advice, assistance, and I was able to get through rough academic and administrative challenges. My acknowledgement also goes to Dr. Cristian Botez for accepting to be part of my thesis committee.

Last but not the least, I would like to thank my family for their unconditional love and support.

All in all, this Thesis Dissertation is not only a of my hard work, but it is also an achievement of those around me.

v Abstract

The theory of inverse semigroups forms a major part of theory. This theory has deep connections with important mathematical disciplines, not only classical ones such as geometry, , number theory, but also with more recent theories: the theory of , graph theory, the mathematical theory of automata, etc. The importance of inverse semigroups, first and foremost, is that they form an abstract of algebraic structures which are isomorphic to semigroups of partial .

This thesis is organized in two parts, inverse semigroups in Part 1, and inverse categories (that arises if we apply a basic property of inverse semigroups to of a ) in Part 2.

In the first chapter of my thesis, I out the basic properties of semigroups and inverse semigroups: this includes the theorem for semigroups, the algebraic properties of inverse semigroups connected with inverses, elements, Green’s relations, etc. The examples presented at the end of the first chapter include the inverse semigroups of partial bijections, the free monogenic and an inverse semigroup-like set which is an analogue of -like sets. The last example was used in our paper [9], and was one of the examples suggested the need for an inverse semigroup-like .

The theory of inverse categories, a natural generalization of the theory of inverse , may be regarded as the theory of partial . In chapter two of this thesis, I present the basic properties of inverse categories which are analogous to the properties of inverse semigroups. Four examples of inverse categories are discussed (given) at the end of my thesis: the inverse category of partial bijections, the inverse category of invertible matrices, the inverse

vi category that represent an equivalence , and an inverse category as a of the category of based sets and based functions.

vii Table of Contents

Acknowledgements ...... v

Abstract ...... vi

Table of Contents ...... viii

Chapter 1: Semigroups ...... 1 1.1 Basics Concepts of Semigroups ...... 1 1.2 Inverse Semigroups ...... 9

Chapter 2: Categories ...... 25 2.1 Basics Concepts of Categories ...... 25 2.2 Inverse Categories ...... 34

References ...... 40

Index ...... 41

Curriculum Vita ...... 42

viii Chapter 1 Semigroups

The foundations of semigroup theory were set by the Russian mathematician A.K.

Suschkewitsch in 1928. At the beginning the theory of semigroups was related and compared with the theory of groups and rings. The study of the semigroups requires also the study of congruences. The first mathematician that introduced the inverse semigroups was V.V. Wagner, in 1952. However, he referred them as generalized groups. In 1954 G.B. Preston reintroduced the same theory, but he called them inverse semigroups.

1.1 Basic Concepts of Semigroups

A semigroup is considered to be an 풮, 퓌2 , where 풮 is a nonempty set and 퓌2 is an associative binary on 풮. The is defined as a

퓌2: 풮 × 풮 → 풮. For simplicity we will denote the binary operation 퓌2 퓍, 푦 with the multiplication symbol 푥푦, for all 퓍, 푦 ∈ 풮. The is 퓍 푦푧 = 퓍푦 푧, for any elements 퓍, 푦, 푧 ∈ 풮.

Definition 1.1. If there exists an element ℯ of a semigroup 풮 with the property that for all 퓍 ∈ 풮, we have 푒퓍 = 퓍푒 = 퓍, then ℯ is called the element of 풮. A semigroup with an is called a .

Observation 1.2. From now on, we will denote the identity element by 1. Therefore, the relation above can be written as follows 1퓍 = 퓍1 = 퓍.

1 Proposition 1.3. A monoid cannot have more than one identity element.

Proof. Let 1 ∈ 풮 be the identity element such that for any 퓍 ∈ 풮 we have 1퓍 = 퓍1 = 퓍.

Suppose there is another element 1′ ∈ 풮 with the same property that for any 퓍 ∈ 풮 we have 1′ 퓍 = 퓍1′ = 푥. Using the identity element we can write 1′ = 1′ 1 . We know that

1′ 퓍 = 퓍, for any 퓍 ∈ 풮. This implies that 1′ = 1′ 1 = 1 . Therefore, the identity element is unique. □

Now, we can define the set of all units. The set of all units of 풮, denoted by 푈 풮 , can be described as follows 푈 풮 = 퓍 ∈ 풮 | there is 푎 퓍′ ∈ 풮 such that 퓍퓍′ = 퓍′ 퓍 = 1 ⊆ 풮.

Definition 1.4. A of a semigroup 풮 is an element ℴ ∈ 풮 with the property that for all 퓍 ∈ 풮, we have 풮 ≠ ℴ 푎푛푑 ℴ퓍 = 퓍ℴ = ℴ .

Observation 1.5. From now on, for simplicity, we will denote the zero element by 0. Therefore, the relation above can be written as follows 0퓍 = 퓍0 = 0.

Proposition 1.6. A semigroup cannot have more than one zero element.

Proof. Let 0 ∈ 풮 be the zero element such that for any 퓍 ∈ 풮 we have 0퓍 = 퓍0 = 0. Suppose there is another element 0′ ∈ 풮 with the property that for any 퓍 ∈ 풮 we have 0′ 퓍 = 퓍0′ = 0′ .

Using the fact that 0′ 퓍 = 퓍0′ = 0′ we can write 0′ = 0′ 0. For any 퓍 ∈ 풮 we know that 퓍0 = 0, thus, we obtain 0′ = 0′ 0 = 0. This implies that 0′ = 0. Therefore, the zero element is unique. □

2 A semigroup is called commutative if for any elements 퓍, y from 풮 we have 퓍y = y퓍.

Let 풮 be a semigroup with 퓍 ∈ 풮 and consider 푘 a positive . We can denote the power of an element by the following relations: 퓍1 = 퓍 and 퓍푘+1 = 퓍푘 퓍.

Observation 1.7. Let 풮 be a monoid and 퓍 ∈ 풮, then by notation 퓍0 = 1.

Proposition 1.8. In a semigroup 풮, for any element 퓍 ∈ 풮 and 푛, 푚 any positive , the following property holds: 퓍푛 퓍푚 = 퓍푛+푚 .

Proof. We will use the induction after 푛. For 푛 = 1 we have 푥푚 ⋅ 푥1 = 푥푚+1 is true. Assume the relation true for 푛 = 푘 . Thus, 푥퓂 ⋅ 푥푘 = 푥퓂+푘 . Let 푛 = 푘 + 1. We have to prove that

푥퓂 ⋅ 푥푘+1 = 푥퓂+푘+1. So, 푥푚 ⋅ 푥푘+1 = 푥푚 ⋅ 푥푘 ⋅ 푥 = 푥푚 ⋅ 푥푘 ⋅ 푥 = 푥푚+푘 ⋅ 푥 = 푥푚+푘+1. □

Definition 1.9. Let 풮 be a semigroup and consider an element ℯ from 풮. The element ℯ is called idempotent, if 푒2 = 푒 ∙ 푒 = ℯ.

Observation 1.10. In any semigroup 풮, if the identity element or the zero element exist, they are idempotents. Also, if ℯ is idempotent, then ℯ푛 = ℯ , for any 푛 positive integer.

Examples of semigroups:

 Any group is a semigroup.

 Any under multiplication is a semigroup.

 ℤ+, + , ℤ+, ⋅ , ℤ, ⋅ , ℤ푛 , ⋅ are semigroups; where ℤ is the set of integers,

3 ℤ+ is the set of non-negative integers, ℤ푛 the complete set of residues modulo n,

and + and ∙ are the normal operations of addition and multiplication.

 Given 풞 a category, and A is an object of 풞, then the morphisms in 풞 from A to A,

ℋℴ퓂풞 퐴, 퐴 , is a semigroup.

 Let 풮 퐴 = 푓 | 푓: 퐴 → 퐴 , where 푓 are functions from A to A, and 풮 퐴 is

closed under the usually ∘ . Then, 풮 퐴 , ∘ is a semigroup

and it is called the symmetric semigroup on A.

Definition 1.11. Take 풮 and 풮′ to be two semigroups such that 풮′ is the of 풮, 풮′ ⊆ 풮. We call 풮′ a subsemigroup if 풮′ is closed under multiplication ( i.e. for any 퓍, 푦 ∈ 풮′, 퓍푦 ∈ 풮′ ).

Definition 1.12. Let 풮 and 풯 be two semigroups and consider the mapping 푓: 풮 → 풯 from 풮 into 풯. We say that 푓 is a semigroup , if for any , 푦 ∈ 풮, 푓 퓍푦 = 푓 퓍 푓 푦 .

A bijective semigroup homomorphism is called a semigroup isomorphism.

Observation 1.13. The composition of two semigroup is a homomorphism.

For instance, if 푓: 풮 → 풯 and 푕 ∶ 풯 → 풱 are homomorphisms, it implies that 푓 ∘ 푕 ∶ 풮 → 풱 is also a homomorphism.

For 풮 and 풯 two monoids and for the mapping 푓: 풮 → 풯 we can say that 푓 is a monoid homomorphism from 풮 to 풯, if 푓 is a homomorphism that preserves identity elements 푓 1 = 1

4 Definition 1.14. Let 풮 be a semigroup and consider 휓 an equivalent relation on 풮. We say that

휓 is a congruence on 풮, if for any elements 푥, 푥′, 푦, 푦′ from 풮, with the properties 퓍 휓 푦 and

푥′휓 푦′ , we have 퓍푥′휓 푦푦′ .

Before we continue with the next properties, we need to introduce more terminology and notations. Given the mapping 푓: 풮 → 풯 a semigroup homomorphism, we will define an on 풮, denoted 푘푒푟푓, where 푘푒푟푓 ⊆ 풮2 as it follows: for any 퓍, y ∈ 풮 ,

퓍 푘푒푟푓 y, if and only if, 푓 퓍 = 푓 푦 .

Furthermore, we will call the set 풮/푘푒푟푓 the quotient of 풮 by 푘푒푟푓 and for any 푥 , 푦 ∈

풮/ ker 푓 we will consider the well defined operation 푥 푦 = 푥 푦. It is easy to verify that 풮/푘푒푟푓,

∙ is a semigroup called the quotient semigroup. We define the 푝: 풮 → 풮/푘푒푟 푓 such that 푝 퓍 = 푥 .

Let 풮 be a semigroup and 풮′ its subsemigroup. The inclusion mapping 푖 ∶ 풮′ ↪ 풮 is defined by 푖 퓍 = 퓍.

Proposition 1.15. Given 푓: 풮 → 풯 a surjective semigroup homomorphism, let 휓 be a congruence on 풯. Let’s define 푓−1 휓 by 퓍푓−1 휓 푦 if and only if, 푓퓍 휓 푓푦. Consequently,

푓−1 휓 is a congruence on 풮 that includes ker 푓.

Theorem 1.16. (The Isomorphism Theorem). Given 푓: 풮 → 풯 a semigroup homomorphism, we have the following statements:

( i ) The of f , im f is a subsemigroup of 풯. ( 푓 풮 = 푖푚 푓).

( ii ) 푘푒푟푓 is a congruence on 풮.

5 ( iii ) There exists an unique isomorphism 휃: 풮/푘푒푟 푓 → 푖푚 푓 such that the diagram is commutative (i.e. 푓 = 푖 ∘ 휃 ∘ 푝).

푓 풮 풯 푝 ↓ ↑ 푖 풮/ 푘푒푟푓 푖푚 푓 휃

Figure 1.1 Commutative Diagram for the Isomomorphism Theorem

Proof. ( i ) We have to show that if 푦1, 푦2 ∈ 푖푚 푓 then 푦1푦2 ∈ 푖푚 푓 .

Given 푦1, 푦2 ∈ 푓 풮 = 푖푚 푓, there exist 퓍1, 퓍2 ∈ 풮 such that 푓 퓍1 = 푦1, 푓 퓍2 = 푦2 and 퓍1퓍2 ∈ 풮. First we take the product of 푦1푦2, and then we substitute 푦1, 푦2 with 푓 퓍1 and

푓 퓍2 , respectively. The next step is to use the fact that f is a homomorphism. Hence, we obtain

푦1푦2 = 푓 퓍1 푓 퓍2 = 푓 퓍1퓍2 . Since we know that 퓍1퓍2 ∈ 풮, it implies that 푓 퓍1퓍2 ∈ 푓 풮 .

So, the product 푦1푦2 ∈ 푓 풮 = 푖푚 푓 . Therefore, 푖푚 푓 is a subsemigroup of 풯.

( ii ) We have to show that 퓍1퓍2 ker 푓 푦1푦2 .

Let 퓍1, 퓍2, 푦1, 푦2 ∈ 풮, such that 퓍1 ker 푓 푦1 and 퓍2 ker 푓 푦2. By the definition of the ker 푓 we have, 퓍1 ker 푓 푦1 , it implies that 푓 퓍1 = 푓 푦1 and 퓍2 ker 푓 푦2 it implies that

푓 퓍2 = 푓 푦2 . In what follows, we will use these equalities as well as the properties of the homomorphism.

푓 퓍1퓍2 = 푓 퓍1 푓 퓍2 = 푓 푦1 푓 푦2 = 푓 푦1푦2 ⟹ 퓍1퓍2 = 푦1푦2 ⟹ 퓍1퓍2 ker 푓 푦1푦2 .

( iii ) We have to show that there exist a isomorphism 휃: 풮/푘푒푟 푓 → 푖푚 푓. We take

휃 퓍 = 푓 퓍 , where 휃 is well defined: for any 푥′ from 퓍 we have 푥′ ker 푓 푥 that implies

푓 푥′ = 푓 푥 .

6 We will verify that 휃 is bijective. Let 퓍 1 , 퓍 2 ∈ 풮/ ker 푓 such that 휃 퓍1 = 휃 퓍2 .

Based on the definition of 휃, we know that 휃 퓍1 = 푓 퓍1 and 휃 퓍2 = 푓 퓍2 . Consequently, from푓 퓍1 = 푓 퓍2 it implies that 퓍1 ker 푓 퓍2, thus 퓍 1 = 퓍 2, hence, 휃 is injective. Given 푦 ∈

푖푚 푓, there exists 퓍 ∈ 풮 such that 푓 퓍 = 푦. Furthermore, there exists 푥 ∈ 풮/ ker 푓 with

휃 퓍 = 푓 퓍 = 푦. So, 휃 is surjective. Since 휃 is injective and surjective, it implies 휃 is bijective.

Now, we will verify that 휃 is a homomorphism. Let’s take 푥 , 푦 ∈ 풮/ ker 푓. Then, we have 휃 퓍 푦 = 휃 퓍 푦 = 푓 퓍푦 = 푓 퓍 푓 푦 = 휃 푥 휃 푦 , this implies that 휃 is a homomorphism. Therefore, 휃 is a isomorphism.

We have to show that 푓 = 푖 ∘ 휃 ∘ 푝 as well as the uniqueness of 휃. Given any 퓍 ∈ 풮, we have:

푖 ∘ 휃 ∘ 푝 퓍 = 푖 ∘ 휃 푝 퓍 = 푖 ∘ 휃 퓍 = 푖 휃 퓍 = 푖 푓 퓍 = 푓 푥 ⟹ 푖 ∘ 휃 ∘ 푝 = 푓.

Suppose there is another 휃′ : 풮/푘푒푟 푓 → 푖푚 푓 with the property that 푖 ∘ 휃′ ∘ 푝 = 푓. Since

푖 ∘ 휃 ∘ 푝 = 푓 and also 푖 ∘ 휃′ ∘ 푝 = 푓, it means that 푖 ∘ 휃 ∘ 푝 = 푖 ∘ 휃′ ∘ 푝. Based on the fact that 푖 is injective, we can imply that 휃 ∘ 푝 = 휃′ ∘ 푝 ⟹ 휃 = 휃′. □

Theorem 1.17. ([7] p.12) If 푓: 풮 → 풯 is an injective homomorphism, the homomorphism

푕 ∶ 풰 → 풯 factors through 푓 푕 = 푓 ∘ 푘, 푓표푟 푠표푚푒 푘: 풰 ⟶ 풮 if and only if, 푖푚 푕 ⊆ im 푓; then 푕 factors uniquely through f ( h is unique).

Theorem 1.18. ([7] p.12) If 푓: 풮 → 풯 is a surjective homomorphism, the homomorphism

푕 ∶ 풮 → 풰 factors through 푓 푕 = 푘 ∘ 푓, 푓표푟 푠표푚푒 푘: 풯 ⟶ 풰 if and only if, 푘푒푟 푓 ⊆ ker 푕; then 푕 factors uniquely through f ( k is unique).

7 Proof. ([7]) If h ∶ 풮 → 풰 factors through 푓 푕 = 푘 ∘ 푓, 푓표푟 푠표푚푒 푘 , then 퓍 ker 푓푦 implies

푓 퓍 = 푓 푦 , 푕 퓍 = 푘 푓 푥 = 푘 푓 푦 = 푕 푦 , and 퓍 ker 푕푦 . So, 푘푒푟 푓 ⊆ ker 푕 .

Conversely, we will assume that 푘푒푟 푓 ⊆ ker 푕. We are looking for a homomorphism 푘 that satisfies 푘 푓 푧 = 푕 푧 , for any 푧 ∈ 풮. We shall prove that there exists a well defined mapping 푘: 풯 ⟶ 풰 which assigns 푕 푧 to 푓 푧 , for any 푧 ∈ 풮. For each 푡 ∈ 풯, there is at least one 푕 푧 to assign to 푡 , since 푓 is surjective. If 푕 푧′ and 푕 푧′′ are assigned to 푡, then 푡 =

푓 푧′ = 푓 푧′′ , 푧′ ker 푓 푧′′ , 푧′ ker 푕 푧′′ , since 푘푒푟 푓 ⊆ ker 푕 and 푕 푧′ = 푕 푧′′ . Thus at most one element is assigned to 푡. Hence, a mapping 푘: 풯 ⟶ 풰 is well-defined by 푘 푓 푧 = 푕 푧 for any 푧 ∈ 풮.

Since 푓 and 푕 are homomorphisms, 푘 푓 퓍 푓 푦 = 푘 푓 퓍푦 = 푕 퓍푦 = 푕 퓍 푕 푦 =

푘 푓 푥 푘 푓 푦 for any 푓 퓍 , 푓 푦 ∈ 풯, and 푘 is a homomorphism. Also, by definition we have 푕 = 푘 ∘ 푓. If 푘′: 풯 ⟶ 풰 is another homomorphism such that 푕 = 푘′ ∘ 푓 , then 푘 and 푘′ agree on 푖푚 푓 = 풯, and 푘 = 푘′. □

8 1.2. Inverse Semigroups

We will begin this section by giving the description of some terms and notations. Let 풮 be a semigroup and 푎 ∈ 풮. We say that 푎 is regular if based on the definition there exist 퓍 ∈ 풮 such that we have 푎 = 푎퓍푎. A semigroup 풮 is called a if each element of 풮 is regular.

Given 풮 a semigroup and 푎, 푏 ∈ 풮, we say that 푏 is an inverse of 푎 if 푎 = 푎푏푎 and

푏 = 푏푎푏 and we say that 푎 and 푏 are mutually inverse. In general, the inverse of an element is not unique in a semigroup.

Definition 1.19. We call 풮 an inverse semigroup if each element of 풮 has an unique inverse.

We will use for the set of all idempotents of the semigroup 풮, the following notation:

퐸 풮 = 푒 ∈ 풮 ∶ 푒2 = 푒

Proposition 1.20. Consider 풮 a semigroup and 푎, 푏 ∈ 풮 ,then,

( i ) If 푎 is regular, then 푎 has an inverse.

( ii ) If 푎, 푏 are mutually inverse, then 푎푏, 푏푎 ∈ 퐸 풮 .

Proof. ( i ) We have to show that there exist an 푏 ∈ 풮 such that 푎 = 푎푏푎 and 푏 = 푏푎푏.

If 푎 is regular it implies that there exists an element 퓍 ∈ 풮 such that 푎 = 푎퓍푎. Let’s take the element 푏 ∈ 풮 that satisfies the relation 푏 = 퓍푎퓍. Next, we will substitute 푏 = 퓍푎퓍 in

푎 = 푎푏푎 and 푏 = 푏푎푏. So, we obtain 푎푏푎 = 푎 퓍푎퓍 푎 = 푎퓍푎퓍푎 = 푎퓍푎 퓍푎 = 푎퓍푎 = 푎 and

푏푎푏 = 퓍푎퓍 푎 퓍푎퓍 = 퓍푎퓍푎퓍푎퓍 = 퓍 푎퓍푎 퓍푎퓍 = 퓍푎퓍푎퓍 = 퓍 푎퓍푎 퓍 = 퓍푎퓍 = 푏. Given

9 the fact that 푎푏푎 = 푎 and 푏푎푏 = 푏, it implies that 푏 is an inverse of 푎.

( ii ) We have to show that 푎푏 2 = 푎푏 and 푏푎 2 = 푏푎.

Since 푎, 푏 are mutually inverse, we have that 푎 = 푎푏푎 and 푏 = 푏푎푏. Thus, using these equalities we get the following 푎푏 2 = 푎푏푎푏 = 푎푏푎 푏 = 푎푏 ⟹ 푎푏 ∈ 퐸 풮 and

푏푎 2 = 푏푎푏푎 = 푏푎푏 푎 = 푏푎 ⟹ 푏푎 ∈ 퐸 풮 . □

If 풮 is an inverse semigroup, with 푎, 푏 ∈ 풮 and 푏 is the unique inverse of 푎, then we can denote 푏 by 푎−1 ( the inverse of 푎 ). As a result, we can write the following observation.

Observation 1.21. Given 풮 an inverse semigroup, then,

( i ) 푎 = 푎푎−1푎 and 푎−1 = 푎−1푎푎−1 , for any 푎 ∈ 풮.

( ii ) 푎−1 −1 = 푎 , for any 푎 ∈ 풮.

( iii ) 푒−1 = 푒 , for any 푒 ∈ 퐸 풮 .

( iv ) 푎푎−1, 푎−1푎 ∈ 퐸 풮 , for any 푎 ∈ 풮.

( v ) 풮 is a regular semigroup.

In the following part of the section, we will talk about the Green’s relations.

Consider 풮 a semigroup, and a nonempty subset 풜 ⊆ 풮, then 풜 is a left of 풮, if

풮풜 ⊆ 풜 ( i.e. for any 퓍 ∈ 풮 and 푎 ∈ 풜 we have 퓍푎 ∈ 풜 ). We say that 풜 is a right ideal of

풮, if 풜풮 ⊆ 풜 ( i.e. for any 퓍 ∈ 풮 and 푎 ∈ 풜 we have 푎퓍 ∈ 풜 ).

Definition 1.22. Let 풮 be a semigroup and 푎 ∈ 풮. The smallest left ideal containing 푎, denoted by 풮1푎 = 풮푎 ∪ 푎 , is called the left principal ideal generated by 푎. Analogous, we can define

10 the right principal ideal generated by 푎 , denoted by 푎풮1 = 푎풮 ∪ 푎 as the smallest right ideal containing 푎 .

Definiton 1.23. Given 풮 a semigroup with 푎, 푏 ∈ 풮, Green’s Relations ℒ and 풭 are defined as follow:

( i ) 푎 ℒ 푏 if 풮1푎 = 풮1푏, ( i.e. there exist 퓍, 푦 ∈ 풮1 such that 퓍푎 = 푏 and 푦푏 = 푎 ).

( ii ) 푎 풭 푏 if 푎풮1 = 푏풮1, ( i.e. there exist 푧, 푡 ∈ 풮1 such that 푎푧 = 푏 and 푏푡 = 푎 ).

Proposition 1.24. The Green’s Relations ℒ and 풭 commute, if for any 푎, 푏 ∈ 풮, there exist

푥, 푦 ∈ 풮 such that 푎 ℒ 푥 풭 푏 if and only if 푎풭 푦 ℒ 푏 .

Proof. Assume that 푎 ℒ 푥 풭 푏 , for some 푥 ∈ 풮, then there exist 푢, 푣, 푝, 푟 ∈ 풮1, such that

푢푎 = 푥, 푣푥 = 푎 and 푥푝 = 푏, 푏푟 = 푥. Since 푣푥 = 푎, then 푣푥푟 = 푎푟. Next we will use the fact that 푣푥 = 푎, 푏푟 = 푥 and 푥푝 = 푏, therefore we have 푎 = 푣푥 = 푣푏푟 = 푣푥푝푟, this implies that

푎 풭 푣푥푝. Since 푥푝 = 푏, we can notice that 푣푥푝 = 푣푏. Given that 푢푎 = 푥, then 푏 = 푥푝 = 푢푎푝, since 푣푥 = 푎, then 푏 = 푥푝 = 푢푎푝 = 푢푣푥푝. This implies that 푣푥푝 ℒ 푏. Since 푎 풭 푣푥푝 and

푣푥푝 ℒ 푏 we can conclude that 푎 풭 푣푥푝 ℒ 푏, therefore, if we denote 푣푥푝 = 푦 ∈ 풮, we have

푎 풭 푦 ℒ 푏. Similarly we can show that if 푏 ℒ 푦 풭 푎, for some 푦 ∈ 풮, implies that 푏 풭 푥 ℒ 푎, where 푥 ∈ 풮. Therefore, ℒ and 풭 commute. □

Both relations, ℒ and 풭 are equivalent relations on 풮 and they commute, as shown in

Proposition 1.24. Hence, we can denote the ℒ ∘ 풭 = 풭 ∘ ℒ = 풟, where 풟 is also an equivalent relation.

11 Next, we will describe some of the notations we will refer later.

ℒ푎 is the ℒ equivalence class containing 푎, 풭푎 is the 풭 equivalence class containing 푎 , and

풟푎 is the 풟 equivalence class containing 푎 .

Given 풮 a regular semigroup, for any 푎 ∈ 풮, then ℒ푎 ( 풭푎 ) contains at least one idempotent.

Theorem 1.25. ([5] p.130) If 풮 semigroup, the following statements are equivalent:

( i ) 풮 is an inverse semigroup.

( ii ) 풮 is regular and the idempotent elements commute.

( iii ) For any 푎 ∈ 풮 , each ℒ– class and each 풭– class of 풮 , contains an unique idempotent.

Proof. 푖 ⟹ 푖푖 Given 푒, 푓 ∈ 퐸 풮 . If 퓍 = 푒푓 −1 ⟹ 푒푓 퓍 푒푓 = 푒푓 and 퓍 푒푓 퓍 = 퓍.

Let 푦 = 푓퓍푒 , then 푦2 = 푓퓍푒푓퓍푒 = 푓 퓍푒푓퓍 푒 = 푓퓍푒 = 푦 ⟹ 푦2 = 푦 ∈ 퐸 풮 ⟹ 푦−1 = 푦 ∈

퐸 풮 . Next, we will show that 푦−1 = 푒푓 by using the inverse property.

푒푓 푦 푒푓 = 푒푓푓퓍푒푒푓 = 푒푓푥푒푓 = 푒푓

푦 푒푓 푦 = 푓퓍푒 푒푓 푓퓍푒 = 푓퓍푒푒푓푓퓍푒 = 푓퓍푒푓퓍푒 = 푦푦 = 푦

From the above equalities it implies that 푦−1 = 푒푓 ∈ 퐸 풮 .

The proof for 푓푒 ∈ 퐸 풮 is similar. It follows,

푓 푓푒 푒푓 = 푒푓푓푒푒푓 = 푒푓푒푓 = 푒푓 2 = 푒푓.

푓푒 푒푓 푓푒 = 푓푒푒푓푓푒 = 푓푒푓푒 = 푓푒 2 = 푓푒.

Thus, 푒푓 −1 = 푓푒. Since 푒푓 and 푓푒 are idempotents, it implies 푒푓 = 푓푒.

푖푖 ⟹ 푖푖푖 If 풮 is regular it means that each ℒ– class and each 풭– class of 풮 contains at least one idempotent. Let 푒, 푓 ∈ 퐸 풮 such that 푒, 푓 ∈ ℒ푎 ⟹ 푒 ℒ 푓 ⟹ there exist 퓍, 푦 ∈ 풮

12 such that 퓍푒 = 푓, 푦푓 = 푒. Using these equalities we get:

푓푒 = 퓍푒 푒 = 푥푒2 = 푥푒 = 푓

푒푓 = 푦푓 푓 = 푦푓2 = 푦푓 = 푒

Given that the idempotents comute, we have = 푒푓 ⟹ 푒 = 푓 .

푖푖푖 ⟹ 푖 We know that each 풟– class contains an idempotent, then for any 푎 ∈ 풮, there is an element 푒 ∈ 퐸 풮 such that 푒 ∈ 풟푎 ⟹ 푒 풟 푎. So, since 푒 ∈ 퐸 풮 ⟹ 푒 is regular and we know that 푒 풟 푎 , thus, for any 푎 ∈ 풮, 푎 is regular ⟹ 풮 is regular semigroup.

풮 regular semigroup with 푎 ∈ 풮 it implies that there exists 푎′ ∈ 풮 such that 푎푎′ 푎 = 푎

푎푎′ 푎 = 푎 푎푛푑 푎′ 푎푎′ = 푎 and 푎′ 푎푎′ = 푎′. Let 푎′ , 푎′′ be inverses of 푎 ⟹ ⟹ 푎′ = 푎′ 푎푎′ . 푎푎′′ 푎 = 푎 푎푛푑 푎′′ 푎푎′′ = 푎′′

′ ′′ ′ ′′ ′ ′′ Given 푎푎 풭푎 and 푎푎 풭푎 it means that 푎푎 풭 푎푎 ⟹ 푎푎 , 푎푎 ∈ 풭푎 . Using the fact that 풭 contains an unique idempotent we have 푎푎′ = 푎푎′′ . Similarly we prove 푎′ 푎 = 푎′′ 푎 .

Hence, 푎′ = 푎′ 푎푎′ = 푎′ 푎 푎′′ = 푎′′푎푎′′ = 푎′′ . □

Proposition 1.26. ([5] p.131) Let 풮 be an inverse semigroup, then,

( i ) 푎푏 −1 = 푏−1푎−1 , for any 푎, 푏 ∈ 풮.

( ii ) 푎−1푒푎 ∈ 퐸 풮 , 푎푒푎−1 ∈ 퐸 풮 , for any 푎 ∈ 풮 and any 푒 ∈ 퐸 풮 .

( iii ) 푎 풭 푏 if and only if 푎푎1 = 푏푏 −1 , and 푎ℒ푏 if and only if 푎−1푎 = 푏 −1푏 , for any

푎, 푏 ∈ 풮.

( iv ) If 푒, 푓 ∈ 퐸 풮 , then 푒 풟 푓 in 풮 if and only if there exists 푎 ∈ 풮 such that

푎푎−1 = 푒 and 푎−1푎 = 푓.

Proof. ( i ) We have to show that for any 푎, 푏 ∈ 풮 the two equalities below occur

푎푏 푏−1푎−1 푎푏 = 푎푏 and 푏−1푎−1 푎푏 푏−1푎−1 = 푏−1푎−1.

We will use the fact that 푏푏−1 and 푎−1푎 are idempotents.

13 푎푏 푏−1푎−1 푎푏 = 푎푏푏−1푎−1푎푏 = 푎 푏푏−1 푎−1푎 푏 = 푎 푎−1푎 푏푏−1 푏 = 푎푏

푏−1푎−1 푎푏 푏−1푎−1 = 푏−1 푎−1푎 푏푏−1 푎−1 = 푏−1 푏푏−1 푎−1푎 푎−1 = 푏−1푎−1.

Therefore, 푎푏 −1 = 푏−1푎−1 , for any 푎, 푏 ∈ 풮.

( ii ) We need to show that 푎−1푒푎 2 = 푎−1푒푎 and 푎푒푎−1 2 = 푎푒푎−1.

푎−1푒푎 2 = 푎−1푒푎 푎−1푒푎 = 푎−1푒 푎 푎−1 푒푎 = 푎−1 푎 푎−1 푒푒푎 = 푎−1푎 푎−1 푒2푎 = 푎−1푒푎

푎푒푎−1 2 = 푎푒푎−1푎푒푎−1 = 푎푒 푎−1푎 푒푎−1 = 푎 푎−1푎 푒푒푎−1 = 푎푎−1푎 푒2푎−1 = 푎푒푎−1 .

So, 푎−1푒푎 ∈ 퐸 풮 , 푎푒푎−1 ∈ 퐸 풮 , for any 푎 ∈ 풮 and any 푒 ∈ 퐸 풮 .

−1 −1 −1 −1 ( iii ) ⟹ Since 푎 풭 푏 , 푎 풭 푎푎 and 푏 풭 푏푏 it implies 푎푎 , 푏푏 ∈ 풭푎 = 풭푏 .

−1 −1 We know that 푎 푎 and 푏푏 are idempotents and that the class of 푎, 풭푎 contains a unique idempotent. Therefore, 푎−1 = 푏푏−1 .

⟸ Given that 푎푎−1 = 푏푏−1, 푎 풭 푎푎−1 and 푏 풭 푏푏−1 it implies that 푎 풭 푏 .

The proof for 푎 ℒ 푏 is similar.

( iv ) From 푒 풟 푓 ⟹ 푒 풭 ∘ ℒ 푓 ⟹ there exists 푎 ∈ 풮 such that 푒 풭 푎 and 푒 ℒ 푓.

Using the statement ( ii ) of the theorem it implies that 푒푒−1 = 푎푎−1 and 푎−1푎 = 푓−1푓 . Since

푒, 푓 ∈ 퐸 풮 ⟹ 푎푎−1 = 푒푒−1 = 푒2 = 푒 and 푎−1푎 = 푓−1푓 = 푓2 = 푓 . □

Example 1.27. The Symmetric Inverse Semigroup on X

Given any two sets 푋 and 푌 , we define a from 푋 to 푌 as a function 푓 from a subset of 푋, called the of 푓, 푑표푚(푓), to a subset of 푌, called the image of 푓, 푖푚 푓

Let 퐴 be a subset of 푋, then the 1퐴 on 퐴 is a partial function from

푋 to 푋, called partial identity. We will denote by 0푋,푌 the empty partial function from 푋 to 푌.

Next, we will define the composition of two partial functions 푓: 푋 ⟶ 푌 and 푔: 푌 ⟶ 푍, as follows: 푔 ∘ 푓 푥 = 푔 푓 푥 , where 푑표푚 푔 ∘ 푓 = 푥 ∈ 푋 | 푓 푥 ∈ 푑표푚 푔 . Furthermore,

14 we can notice that if 푖푚 푓 and 푑표푚 푔 are disjoint, then the composition of 푓 and 푔 is equal to the empty partial function, i.e. 푔 ∘ 푓 = 0푋,푌 .

If a partial function 푓: 푑표푚 푓 ⟶ 푖푚 푓 is bijective, then 푓 is called a partial .

One of the properties of partial bijections is that the composition of partial bijections is also a partial bijection.

Consider the set ℐ 푋 of all partial bijections from a set 푋 to 푋. ℐ 푋 is closed under the composition (i.e. if 푓, 푔 ∈ ℐ 푋 then 푔 ∘ 푓 ∈ ℐ 푋 ). Remark that the empty partial function,

0푋,푌 is included in ℐ 푋 . If 푓: 푋 ⟶ 푌 is a partial bijection, then 푓 has an inverse, denoted by 푓−1: 푌 ⟶ 푋, such that 푓−1 is also a partial bijection and it follows that 푑표푚 푓−1 = 푖푚 푓,

푖푚 푓−1 = 푑표푚 푓 and 푓−1 푦 = 푥 if and only if 푓 푥 = 푦. Based on these equalities we can

−1 −1 −1 −1 −1 −1 −1 conclude that 푓 ∘ 푓 = 1푑표푚 푓 , 푓 ∘ 푓 = 1푖푚 푓 , 푓 = 푓 , and 푔 ∘ 푓 = 푓 ∘ 푔 .

−1 −1 −1 −1 −1 Thus, 푓 ∘ 푓 ∘ 푓 = 푓 ∘ 1푑표푚 푓 = 푓 and 푓 ∘ 푓 ∘ 푓 = 푓 ∘ 1푖푚 푓 = 푓 , hence, ℐ 푋 is a regular semigroup. As a result, we can verify that ℐ 푋 is an inverse semigroup called the symmetric inverse semigroup on X.

In order to prove that ℐ 푋 is an inverse semigroup, we must show that its itempotents commute. First, we need to be able to recognize the idempotents. The idempotents of ℐ 푋 are the partial identities on 푋. Let 퐸 ℐ 푋 be the set of all itempotents of ℐ 푋 . If 1퐴, 1퐵 ∈

퐸 ℐ 푋 , then we have that 1퐴 ∘ 1퐵 = 1퐴∩퐵 and 1퐵 ∘ 1퐴 = 1퐴∩퐵 , so, 1퐴 ∘ 1퐵 = 1퐵 ∘ 1퐴 =

1퐴∩퐵, it implies that the idempotents commute.

Since we showed that ℐ 푋 is a regular semigroup and that the idempotents commute, we can conclude that ℐ 푋 is an inverse semigroup.

One of the earliest result proved in the inverse semigroup theory is the Wagner-Preston

Theorem: “Any inverse semigroup can be embedded in a symmetric inverse semigroup”. The

15 Wagner-Preston theorem is an analogue of Cayley’s Theorem for groups.

Example 1.28. The free monogenic inverse semigroup

The free inverse semigroups are one of the most important and interesting classes of inverse semigroups. In [11] there are presented five isomorphic copies of the free monogenic inverse semigroup. The first isomorphic copy is developed in what follows and is partially related to results from [11]. We give the details of our calculations.

Let ℤ− be the set of non-positive integers, ℤ+ the set of non-negative integers, and ℤ the set of all integers. Consider the set

푆 = 푖, 푢, 푎 ∈ ℤ− × ℤ × ℤ+| 푖 ≤ 푢 ≤ 푎 equipped with a multiplication defined by

푖, 푢, 푎 ∙ 푗, 푣, 푏 = min 푖, 푗 + 푢 , 푢 + 푣, max 푎, 푏 + 푢

1 The multiplication is associative:

We have:

푖, 푢, 푎 ∙ 푗, 푣, 푏 ∙ 푘, 푤, 푐 =

= min min 푖, 푗 + 푢 , 푘 + 푢 + 푣 , 푢 + 푣 + 푤, max max 푎, 푏 + 푢 , 푐 + 푢 + 푣 and

푖, 푢, 푎 ∙ 푗, 푣, 푏 ∙ 푘, 푤, 푐 =

= min 푖, 푢 + min 푗, 푘 + 푣 , 푢 + 푣 + 푤, max 푎, 푢 + max 푏, 푐 + 푣

In order to show that the multiplication is associative we need to prove that

푖, 푢, 푎 ∙ 푗, 푣, 푏 ∙ 푘, 푤, 푐 = 푖, 푢, 푎 ∙ 푗, 푣, 푏 ∙ 푘, 푤, 푐 .

This equality is true only if:

 min min 푖, 푗 + 푢 , 푘 + 푢 + 푣 = min 푖, 푢 + min 푗, 푘 + 푣

16 and

 max max 푎, 푏 + 푢 , 푐 + 푢 + 푣 = max 푎, 푢 + max 푏, 푐 + 푣 .

We have to consider the following four cases. a) If 푖 ≤ 푗 + 푢 and 푖 ≤ 푘 + 푢 + 푣 , then

min min 푖, 푗 + 푢 , 푘 + 푢 + 푣 = 푖

and

푖, 푖푓 푗 ≤ 푘 + 푣 min 푖, 푢 + min 푗, 푘 + 푣 = . 푖, 푖푓 푗 ≥ 푘 + 푣 b) If 푖 ≤ 푗 + 푢 and 푖 ≥ 푘 + 푢 + 푣 ( that is 푗 ≥ 푖 − 푢 ≥ 푘 + 푣), then

min min 푖, 푗 + 푢 , 푘 + 푢 + 푣 = 푘 + 푢 + 푣 and

min 푖, 푢 + min 푗, 푘 + 푣 = 푘 + 푢 + 푣. c) If 푖 ≥ 푗 + 푢 and 푗 + 푢 ≤ 푘 + 푢 + 푣 (therefore 푗 ≤ 푘 + 푣), then

min min 푖, 푗 + 푢 , 푘 + 푢 + 푣 = 푗 + 푢

and

min 푖, 푢 + min 푗, 푘 + 푣 = 푗 + 푢. d) If 푖 ≥ 푗 + 푢 and 푗 + 푢 ≥ 푘 + 푢 + 푣 (consequently 푗 ≥ 푘 + 푣), then

min min 푖, 푗 + 푢 , 푘 + 푢 + 푣 = 푘 + 푢 + 푣 and

min 푖, 푢 + min 푗, 푘 + 푣 = 푘 + 푢 + 푣.

We need to check the second equality for the following four cases. a) If 푎 ≥ 푏 + 푢 and 푎 ≥ 푐 + 푢 + 푣 , then

max max 푎, 푏 + 푢 , 푐 + 푢 + 푣 = 푎

17 and

max 푎, 푢 + max 푏, 푐 + 푣 = 푎 in both cases 푏 ≥ 푐 + 푣 and 푏 ≤ 푐 + 푣. b) If 푎 ≥ 푏 + 푢 and 푎 ≤ 푐 + 푢 + 푣 (that is 푏 ≤ 푎 − 푢 ≤ 푐 + 푣), then

max max 푎, 푏 + 푢 , 푐 + 푢 + 푣 = 푐 + 푢 + 푣 and

max 푎, 푢 + max 푏, 푐 + 푣 = 푐 + 푢 + 푣 . c) If 푎 ≤ 푏 + 푢 and 푏 + 푢 ≥ 푐 + 푢 + 푣 (therefore 푏 ≥ 푐 + 푣), then

max max 푎, 푏 + 푢 , 푐 + 푢 + 푣 = 푏 + 푢 and

max 푎, 푢 + max 푏, 푐 + 푣 = 푏 + 푢 . d) If 푎 ≤ 푏 + 푢 and 푏 + 푢 ≤ 푐 + 푢 + 푣 (consequently 푏 ≤ 푐 + 푣), then

max max 푎, 푏 + 푢 , 푐 + 푢 + 푣 = 푐 + 푢 + 푣 and

max 푎, 푢 + max 푏, 푐 + 푣 = 푐 + 푢 + 푣 .

2 The set of the idempotents of 풮 is

퐸 풮 = 푖, 0, 푎 | 푖 ∈ ℤ−; 푎 ∈ ℤ+ .

Let’s take an element 푖, 0, 푎 from 퐸 풮 . We can apply the multiplication as follows:

푖, 0, 푎 2 = 푖, 0, 푎 ∙ 푖, 0, 푎 = min 푖, 푖 , 0, max 푎, 푎 = 푖, 0, 푎 .

Hence 푖, 0, 푎 is an idempotent. Now, let 푖, 푢, 푎 be an element of 풮 such that

푖, 푢, 푎 2 = 푖, 푢, 푎 . Then 푢 + 푢 = 푢, and therefore 푢 = 0. It follows that 푖, 0, 푎 | 푖 ∈

ℤ−; 푎 ∈ ℤ+ is the set of all idempotents of 풮.

18 2 ′ The identity element is 1 = 0,0,0 .

Let’s consider 푥 = 푖, 푢, 푎 to be an elements of 풮. The element 1 is an identity element, if 1 ∙ 푥 = 푥 ∙ 1 = 푥.

1 ∙ 푥 = 0,0,0 ∙ 푖, 푢, 푎 = min 0, 푖 + 0 , 0 + 푢, max 0, 푎 + 0 = min 0, 푖 , 푢, max 0, 푎 =

= 푖, 푢, 푎 .

푥 ∙ 1 = 푖, 푢, 푎 ∙ 0,0,0 = min 푖, 0 + 푢 , 푢 + 0, max 푎, 0 + 푢 = min 푖, 푢 , 푢, max 푎, 푢 =

= 푖, 푢, 푎 .

Hence, the element 1 = 0,0,0 is the identity element of 풮 .

3 The idempotents commute.

Let’s take two elements, 푖, 0, 푎 and 푗, 0, 푏 from 퐸 풮 . In order to prove that the idempotents commute, we need to show that 푖, 0, 푎 ∙ 푗, 0, 푏 = 푗, 0, 푏 ∙ 푖, 0, 푎 .

We have:

푖, 0, 푎 ∙ 푗, 0, 푏 = min 푖, 푗 , 0, max 푎, 푏 = min 푗, 푖 , 0, max 푏, 푎 = 푗, 0, 푏 ∙ 푖, 0, 푎

It follows that the idempotents commute.

4 The inverse of an element.

−1 Let 푥 = 푖, 푢, 푎 and 푥 = 푖 − 푢, −푢 , 푎 − 푢 . Then, 푥푥−1푥 = 푖, 0, 푎 푖, 푢, 푎 = 푖, 푢, 푎 = 푥 ;

푥−1푥푥−1 = 푖 − 푢, −푢, 푎 − 푢 푖, 0, 푎 = 푖 − 푢, −푢, 푎 − 푢 = 푥−1 .

It follows that 푥 = 푖, 푢, 푎 푎푛푑 푥−1 = 푖 − 푢, −푢 , 푎 − 푢 are mutually inverse.

The conclusion is that 풮 is an inverse semigroup. This semigroup is the free inverse

semigroup generated by a singleton.

19 Remark. Since 푥−1푥 = 푖 − 푢, 0 , 푎 − 푢 푎푛푑 푥푥−1 = 푖, 0 , 푎 , it follows:

푖, 푢, 푎 ℒ 푗, 푣, 푏 ⟺ 푖 − 푢 = 푗 − 푣 푎푛푑 푎 − 푢 = 푏 − 푣;

푖, 푢, 푎 풭 푗, 푣, 푏 ⟺ 푖 = 푗 푎푛푑 푎 = 푏 ,

where ℒ and 풭 are the Green’s relations.

The following example is special in the sense that it is not a semigroup. The operation is defined

partially with “inverse semigroup properties”.

Example 1.29. An inverse semigroup - like set

If 휔: 푋 × 푋 → 푋 is a partial function (i.e. a function 휔: 퐷 → 푋 where D is a subset of 푋 × 푋) then it is called a partial binary operation on 푋. In the following we will refer to an example of partial binary operation with “inverse semigroup” properties.

If 풮 is a semigroup with zero, then 풮∗ = 풮 − {0} has a partial binary operation induced by the semigroup product. The set 풮∗ equipped with this partial binary operation is called a presemigroup. If for any non-zero elements 푥, 푦 ∈ 풮 we have 푥푦 ≠ 0 then the presemigroup 풮∗ is just a semigroup. Now let 푃 be a set equipped with a partial product. When a 0 is adjoined and all product are defined to be zero then the set 푃0 = 푃 ∪ {0} become a set equipped with a binary operation. This binary operation can be associative or not. In the first case 푃 is a presemigroup, and the semigroup 푃0 can be, in particular, an inverse semigroup (with zero).

In this section we will consider a set Q equipped with a partial binary product. Although this is not a presemigroup, we will inspect the “inverse semigroup” properties of this algebraic structures. Recall that if x and y are elements of Q and the product xy is defined, we write ∃푥푦.

Let

3 풬 = 푎, 푏, 푚 ∈ ℤ+ | 푎, 푏 ≤ 푚

20 equipped with a partial product defined by:

푎 + 푎′ − 푏, 푏′ , 푚′ 푖푓 푚 ≤ 푚′, 푏 ≤ 푎′ 푎푛푑 푎′ − 푏 ≤ 푚′ − 푚 푎, 푏, 푚 ∙ 푎′, 푏′, 푚′ = 푎, 푏 + 푏′ − 푎′, 푚 푖푓 푚 ≥ 푚′, 푏 ≥ 푎′ 푎푛푑 푏 − 푎′ ≤ 푚 − 푚′

This partial product is not associative.

For example, if 푥 = 2,4,4 , 푦 = 3,2,6 , 푧 = 7,8,20 then 푥 ∙ 푦 ⋅ 푧 = 6,8,20 but 푥 ∙ 푦 ∙ 푧 is undefined since 푥 ∙ 푦 is undefined. For this reason, 풬 is not converted into a semigroup (with zero) by adjoining a zero such that 푥 ∙ 푦 = 0 if 푥 ∙ 푦 is undefined and 0 ∙ 푥 = 푥 ∙ 0 = 0 ∙ 0 = 0 .

Now, we will study the inverse semigroup properties of 풬, ∙ .

1) Partial associativity:

Let 푥 = 푎, 푏, 푚 , 푦 = 푎′ , 푏′ , 푚′ , 푧 = 푎", 푏", 푚" three elements in 풬. If ∃푥푦, ∃푦푧 then

∃ 푥푦 푧, ∃푥 푦푧 and 푥푦 푧 = 푥(푦푧).

We consider the following cases:

Case 1: 푚 ≤ 푚′ , 푏 ≤ 푎′ , 푎′ − 푏 ≤ 푚′ − 푚 푎푛푑 푚′ ≤ 푚′′ , 푏′ ≤ 푎′′ , 푎′′ − 푏′ ≤ 푚′′ − 푚′.

푥 ∙ 푦 ∙ 푧 = 푎, 푏, 푚 ∙ 푎′ , 푏′ 푚′ ∙ 푎′′ , 푏′′ , 푚′′ = 푎 + 푎′ − 푏, 푏′ , 푚′ ∙ 푎′′ , 푏′′ , 푚′′

= 푎 + 푎′ − 푏 + 푎′′ − 푏′ , 푏′′ , 푚′′ = 푎 + 푎′ + 푎′′ − 푏′ − 푏, 푏′′ , 푚′′

= 푎, 푏, 푚 ∙ 푎′ + 푎′′ − 푏′ , 푏′′ , 푚′′ = 푎, 푏, 푚 ∙ 푎′ , 푏′ , 푚′ ∙ 푎′′ , 푏′′ , 푚′′

= 푥 ∙ 푦 ∙ 푧

Case 2: 푚 ≤ 푚′ , 푏 ≤ 푎′ , 푎′ − 푏 ≤ 푚′ − 푚 푎푛푑 푚′ ≥ 푚′′ , 푏′ ≥ 푎′′ , 푏′ − 푎′′ ≤ 푚′ − 푚′′.

푥 ∙ 푦 ∙ 푧 = 푎, 푏, 푚 ∙ 푎′ , 푏′ 푚′ ∙ 푎′′ , 푏′′ , 푚′′ = 푎 + 푎′ − 푏, 푏′ , 푚′ ∙ 푎′′ , 푏′′ , 푚′′

= 푎 + 푎′ − 푏, 푏′ + 푏′′ − 푎′′ , 푚′ = 푎, 푏, 푚 ∙ 푎′ , 푏′ + 푏′′ − 푎′′ , 푚′

= 푎, 푏, 푚 ∙ 푎′ , 푏′ , 푚′ ∙ 푎′′ , 푏′′ , 푚′′ = 푥 ∙ 푦 ∙ 푧 = 푥 ∙ 푦 ∙ 푧

21 Case 3: 푚 ≥ 푚′ , 푏 ≥ 푎′ , 푏 − 푎′ ≤ 푚 − 푚′ 푎푛푑 푚′ ≥ 푚′′ , 푏′ ≥ 푎′′ , 푏′ − 푎′′ ≤ 푚′ − 푚′′.

푥 ∙ 푦 ∙ 푧 = 푎, 푏, 푚 ∙ 푎′ , 푏′ 푚′ ∙ 푎′′ , 푏′′ , 푚′′ = 푎, 푏 + 푏′ − 푎′, 푚 ∙ 푎′′ , 푏′′ , 푚′′

= 푎, 푏 + 푏′ − 푎′ + 푏′′ − 푎′′ , 푚 = 푎, 푏 + 푏′ + 푏′′ − 푎′′ − 푎′, 푚

= 푎, 푏, 푚 ∙ 푎′ , 푏′ + 푏′′ − 푎′′, 푚′ = 푎, 푏, 푚 ∙ 푎′ , 푏′ , 푚′ ∙ 푎′′ , 푏′′ , 푚′′

= 푥 ∙ 푦 ∙ 푧

Case 4: 푚 ≥ 푚′ , 푏 ≥ 푎′ , 푏 − 푎′ ≤ 푚 − 푚′ 푎푛푑 푚′ ≤ 푚′′ , 푏′ ≤ 푎′′ , 푎′′ − 푏′ ≤ 푚′′ − 푚′ .

This particular case can be divided in additional two cases:

 If 푚 ≤ 푚′′ , 푏 + 푏′ − 푎′ ≤ 푎′′ 푎푛푑 푎′′ − 푏 − 푏′ + 푎′ ≤ 푚′′ − 푚 , then

푥 ∙ 푦 ∙ 푧 = 푎, 푏, 푚 ∙ 푎′ , 푏′ 푚′ ∙ 푎′′ , 푏′′ , 푚′′ = 푎, 푏 + 푏′ − 푎′ , 푚 ∙ 푎′′ , 푏′′ , 푚′′

= 푎 + 푎′′ − 푏 − 푏′ + 푎′ , 푏′′ , 푚′′ = 푎, 푏, 푚 ∙ 푎′′ − 푏′ + 푎′ , 푏′′ , 푚′′

= 푎, 푏, 푚 ∙ 푎′ , 푏′ , 푚′ ∙ 푎′′ , 푏′′ , 푚′′ = 푥 ∙ 푦 ∙ 푧

 If 푚 ≥ 푚′′ , 푏 + 푏′ − 푎′ ≥ 푎′′ 푎푛푑 푏 + 푏′ − 푎′ − 푎′′ ≤ 푚 − 푚′′ , then

푥 ∙ 푦 ∙ 푧 = 푎, 푏, 푚 ∙ 푎′ , 푏′ 푚′ ∙ 푎′′ , 푏′′ , 푚′′ = 푎, 푏 + 푏′ − 푎′ , 푚 ∙ 푎′′ , 푏′′ , 푚′′

= 푎, 푏 + 푏′ − 푎′ + 푏′′ − 푎′′ , 푚 = 푎, 푏, 푚 ∙ 푎′ + 푎′′ − 푏′ , 푏′′ , 푚′′

= 푎, 푏, 푚 ∙ 푎′ , 푏′ , 푚′ ∙ 푎′′ , 푏′′ , 푚′′ = 푥 ∙ 푦 ∙ 푧 .

2) The set of the idempotents of 풬 is

퐸 풬 = 푎, 푎, 푚 ∈ 풬 .

3) The idempotents commute:

If 푒, 푓 ∈ 퐸 풬 such that ∃ 푒 ∙ 푓 then ∃ 푓 ∙ 푒 and 푒 ∙ 푓 = 푓 ∙ 푒 ∈ 퐸 풬 .

22 4 The inverse:

Let 푥 = 푎, 푏, 푚 푎푛푑 푥−1 = 푏, 푎, 푚 . Then,

푎 푥−1 −1 = 푥 ; 푖푓 ∃ 푥 ∙ 푦 푡푕푒푛 ∃ 푦−1 ∙ 푥−1 푎푛푑 푥 ∙ 푦 −1 = 푦−1 ∙ 푥−1 ;

푏 ∃ 푥−1 ∙ 푥 , ∃ 푥 ∙ 푥−1 푎푛푑 푥−1 ∙ 푥 = 푏, 푏, 푚 , 푥 ∙ 푥−1 = 푎, 푎, 푚 .

Moreover:

푐 ∃ 푥 ∙ 푥−1 ∙ 푥 , ∃ 푥 ∙ 푥−1 ∙ 푥 푎푛푑 푥 ∙ 푥−1 ∙ 푥 = 푥 ∙ 푥−1 ∙ 푥 = 푥.

Remarks.

1. Consulting the properties 1) - 4) of 1.28 and 1.29, we find a correlation between these

properties. Therefore we used the name “inverse semigroup – like set” in the title of

Example 1.29.

2. Following Jekel [4] and Lawson [7], a group-like set 푃 is a set equipped with a partial binary operation ∘ , with a distinguished element 1, and an 푥 ⟼ 푥−1 푥−1 −1 = 푥 , satisfying the following axioms:

푃1 ∃ 푥, 1 , ∃ 1, 푥 and 1 ∘ 푥 = 푥 ∘ 1 = 푥 for all 푥 ∈ 푃;

−1 −1 −1 −1 푃2 ∃ 푥, 푥 , ∃ 푥 , 푥 and 푥 ∘ 푥 = 푥 ∘ 푥 = 1 for all 푥 ∈ 푃;

−1 −1 −1 −1 −1 푃3 푖푓 ∃ 푥, 푥 , ∃ 푥 , 푥 and 푥 ∘ 푦 = 푦 ∘ 푥 .

An example of a group-like set is the set of integers ℤ with the usual addition on the set

퐷 = 푥, 푦 ∈ ℤ × ℤ | 푥푦 ≤ 0 as the partial binary operation, 0 is the distinguished element, and 푥 ⟼ −푥 is the involution.

We can see that 3 + −5 + 7 = 5 but 3 + −5 + 7 is not defined and therefore,

3 + −5 + 7 ≠ 3 + −5 + 7 . Thus in the group-like set of integers the usual addition is not

23 associative and the group-like set of integers cannot be converted into a semigroup (with zero) by adjoining a zero such that 푥 ∘ 푦 = 0 if 푥 ∘ 푦 is undefined.

24 Chapter 2

Categories

The foundations of the theory of categories were set in 1945 when the article “General

Theory of Natural Equivalences” of S. Eilemberg and S. MacLane was published. The theory of categories is a branch of mathematics that studies the properties of particular mathematical concepts in an abstract approach. Various fields of mathematics can be characterized as categories. By using the theory of categories several mathematical results were stated and proved, much easier than with any other method.

2.1. Basic Concepts of Categories

Definition 2.1. A category 풞 is a mathematical concept given by:

( i ) A class of objects of 풞, denoted 푂푏 풞, whose elements are called objects of 풞.

( ii ) For each pair of objects 퐴, 퐵 from 푂푏 풞, a set denoted ℋℴ퓂풞 퐴, 퐵 and called the set of morphisms from A to B.

( iii ) For each three objects A, B, C from 푂푏 풞, a mapping

휇퐴퐵퐶 : ℋℴ퓂풞 퐴, 퐵 × ℋℴ퓂풞 퐵, 퐶 ⟶ ℋℴ퓂풞 퐴, 퐶 is called the composition of morphisms.

The concept of category complies with the following axioms:

Ax1. If 퐴, 퐵 and 퐶, 퐷 are two distinct pairs of objects from 풞, then ℋℴ퓂풞 퐴, 퐵 and ℋℴ퓂풞 퐶, 퐷 are disjoint.

Ax2. If 푓 ∈ ℋℴ퓂풞 퐴, 퐵 , 푔 ∈ ℋℴ퓂풞 퐵, 퐶 , 푕 ∈ ℋℴ퓂풞 퐶, 퐷 are arbitrary, then the composition of morphisms is associative: 휇퐴퐶퐷 휇퐴퐵퐶 푓, 푔 , 푕 = 휇퐴퐵퐷 푓, 휇퐵퐶퐷 푔, 푕 .

25 Ax3. For each object 퐴 ∈ 푂푏 풞, there exists an element 1퐴 ∈ ℋℴ퓂풞 퐴, 퐴 , such that, for each 푋 ∈ 푂푏 풞: 푓 ∈ ℋℴ퓂풞 퐴, 푋 then 휇퐴퐴푋 1퐴, 푓 = 푓 and

푔 ∈ ℋℴ퓂풞 푋, 퐴 then 휇푋퐴퐴 푔, 1퐴 = 푔.

Definition 2.2. For any 퐴 ∈ 푂푏 풞, the 1퐴 , is called the identity morphism.

The identity morphism is unique.

Definition 2.3. We call 풞 a small category if 푂푏 풞 is a set.

Definition 2.4. A category 풞 is called subcategory of the category 풟, denoted by 풞 ⊆ 풟, if:

( i ) 푂푏 풞 ⊆ 푂푏 풟

( ii ) For each A, B from 푂푏 풞, ℋℴ퓂풞 퐴, 퐵 ⊆ ℋℴ퓂풟 퐴, 퐵

( iii ) The composition of morphisms from 풞 is induced by the composition of morphisms from 풟.

( iv ) For each A ∈ 푂푏 풞 the morphism 1퐴 ∈ ℋℴ퓂풟 퐴, 퐴 belongs to ℋℴ퓂풞 퐴, 퐴 .

There are many examples of categories. Next, we will describe a few of them.

 Rel (the category of binary relations) – The objects are sets, the morphisms,

ℋℴ퓂푅푒푙 퐴, 퐵 = 퐴, 퐵, 푅 | 푅 ⊆ 퐴 × 퐵 are binary relations from A to B and 휇 is the usual composition of binary relations.

 Ens (the ) – The objects of Ens are sets, the morphisms from A to B are functions from A to B and the product of morphisms is the usual composition of functions.

 Grp ( the category of groups) – The objects are groups, the morphisms are group

26 homomorphisms and the product of morphisms is the usual composition.

 Ord (the category of the ordered sets) – The objects are the ordered sets (posets), the morphisms are the isotone functions and the product of morphisms is the usual composition of functions.

(the ) − The objects are rings, the morphisms are ring homomorphisms and the product of morphisms is the usual composition. Rngu is the category of rings with unity.

 푅푀표푑 (the category of R−modules) − The objects are R-modules, the morphisms are

R−modules homomorphisms and the product of morphisms is the usual composition of functions.

 퓂ℝ (the category of matrices with real elements) − The objects are the positive integers

푂푏 퓂ℝ = ℤ+ = 1, 2, 3, … , the morphisms are the sets of matrices ℋℴ퓂 퓃, 퓂 =

푎 | 푎 ∈ ℝ and the product of morphisms is the usual product of matrices. 푖푗 퓃×퓂 푖푗

 풞푝 (the ordered category of a poset 풫, ≤ ) – The objects are the elements of a poset,

푥, 푦 , 푖푓 푥 ≤ 푦 풫, 푂푏 풞 = 풫, the morphisms are ℋℴ퓂 푥, 푦 = , and the composition is 푝 풞 ∅, 푖푓 푥 ≰ 푦

푥 ≤ 푦, 푦 ≤ 푧 ⟹ 푥 ≤ 푧. For the case of the composition of morphisms we use the notation

푥, 푦 푦, 푧 = 푥, 푧 .

 Latt – The objects are lattices, the morphisms are homomorphisms of lattices and the product 휇 is the usual composition of functions.

Before we continue with the next theorems, for simplicity, we will use the notation :

푓 푓: 퐴 ⟶ 퐵 or 퐴 퐵 , as a replacement for 푓 ∈ ℋℴ퓂풞 퐴, 퐵 . In this case A is called the

27 domain and B is called the . If 푓: 퐴 ⟶ 퐵 and : 퐵 ⟶ 퐶 , then we will denote

휇퐴퐵퐶 푓, 푔 = 푔푓.

Definition 2.5. Let 풞 be a category, 푓 ∈ ℋℴ퓂풞 퐴, 퐵 , then:

( i ) 푓 is a monomorphism, if for any 푢, 푣 ∈ ℋℴ퓂풞 푋, 퐴 , 푓푢 = 푓푣 ⟹ 푢 = 푣.

( ii ) 푓 is an epimorphism, if for any 푢, 푣 ∈ ℋℴ퓂풞 퐵, 푌 , 푢푓 = 푣푓 ⟹ 푢 = 푣.

( iii ) 푓 is a bimorphism, if 푓 is a monomorphism and 푓 is an epimorphism.

In the following, we will give a few examples of monomorphisms, epimorphisms and bimorphisms.

Proposition 2.6. Let 푓: 퐴 ⟶ 퐵 be a morphism in Ens, then:

( i ) 푓 is monomorphism if and only if 푓 is an . Also, this statement is true in the case of the categories Grp, Rngu, 푅푀표푑 .

( ii ) 푓 is epimorphism if and only if 푓 is a . Also, this statement is true in the case of the categories Grp and 푅푀표푑 .

Proof. ( i ) ⟹ Given that 푓 is a monomorphism, suppose 푥1, 푥2 ∈ 퐴 such that 푥1 ≠ 푥2

푓 푥1 = 푓 푥2 . Let 푋 = 푥1, 푥2 , 푢: 푋 ⟶ 퐴 and 푣: 푋 ⟶ 퐴 such that 푢 푥 = 푥1 and 푣 푥 = 푥2 for any 푥 ∈ 푋.

푓푢 푥 = 푓 푢 푥 = 푓 푥1 = 푓 푥2 = 푓 푣 푥 = 푓푣 푥 ⟹ 푓푢 = 푓푣 , but 푢 ≠ 푣.

Contradiction with 푓 monomorphism.

28 ⟸ Let 푓 be an injective function and consider 푢: 푋 ⟶ 퐴 and 푣: 푋 ⟶ 퐴 such that 푓푢 = 푓푣 ⟹ 푓푢 푥 = 푓푣 푥 ⟹ 푓 푢 푥 = 푓 푣 푥 , for any 푥 ∈ 푋.

Since 푓 is an injective function, it implies that 푥 = 푣 푥 ⟹ 푢 = 푣 , for any 푥 ∈ 푋.

( ii ) ⟹ Given that 푓: 퐴 ⟶ 퐵 is an epimorphism, suppose that 푓 is not surjective.

Let’s take an element 푏0 ∈ 퐵 such that 푏0 ∉ 퐼푚푓. Consider 푌 = ∗, 푏0 , where ∗ = is not an

∗ , 푖푓 푦 ≠ 푏 element of B. Let 푢: 퐵 ⟶ 푌 and 푣: 퐵 ⟶ 푌 such that 푢 푦 = ∗ and 푣 푦 = 0 , 푏0, 푖푓 푦 = 푏0 for any 푦 ∈ 퐵.

푢푓 푥 = 푢 푓 푥 = ∗ = 푣 푓 푥 = 푣푓 푥 ⟹ 푢푓 = 푣푓, but 푢 ≠ 푣 , for any 푥 ∈ 퐴.

Contradiction with 푓 epimorphism.

⟸ Assume 푓 is a surjective function, it implies that for any 푦 ∈ 퐵, there exists 푥 ∈ 퐴 such that 푓 푥 = 푦. Let 푢: 퐵 ⟶ 푌 and 푣: 퐵 ⟶ 푌 such that 푢푓 = 푣푓.

푢 푦 = 푢 푓 푥 = 푢푓 푥 = 푣푓 푥 = 푣 푓 푥 = 푣 푦 , for any 푦 ∈ 퐵. Therefore, 푢 = 푣.

□

Theorem 2.7. Let 퓂ℝ be a category of matrices. Consider 퐴: 푛 ⟶ 푚 a morphism in 퓂ℝ and

퐴 = 푎 . Then 푖푗 푚×푛

( i ) A is a monomorphism if and only if 푟푎푛푘 퐴 = 푛 .

( ii ) A is an epimorphism if and only if 푟푎푛푘 퐴 = 푚 .

Proof. ( i ) We have 퐴: 푛 ⟶ 푚 in 퓂 , 퐴 = 푎 .We will consider the system below. ℝ 푖푗 푚×푛

푎11푥1 + 푎12푥2+. . . +푎1푛 푥푛 = 0 푎 푥 + 푎 푥 +. . . +푎 푥 = 0 푆 = 21 1 22 2 2푛 푛 . … … … … … … … … … … … … … … . 푎푚1푥1 + 푎푚2푥2+. . . +푎푚푛 푥푛 = 0

29 The system S has one solution if 푟푎푛푘 퐴 = 푛, where 푛 is the number of columns. If 퐴 > 푛 , then we have more than one solution.

⟹ A is monomorphism if and only if, for any 푈: 푝 ⟶ 푛 and 푉: 푝 ⟶ 푛 such that

0 0 0 퐴푈 = 퐴푉 ⟹ 푈 = 푉. Suppose 푟푎푛푘 퐴 ≠ 푛, then there exists 푥1 , 푥2 , … , 푥푛 a nontrivial solution for the system S. We can represent U, V as follows:

0 푥1 0 … 0 0 0 0 ⋯ 푈 = 푥2 0 … 0 , 푉 = 0 0 . ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ 0 푛×푝 푥푛 0 … 0 푛×푝

So, 퐴푚×푛 ∙ 푈푛×푝 = 0푚×푝 = 퐴푚×푛 ∙ 푉푛×푝 = 0푚×푝 . But we have 푈 ≠ 푉. This is a contradiction with the fact that A is a monomorphism, thus, 푟푎푛푘 퐴 = 푛.

⟸ We know that 푟푎푛푘 퐴 = 푛. Let 푈: 푝 ⟶ 푛 and 푉: 푝 ⟶ 푛 such that 퐴푈 = 퐴푉 ⟹

퐴푈 − 퐴푉 = 0 ⟹ 퐴 푈 − 푉 = 0. Let’s denote 푈 − 푉 = 푍. Therefore, 퐴 ∙ 푍 = 0. This implies that the p columns of Z are the solutions for the system S. But, since 푟푎푛푘 퐴 = 푛, it means that the system has only the trivial solution. Furthermore, it implies that 푍 = 0 ⟹ 푈 − 푉 = 0 ⟹

푈 = 푉 ⟹ 퐴 is a monomorphism.

The proof for part ( ii ) is similar. □

Theorem 2.8. Let 풞 be a category and 푓 ∈ ℋℴ퓂풞 퐴, 퐵 , 푔 ∈ ℋℴ퓂풞 퐵, 퐶 . Then:

( i ) If 푓 and 푔 are monomorphisms then 푔푓 is a monomorphism.

( ii ) If 푓 and 푔 are epimorphisms then 푔푓 is an epimorphism.

( iii ) If 푔푓 is a monomorphism then 푓 is a monomorphism.

( iv ) If 푔푓 is an epimorphisms then 푔 is an epimorphism.

30 Proof. Let 푕, 푘 ∈ ℋℴ퓂풞 퐴, ∙ .

푔 푚표푛표푚표푟푝 푕푖푠푚 푓 푚표푛표푚표푟푝 푕푖푠푚 ( i ) 푔푓 푕 = 푔푓 푘 ⟹ 푔 푓푕 = 푔 푓푘 푓푕 = 푓푘 푕 = 푘.

푓 푒푝푖푚표푟푝 푕푖푠푚 푔 푒푝푖푚표푟푝 푕푖푠푚 ( ii ) 푕 푔푓 = 푘 푔푓 ⟹ 푕푔 푓 = 푘푔 푓 푕푔 = 푘푔 푕 = 푘.

푔푓 푚표푛표푚표푟푝 푕푖푠푚 ( iii ) 푓푕 = 푓푘 ⟹ 푔 푓푕 = 푔 푓푘 ⟹ 푔푓 푕 = 푔푓 푘 푕 = 푘.

푔푓 푒푝푖푚표푟푝 푕푖푠푚 ( iv ) 푕푔 = 푘푔 ⟹ 푕푔 푓 = 푘푔 푓 ⟹ 푕 푔푓 = 푘 푔푓 푕 = 푘. □

Definition 2.9. Consider 풞 a category and 푓: 퐴 ⟶ 퐵 a morphism in 풞 .

( i ) 푓 is called coretraction if there exists a morphism 푔: 퐵 ⟶ 퐴 such that 푔 ∘ 푓 = 1퐴

(has left inverse ).

( ii ) 푓 is called if there exists a morphism 푔: 퐵 ⟶ 퐴 such that f ∘ 푔 = 1퐵

(has right inverse ).

( iii ) 푓 is called isomorphism if 푓 is both coretraction and retraction, it means there exists a morphism 푔: 퐵 ⟶ 퐴 such that 푔 ∘ 푓 = 1퐴 and f ∘ 푔 = 1퐵 . We will say that A and B are isomorphisms and we denote it by 퐴 ≃ 퐵.

Proposition 2.10. Let 풞 be a category and 푓: 퐴 ⟶ 퐵 a morphism in 풞.

( i ) if 푓 is a coretraction, then 푓 is a monomorphism.

( ii ) if 푓 is a retraction, then 푓 is an epimorphism.

( iii ) if 푓 is an isomorphism, then 푓 is a bimorphism.

Proof. ( i ) If 푓 is a coretraction, then there exists 푔: 퐵 ⟶ 퐴 , such that 푔푓 = 1.

Let’s consider 푢: 퐵 ⟶ 푌 and 푣: 퐵 ⟶ 푌 such that 푓푢 = 푓푣 ⟹ 푔 푓푢 = 푔 푓푣 ⟹ 푔푓 푢 =

31 푔푓 푣 ⟹ 푢 = 푣 , hence, 푓 is a monomorphism.

( ii ) If 푓 is a retraction, then there exists 푔: 퐵 ⟶ 퐴 , such that 푓푔 = 1. Let’s consider

푢: 퐵 ⟶ 푌 and 푣: 퐵 ⟶ 푌 such that 푢푓 = 푣푓 ⟹ 푢푓 푔 = 푣푓 푔 ⟹ 푢 푓푔 = 푣 푓푔 ⟹ 푢 = 푣, thus, 푓 is an epimorphism.

The proof of part ( iii ) is similar. □

Proposition 2.11. Consider the category Ens and 푓: 퐴 ⟶ 퐵 a morphism from Ens.

( i ) 푓 is a coretraction, if and only if, 푓 is a monomorphism.

( ii ) 푓 is a retraction, if and only if, 푓 is an epimorphism.

( iii ) 푓 is an isomorphism, if and only if, 푓 is a bimorphism.

Proof. ( i ) ⟹ See Proposition 2.11.

Ens ⟸ 푓 is a monomorphism f is injective ⟹ there exists 푔: 퐵 ⟶ 퐴 such that

푓−1 푦 , 푖푓 푦 ∈ 푓(퐴) 푔 푦 = , where 푎 ∈ 퐴. 푎, 푖푓 푦 ∉ 푓(퐴)

Thus, 푔푓 = 1퐴 ⟹ 푓 coretraction.

( ii ) ⟹ See Proposition 2.11.

Ens ⟸ 푓 is an epimorphism f is surjective ⟹ for any 푦 ∈ 퐵, 푓−1 푦 ≠ ∅ .

Consider 푔: 퐵 ⟶ 퐴 such that 퐵 ∋ 푦 ⟼ select only one 푥 ∈ 푓−1 푦 ⊆ 퐴.

Therefore, 푓푔 = 1퐵 ⟹ 푓 retraction.

( iii ) The proof is derived from parts ( i ) and ( ii ). □

Definition 2.12. A category 풞 is called balanced if any bimorphism is an isomorphism.

32 Definition 2.13. Let 풞 a category and 퐴 ∈ 푂푏 풞.

A is called initial object if for any ∈ 푂푏 풞 , ℋℴ퓂풞 퐴, 푋 = 1 .

A is called terminal object if for any ∈ 푂푏 풞 , ℋℴ퓂풞 푋, 퐴 = 1 .

Here are some examples of initial and terminal objects in some categories.

 In Ens, we have the , ∅ , as the initial object and the singleton (set with one element) as the terminal object.

 In Grp, the initial and the terminal object are the same: the singleton group 푒 .

 In Rngu, the initial object is ℤ, +, ∙ , where ℤ is the set of integers and +, ∙ are the usual operations of addition and multiplication.

33 2.2. Inverse Categories

Definition 2.14. Given a category 풞, we say that 풞 is a regular category,if for any 푓 ∈

ℋℴ퓂풞 퐴, 퐵 , there exists 푔 ∈ ℋℴ퓂풞 퐵, 퐴 such that 푓푔푓 = 푓.

Definition 2.15. Given a category 풞, we say that 풞 is an inverse category,if for any 푓 ∈

ℋℴ퓂풞 퐴, 퐵 , there exists an unique 푔 ∈ ℋℴ퓂풞 퐵, 퐴 such that 푓푔푓 = 푓 and 푔푓푔 = 푔.

If we denote 푔 = 푓−1 then we have 푓푓−1푓 = 푓 and 푓−1푓푓−1 = 푓−1 , where 푓: 퐴 ⟶ 퐵 and

푓−1: 퐵 ⟶ 퐴.

Definition 2.16. If 풞 is a category, then 푓 ∈ ℋℴ퓂풞 퐴, 퐴 is called idempotent if 푓푓 = 푓.

Observation 2.17. If 풞 is an inverse category, then ℋℴ퓂풞 퐴, 퐴 , ∙ is an inverse semigroup.

−1 −1 Observation 2.18. If 풞 is an inverse category and 푓 ∈ ℋℴ퓂풞 퐴, 퐵 푡푕푒푛 푓푓 푎푛푑 푓 푓 are idempotents.

Proof: 푓푓−1 푓푓−1 = 푓푓−1푓푓−1 = 푓푓−1푓 푓−1 = 푓푓−1.

Proposition 2.19. Let 푓: 퐴 ⟶ 퐵, and 푔: 퐵 ⟶ 퐶. Given 풞 an inverse category, then:

( i ) 푔푓 −1 = 푓−1푔−1 .

−1 ( ii ) 1퐴 = 1퐴, for any 퐴 ∈ 푂푏 풞.

34 푓 푔−1푔 푓푓−1 푔 −1 −1 Proof. ( i ) We have 퐴 퐵 퐵 퐵 퐶 . We know that 푔 푔, 푓푓 ∈ ℋℴ퓂풞 퐵, 퐵 are idempotents. Using Observation 2.23, it implies that 푔−1푔 and 푓푓−1 commute. Furthermore,

푔−1푔 , 푓푓−1푐표푚푚푢푡푒 푔푓 푓−1푔−1 푔푓 = 푔 푓푓−1 푔−1푔 푓 푔푔−1푔 푓푓−1푓 = 푔푓

Similarly, 푓−1푔−1 푔푓 푓−1푔−1 = 푓−1푔−1.

( ii ) 1퐴 ∙ 1퐴 ∙ 1퐴 = 1퐴 , because 1퐴 is unique.

Theorem 2.20. Given 풞 a category, 풞 is an inverse category, if and only if, 풞 is a regular category and its idempotents commute.

Proof. ⟹ We have 풞 an inverse category and let 푖 ∈ ℋℴ퓂풞 퐴, 퐴 be an idempotent

푖푥푖 = 푖 morphism. Then, the unique solution of the system is 푥 = 푖. Consequently, we can 푥푖푥 = 푥

−1 denote 푖 = 푖. If we consider 푖1, 푖2 ∈ ℋℴ퓂풞 퐴, 퐴 to be idempotent morphisms from the

−1 inverse category 풞 and 푖1푖2 is the solution of the system

푖1푖2푥푖1푖2 = 푖1푖2 푆1 = 푥 푖1푖2 푥 = 푥

−1 −1 Hence, 푖1푖2 푖1 and 푖2 푖1 푖2 also satisfy the system 푆1 .

−1 −1 −1 Therefore, 푖1푖2 = 푖1푖2 푖1 = 푖2 푖1푖2 .

−1 −1 −1 −1 −1 Since 푖1푖2 푖1 푖2 = 푖1 푖2 푖1푖2 푖1 푖2 = 푖1 푖2 it implies that the morphism

−1 −1 푖1 푖2 is idempotent. In conclusion, 푖1 푖2 = 푖1 푖2 .

Using the same method we show that the morphism 푖2푖1 is also idempotent.

Given that 푖1 푖2 푖2 푖1 푖1 푖2 = 푖1 푖2푖1 푖2 = 푖1 푖2 and 푖2 푖1 푖1 푖2 푖2 푖1 = 푖2 푖1푖2 푖1 = 푖2 푖1 ,

−1 it implies that 푖2 푖1 is a solution of the system 푆1 . Therefore, 푖1 푖2 = 푖2 푖1 .

35 ⟸ Given 풞 regular it implies that for any 푓 ∈ ℋℴ퓂풞 퐴, 퐵 , there exists 푔 ∈

ℋℴ퓂풞 퐵, 퐶 such that 푓푔푓 = 푓. Moreover, there exists a morphism 푕 ∈ ℋℴ퓂풞 퐵, 퐶 such that 푕 = 푔푓푔 . Therefore, we have the results:

 푓푕푓 = 푓 푔푓푔 푓 = 푓푔푓 푔푓 = 푓푔푓 = 푓 .

 푕푓푕 = 푔푓푔 푓 푔푓푔 = 푔 푓푔푓 푔푓푔 = 푔푓푔푓푔 = 푔 푓푔푓 푔 = 푔푓푔 = 푕 .

Next, suppose there exists another morphism 푘 ∈ ℋℴ퓂풞 퐵, 퐶 such that 푓푘푓 = 푓 and

푘푓푘 = 푘. We showed earlier that 푓푕푓 = 푓 and 푕푓푕 = 푕. Thus, we have

푖푑푒푚푝표푡푒푛푡푠 푐표푚푚푢푡푒 푘 = 푘푓푘 = 푘 푓푕푓 푘 = 푘푓푕푓푘 = 푘 푓푕 푓푘 푘 푓푘 푓푕 = 푘푓푕 푖푑푒푚푝표푡푒푛푡푠 푐표푚푚 푢푡푒 ⟹ 푕 = 푘 푕 = 푕푓푕 = 푕 푓푘푓 푕 = 푕푓푘푓푕 = 푕푓 푘푓 푕 푘푓 푕푓 푕 = 푘푓푕

□

We now give some examples of inverse categories:

Example 2.21. The inverse category of partial bijections Bij.

Recall that partial functions and partial bijections were defined in 1.27. It is straightforward to see that the category of partial bijections Bij defined by: the objects are sets and the morphisms are partial bijections, is a subcategory of Rel. The category Rel is not an inverse category, but the category Bij is an inverse category. As well as inverse semigroups, the inverse categories are fully described by categories of partial bijections. Any inverse category is isomorphic with a subcategory of the category Bij. Thus it is obtained a generalization of the

Wagner-Preston Theorem relating to inverse semigroups. In fact we can see that some concepts of inverse semigroups can be introduced in inverse categories, and the properties of inverse categories are analogues to the properties of inverse monoids.

Let 푓: 푋 ⟶ 푌 be a partial bijection and 푔 ∶ 푖푚푓 ⟶ 푑표푚푓 defined by:

36 푔 푦 = 푥, where 푥 ∈ 푑표푚푓 such that 푓 푥 = 푦.

Then g is a partial bijection from Y to X and

푓 ∘ 푔 ∘ 푓 = 푓 ; 푔 ∘ 푓 ∘ 푔 = 푔 .

It follows that Bij is a regular category.

Note that,

푑표푚푔 = 푖푚푓 푖푚푔 = 푑표푚푓.

Now, for any subset A of X the identity function 1퐴 on A is a partial function from X to itself. It is straightforward to check that in Bij the idempotents are the partial identities. Since,

1퐴 ∘ 1퐵 = 1퐴∩퐵 = 1퐵∩퐴 = 1퐵 ∘ 1퐴 it follows that the idempotents commute. By Theorem 2.20, the category Bij is an inverse category.

Example 2.22. The inverse category of invertible matrices

An n by n A is called invertible (nonsingular or nondegenerate) if there exists an n by n matrix B such that 퐴퐵 = 퐵퐴 = 퐼푛 , where 퐼푛 denotes the n by n identity matrix. Note that non-square matrices do not have an inverse.

The category of matrices 퓂ℝ is a regular category but it is not an inverse category.

The subcategory of 퓂ℝ with the positive integers as objects and with invertible matrices as morphisms is an inverse category. The composition of morphisms is the multiplication of matrices. The set of morphisms Hom(m,n) is empty unless m = n, in which case it is the set of all n by n invertible matrices.

37 Example 2.23. The inverse category that represents an equivalence relation

A partition of a set 푋 is a set of nonempty of 푋 such that every element x in 푋 is in exactly one of these subsets. The nonempty subsets of a partition are called cells (or block or part) of the partition. An equivalence relation on 푋 is a relation that partitions the set 푋 so that two elements of the set are considered equivalent if and only if they are element of the same cell.

If 푋 is a set with an equivalence relation denoted by ~ then an inverse category representing this equivalence relation can be formed as follows:

 The objects are the elements of 푋;

 For any two elements 푥 and 푦 in 푋 , there is a single morphism from 푥 to 푦 if and only if 푥 ~ 푦.

Example 2.24. An inverse category as a subcategory of the category of based sets and based functions

The category of sets with base point and functions preserving base points is defined as follows. Its objects are pairs 푋, 푥 , where 푋 is a nonempty set and 푥 is an element of 푋. The morphisms are functions of 푋 into Y carrying the base point of 푋 into the base point of 푌 for each pair of sets 푋, 푌. The composition of morphisms is the usual composition of functions.

Let 풞 be the subcategory of the category of based sets and based functions defined by:

 The objects are the sets with base point;

 A morphism from 푋, 푥 to 푌, 푦 is a set function 푓 from 푋 to 푌 so that 푓 푥 = 푦, and the set {푢 ∈ 푋|푓 푢 = 푣} contains at most one element for each 푣 ∈ 푌 − {푦}. Then 푓−1 defined by:

푢 푖푓 푓 푢 = 푣 ≠ 푦 푓−1(푣) = 푥 표푡푕푒푟푤푖푠푒

38 is the unique morphism from 푌, 푦 to 푋, 푥 such that:

푓 ∘ 푓−1 ∘ 푓 = 푓 ; 푓−1 ∘ 푓 ∘ 푓−1 = 푓−1.

It follows that this subcategory of the category of based sets and based functions is an inverse category.

39 References

[1] O. Ciucurel, E. Schwab, Introduction to the Theory of Categories, Mirton, Timisoara,

1999.

[2] P. A. Grillet, Semigroups: An Introduction to the Structure Theory. Marcel Dekker Inc.,

1995.

[3] C. D. Hollings, Some First Tantalizing Steps into Semigroup Theory, Mathematics

Magazine, Vol. 80, No 5 (2007), 331-344.

[4] S. M. Jekel, Simplicial K(G,1)’s, Manuscripta Math. 21 (1977), 189-203.

[5] J. M. Howie, An Introduction to Semigroup Theory. Academic Press, 1976.

[6] M. V. Lawson, Inverse Semigroups. The Theory of Partial Symmetries. World Scienti-

fic, Singapore, 1998.

[7] M. V. Lawson, E*-unitary Inverse Semigroups, in Semigroups, Algorithms, Auromata

and Languages, World Scientific Publishing (2002), 195-214.

[8] S. Lipscomb, Symmetric Inverse Semigroups. American Mathematical Society, 1996.

[9] A. Macedo, E. D. Schwab, Maximal Elements and their Group-like Set, Annals Tibiscus,

Computer Science Series, Vol.9, No.1 (2011), 107-114.

[10] B. Mitchell, Theory of Categories. Academic Press, 1965.

[11] M. Petrich, Inverse Semigroups, John Wiley & Sons, Inc., 1984.

40 Index

balanced, 33 monoid, 1 bimorphism, 28 monoid homomorphism, 4 monomorphism, 28 category, 25 cell, 38 partiak bijection, 15 commutative, 3 partial function, 14 composition of morphisms, 25 congruence, 5 quotient semigroup, 5 coretraction, 31 regular, 9 epimorphism, 28 regular category, 34 regular semigroup, 9 Green’s Relations, 11 retraction, 31 right ideal, 10 homomorphism, 4 right principal ideal, 11 idempotent, 3, 34 semigroup, 1 identity element, 1 small category, 26 identity morphism, 26 subcategory, 26 initial object, 33 subsemigroup, 4 inverse, 9 symmetric semigroup, 4 inverse category, 34 symmetric inverse semigroup, 15 inverse semigroup, 9 invertible, 37 terminal object, 33 isomorphism, 4,31 zero element, 2 left ideal, 10 left principal ideal, 11

41 Curriculum Vita

Alexandra Macedo was born in May 25, 1984 in Oradea, Romania. Proud daughter of

Marta and Alexandru Bogdan, Alexandra graduated with a Bachelor’s Degree in Mathematics and Informatics, in July 2007. The following year, in 2008 she decides to leave her home town to come to the United States to pursue a graduate degree.

As a graduate student of the University of Texas at El Paso, Alexandra participated in grant funded programs and attended various conferences. Her dedication, work ethic, and drive gave her the tools to complete a Master's degree in Statistics in 2009.

To further her education, she decided to enroll once again as a graduate student to pursue a Master's degree in Mathematics. Currently, she is about to complete this degree.

Alexandra is also working towards becoming an educator for a College or University. As a means to meet this goal, Alexandra has strengthened her pedagogical ideas through research, training and practice.

In addition, throughout this journey, Alexandra met her husband, Oscar Macedo.

Together, they have started a wonderful family and are proud parents of their beautiful daughter

Alexa Macedo.

Permanent address: 14236 Honey Point Dr El Paso, Texas, USA, 79938

This thesis was typed by the author.

42