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UNIT 7 APPLICATIONS of SEMIGROUPS Semigroups

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UNIT 7 APPLICATIONS of SEMIGROUPS Semigroups Applications of UNIT 7 APPLICATIONS OF SEMIGROUPS Semigroups Structure Page No. 7.1 Introduction 77 Objectives 7.2 Semigroups − Some Basics 78 7.3 Free Semigroups 81 7.4 Connections with (Semi)automata 84 7.5 Application to Formal Languages 87 7.6 Summary 91 7.7 Solutions / Answers 92 7.1 INTRODUCTION So far you have studied several aspects of groups. Now we look at simpler algebraic structures, i.e., semigroups and monoids. In your undergraduate studies, you would have come across these algebraic objects in passing. The algebraic theory of semigroups came into its own during the 20th century. For a historical overview you must look at https://www.researchgate.net/publication/226480216_The_Early_Development _of_the_Algebraic_Theory_of_Semigroups. In this unit you will find brief historical remarks, as you go along. To start with, in Sec.7.2, we recall the definition and some properties, of semigroups. In the next section, Sec.7.3, you will study the semigroup analogue of a free group. The importance of semigroups also lies in the applications of its theory to genetics, sociology, psychology, engineering, etc. To give you a feel of the applications, in Sec.7.4 and Sec.7.5, we focus on the applications of semigroups in two linked areas, namely, the study of automata and formal languages. If you are interested in studying more about the ideas discussed in this unit, you can refer to 1) ‘Applied Abstract Algebra’, by Lidl and Pilz, UTM, Springer-Verlag; 2) ‘Algebra’, by PM Cohn, Wiley. Objectives After studying this unit, you should be able to: • define, and give examples of, a semigroup and a monoid; • explain what a finitely generated semigroup/monoid is; • define, and give examples of, a free semigroup; • prove that different bases of a free semigroup must have the same cardinality; • explain what a semiautomaton/automaton is, and its relationship with a semigroup/monoid; 77 Special Groups and • explain what a formal language and a grammar is, and how semigroups Semigroups and monoids are useful for studying them. 7.2 SEMIGROUPS – SOME BASICS Let us begin with a quick review of what a semigroup is. We will also present a few of its properties that we would be using in the later sections. Definitions: 1) A semigroup is an ordered pair (S,∗ ), where S is a non-empty set and ∗ is an associative binary operation on .S 2) A monoid is a semigroup (S,∗ ) with an identity element w.r.t. ∗ , i.e., ∃ e∈ S such that e∗ x= x∗ e= x∀ x∈ S. If asked for examples of semigroups, the following would definitely come to your mind: (,,N +)(,N • )(,,Z +)(,)Z • and (R ,• ) . Further, (N ,• ) is a monoid, while (N ,+ ) is not. However, (N ∪ {0},+ ) is a monoid, with identity .0 Let us look, in some detail, at a few other examples. Example 1: Let S≠ « . Show that the set of all mappings from S to ,S Map(S, S) , is a monoid w.r.t. the composition of mappings. Solution: Firstly, Map(S, S)≠ « since S≠ « . Next, if f, g∈ Map(S, S), then fo g ∈ Map(S, S). Thirdly, the composition of mappings is associative, in general. Finally, the identity map,I:S→ S:I(s)= s, is the identity w.r.t. o. Therefore, (Map(S, S),o ) is a monoid. *** Example 2: Show that every non-empty set can be turned into a semigroup. Solution: Let S≠ « . Define ∗:S × S → S:s1 ∗ s 2 = s 1 . Then, you should check that (S,∗ ) is a semigroup. *** Here is a brief remark about notation. Remark 1: We will sometimes denote the semigroup/monoid (S,∗ ) by only ,S if the underlying operation is understood. Try some related exercises now. M R R • Z E1) Check whether or not (n ( ),+ ),( [x],) and ( ,− ) are semigroups. E2) Let S be a non-empty set, and Re l(S) the set of all relations on ,S i.e., subsets of S× S. Define 78 ∗: Rel(S) × Rel(S) → Rel(S) : (R , R )a R∗ R , defined by Applications of 1 2 1 2 Semigroups ‘ x(R1∗ R 2 )y iff ∃ z∈ S s.t. x R1 z and z R2 y ’, i.e., ‘(x , y)∈RR1 ∗ 2 iff ∃ z∈ S s.t. (x,z)∈ R1 and (z, y)∈ R 2 ’. Show that (Rel(S),∗ ) is a semigroup. [This is called the relation semigroup of S.] E3) If (X,∗ ) is a semigroup, then show that (℘(X), ⊗) is a semigroup, where ℘(X) is the power set of X and A⊗ B = {a ∗ ba ∈ A,b ∈ B}A,B ∀ ∈℘(X). [This is called the power semigroup of X.] Now, you know that a semigroup (S,∗ ) is a group if (S,∗ ) is a monoid and every element in S is invertible w.r.t. ∗. Also, given a monoid, there is a very natural group lying within it. Can you guess what it is? Definition: Let (S,∗ ) be a monoid. Define GS = {x ∈ S | x is invertible}. (GS ,∗ ) is a group, called the unit group of S. You can check that (GS ,∗ ) is a group, and hence the name ‘unit group’ is appropriate. Let us consider some examples. M R • o Example 3: Find the unit groups of (n ( ), ), (Map(S, S), ) and (G,∗ ), where G is a group and S is a non-empty set. M R • M R Solution: The unit group of (n ( ), ) is {A∈ n ( ) A is invertible} R = GLn ( ). The unit group of (Map(S, S),o ) is the group of bijective functions from S to .S The unit group of (G,∗ ) is the whole of G, since every element of G is invertible. *** Now, in the case of groups you studied subgroups and group homomorphisms. We can define analogous objects for semigroups too. Definition: Let (S,∗ ) be a semigroup. A non-empty subset ,T of ,S is called a subsemigroup of S if t1∗ t 2 ∈ T ∀ t1 , t 2 ∈ T. We denote this fact by TS≤≤≤ . The next lot of definitions should not surprise you either. Definitions: 1) Let (S1 ,∗ 1 ) and (S2 ,∗ 2 ) be two semigroups. A function f : SS1→ 2 is a semigroup homomorphism if f(a∗1 b) = f(a) ∗2 f(b) ∀ a,b ∈ S.1 2) A semigroup homomorphism is a monomorphism (respectively, epimorphism) if it is 1-1 (respectively, onto). 3) If a semigroup homomorphism is both 1-1 and onto, we call it an isomorphism of semigroups. 79 Special Groups and As an example, consider f:(Z ,)• → (N ∪ {0},):f(x)• = x. As you can check, Semigroups f is an epimorphism, but not a monomorphism. Here are some exercises now. E4) Show that (N ,• )≤ (Z ,• ). E5) Find the unit groups of (N ∪ {0},+ ),(Z ,),• (℘(S),∩ ) and (Map(S, S),o ) where S≠ « . E6) Check whether f:(Z ,)• → (N ∪ {0},+ ):f(x)= 0 is a semigroup homomorphism. E7) Prove that (℘({1, 2, 3}),∩ ) and (℘ ({a, b, c}),∩ ) are isomorphic semigroups, where a, b, c are distinct symbols. Subsemigroups have several properties analogous to those of subgroups. Let us consider some of them. Theorem 1: A non-empty intersection of any number of subsemigroups of a semigroup (S,∗ ) is a subsemigroup of (S,∗ ). Proof: We leave the proof to you (see E8). This theorem allows us to give the following definition, analogous to that in Unit 5. Definition: Let (S,∗ ) be a semigroup, and let T be a non-empty subset of .S Then the subsemigroup of S generated by T is the intersection of all the subsemigroups of S containing .T It is denoted by < T.> You can check that i) < T > is the smallest subsemigroup of S containing .T K N ii) <T > = {tt1 2 tt n i ∈ T,n ∈ }. Can you think of some examples of generating sets? For instance, given any semigroup (S,∗ ), is S=< S> ? In fact, it is, since S is the smallest subsemigroup of S containing every element of !S So, every semigroup has a generating set, but the fewer the generators, the easier it is for us to ‘see’ the elements of the semigroup. For instance, (,).N + =< N > But (N ,+ )=< 1 > also, since any element of N is a finite sum 1+ 1+L+ 1. In fact, (N ,+ ) is an example of a finitely generated semigroup, as you will just see. Definition: A semigroup (S,∗ ) is called finitely generated if S= < T> , where T is a finite subset of .S If T= 1, then (S,)∗ is called cyclic. 80 For example, (N ,+ ) is cyclic. Let us consider some more examples. Applications of Semigroups Example 4: Which of the following statements are true? Give reasons for you answers. i) (N ∪ {0},+ ) is cyclic. ii) (N ,• ) is not finitely generated. Solution: i) This is false. If it were cyclic, then the only generator possible would be 1, as N =<1> . However, since 0∉< 1> , (N ∪ {0},+ )= < {0, 1}> , and hence it is finitely generated but not cyclic. ii) This is true. By the unique factorisation theorem, every natural number, except ,1 is a product of prime numbers. Also, the set of primes, ,P is infinite. So (N ,• )= < P∪ {1}> . Further, no finite subset of P∪ {1} can generate (N ,• ). Therefore, it is not finitely generated. *** Here are some exercises now. E8) Prove Theorem 1. (Note that an analogous statement is true for monoids.) E9) Give an example of a semigroup S and two subsemigroups S,S of S The union of 1 2 subsemigroups need such that SS1∪ 2 is not a subsemigroup of .S not be a semigroup. E10) Show that every subsemigroup of a finite group G is a subgroup of .G Is the same true for an infinite group? Give reasons for your answers.
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