WHY STUDY SEMIGROUPS? John M. Howie Lecture given to the New Zealand Mathematical Colloquium (Received June 1986) 1. Introduction Before tackling the question in my title I should perhaps begin by saying what a semigroup is. operation . A non-empty set (,xy)z If in addition there exists 1 S - in lx we say that S is called a semigroup if, for all = endowed with a single binary x , y , z in S , x(yz) . S Xl such that, for all = x in S , X is a semigroup with identity or (more usually) a monoid. I shall be confining myself today to semigroups that have no additional structure. Thus, though semigroups feature quite prominently in parts of functional analysis, the algebraic structure of those semigroups is usually very straightforward and so they scarely rate a mention in any algebraic theory. Equally, although they are often of greater algebraic interest, I shall say nothing about topological semigroups. Let me begin by answering a slightly different question: Who studies semigroups? Section 20 in Mathematical Reviews is entitled 'Groups and generalizations' and has two. 'leper colonies' at the end, called 20M Semigroups and 20N Other generalizations. In the Introduction to their book The algebraic theory of semigroups in 1961 Clifford and Preston [2 ] remarked that about thirty papers on semigroups per year were currently appearing. A brief look at recent annual index volumes of Mathematical Reviews shows that the current figures are 1982 321 1983 310 1984 311 . Math. Chronicle 16(1987), 1 -14 . 1 Incidentally, comparable figures for 'other generalizations' are about one third of these. So it is clear that of all generalizations of the group concept the semigroup is the one that has attracted the most interest by far. I shall in due course hazard a guess as to why this is so. Mathematicians are rightly a bit suspicious of theories whose only motive seems to be to generalize existing theories - and if the only motiva tion for semigroup theory were to examine group-theoretical results with a view to generalization, then I would have no very convincing answer to the question of my title. The test proposed by Michael Atiyah for a generaliza tion - that it should have at least two distinct and interesting special cases - is a reasonable one provided it is not applied too dogmatically; and the semigroup concept passes the test, since from the outset semigroup theory drew its ideas partly from group theory and partly from ring theory. For clearly every ring the operation + . are semigroups operation + (#,+,.) is a semigroup if we simply neglect The converse is certainly not true: (S,.) so as to create a ring (5,+,.) . this is to recall the known result that a ring that x2 = x for all tive - i.e. satisfies S = (4 xB) U {0} , and make S that is, there with zero on which it is not possible to define an x in R xy where with the property (a Boolean ring) is necessarily commuta = yx A , The easiest way to see (fl,+,.) B for all x in R . Now let are non-empty sets and 0 £ A * B , into a semigroup with zero by defining (a,i) 0 = 0 (a,£) (aj.fcj) (a2,&2) 00 = = (altb2) . for all 0 x , Then S has the property that hand S is certainly not commutative and so cannot be made into a ring. 2. x1 = x = in S . On the other Pure mathematical reasons Let me now turn to my question 'Why study semigroups?' I am a pure mathematician by instinct and so I begin by offering some pure mathematical reasons. I shall come later to what might be called 'applied mathematical' reasons. 2 My first point, not a fashionable one in these utilitarian times, is that they are fun, that they provide an elegant theory, with arguments that any mathematician can actively enjoy. Let me give an example. The concept of regularity, introduced for rings by no less a person than von Neumann, plays a much more central role in semigroup theory. We say that (5,.) (V a theory than it does in ring is regular if € S)(] i (. S') axa = a As in ring theory, idempotent elements - elements are very important. xa From equation e such that e1 = e we readily see that both ax - and are idempotent. Next, note that a semigroup (V a It is clear that a' (1) (1 ) = xax ; The element € S) (3 a' € S ) (2) —= (1) . then from a' (5,.) (1 ) aa'a is regular if and only if = a , To show that a'aa' (1) = a' . (2 ) (2) simply take we have is usually called an inverse of a , but it should be noted that this is a weaker concept of inverse than the one used in group theory: for example in the four element semigroup with Cayley table it is easy to check that every element is an inverse of every other element. Theorem 1. (1) The following ooniditons on a regular semigroup Idempotents cormrute; 3 S are equivalent (2) Inverses are unique Proof. (1) ==* (2) . Suppose that idempotents commute. inverses of a . a' = a'aa' = a*aa'aa*aa' = a*aa* (2) =» (1) . of = a'aa*aa' = Let = = a* be by by commuting idempotents a* e , (2 ) by f be idempotents and let (2 ) . x be the unique inverse ef : fxe = ef , ef xefx = x . is idempotent, since is an inverse of But an idempotent so a'aa*cua*aa' a*aa'aa'aa* (fxe)2 and a' , Then efxef Then Let ef = fxe , fxe ; = f(xefx)e = fxe , (ef) (fxe) (ef) = efxef = ef . is its own unique inverse The unique inverse of ef , ef = (fxe)(ef) Cfxe) i (ef)(fe)(ef) f(xefx)e fxe : an idempotent. is an inverse of It follows that = = = ef Similarly is thus ef fe (iii = i , H i = i) and is idempotent. itself. On the other hand fe since (ef)2 fe , = ef , (fe) (ef) (fe) = (fe)2 = fe . as required. That argument goes back to the early 1950s, to some fundamental work by Vagner [16 , 17] and Preston [12 , 13 , 14] . A regular semigroup satisfying either one (and hence both) of the conditions in Theorem 1 4 is called an inverse semigroup. A very impressive theory has been created for such semigroups, as is evidenced by the publication in 1984 of a monograph by Petrich [11] But do semigroups of this kind occur 'in nature'? very well known example due to Schein [15] be a group and let includes G K{G) itself and also the cosets of the subgroup G . Ha * Kb This is a natural definition: Define an operation = = Conversely, suppose that 6 Pa and so HaKb 1 , * Let G This which are on by it is not hard to check that HaKb . (HaKa~l)ab £ Pc = Pab . G . (H vaKa~l)ab . smallest coset containing the product ab Let me give a not and McAlister [10] . be the set of all right cosets of effectively the elements of HaKb 674-page devoted entirely to semigroups of this kind. £ Pa Ha * Kb is the [Certainly [H vaKa~l)ab . (€ K(G)) . Then in particular Now Hab £ HaKb c Pab and so H c P ; also (iaKa~1)ab and so aKa"1 £ P . Thus = aKb £ HaKb £ Pab H v aKa~l £ P (fl \iaKa"^)ab c Pab It is a routine matter to check that and that (a"lHa)a~1 Now suppose that Ha is an inverse of = Pa .] * is an associative operation in the semigroup (K(G) , *) . is idempotent: Ha "hen in particular Ha and so = a 2 = la2 Ha * Ha Ha ; = (fl v a & M )a 2 . i.e. 5 a 2 = ha for some h in H . Hence a = h € H and so Ha = H . are precisely the subgroups of H- * K = N of G are such that, for all (ffvff)a = (NH)a . Ha * N = (# vaNa~l)a = (ff vtf)a = N * la = (K(G) , *) . a in (K(G) , *) , K of G is an inverse semigroup. = Na H K * H . N * Ha central idempotent then for all N = (K(G) , *) and so are central idempotents in and so For any two subgroups Svif Thus idempotents commute and so The normal subgroups In fact the idempotents of G . = Ha in K(G) , (M)a , Conversely, if N is a G la * N = (1 vatfa = aN is normal. That is a slightly quaint example, but I mention it because I have the feeling that it has not yet been adequately exploited. The main reason that semigroups turn up in mathematics is that one is very often interested in self-maps of a set of one kind or another, and whenever f , g , h are such maps it is automatically the case that (/ O g) O h = / O {g O h) . If the maps are bijections then the appropriate abstract idea is that of a group; if not then inevitably we must consider a semigroup. It is this connection with maps (arising from the associative axiom) that is the strongest reason why semigroups are more important both theoretically and in applications than the various non-associative generalizations of groups. There is another pure mathematical reason for being interested in semigroups. It is possible to take a very general standpoint in algebra and to discuss a so-called of operations, where n-algebra : Ani — *■ A A having a family ft = {uk is an n^-ary operation. 6 : i € 1} [For example, in a group one can take I = {1 ,2 }, [(o1(a1,a2) If : A — +■ 3 0 = a :a2 , a is map between morphism if (for all i in n1 = J , 2 « 2 ^(aj) = = , 1 a"1 .] ft-algebras then we say that and for all a cu, i 1 <(>(aii (a1 , ... , an )) i If we regard with <j> as applying to = <f> is a in , ••• , A) )) . i £ 1'1 in an obvious way then we can express this property succinctly as a commuting condition <(> O OK A congruence on that (for all A is = Consider the quotient set an equivalence relation “ i4/~ [a] , with the property = W whose elements are equivalence classes {x € A : x ~ a} . A/~ inherits the ft-algebra structure : we simply define w.CCaj] , ... , [a ]) = [u»i (a1 , i and the compatibility condition sense. ~ ..... an ) ~ “i K ..... • ^ t The congruence property means that A (3) i) a l ~ 2 I .....’ an . ~ an. v ^ from U). O $ . Lj(a) and the definition (5) ensures that the definition makes : A — = )] £ (4) There is a natural map ,a /5/~ [a] (a defined by 6 A) can be interpreted as saying that id• ° H 7' 1 = □ H 7 ; 0 * (5) hence, comparing with (3) , Now suppose that $ is a morphia image of is a morphism. is a morphism from A . a ~ a' Define ~ on if and only if It is easy to verify that for we see that ~ A A onto B ; we say that B by the rule that $(a) = is a congruence. . The first isomorphism theorem ft-algebras is then as follows: Theorem 2. Let im $ - B , A , B be and let ~ be the congruence on Then there is an isomorphism n-algebras, let a : A/--- - B $ —*■B : A A be a morphism with defined by (6 ) . such that the diagram A ./~ commutes. This is, in one form or another, one the cornerstones of abstract algebra. It says in effect that an n-algebra A carries its morphic images 'within itself' and that to reveal them we need only consider the quotients of A by its various congruences. The result applies to groups, of course, but it is not usually stated in quite this way. This is because for a group correspondence between congruences N a ~ b The quotient -4/~ = {a € A ~ A there is a one-one and normal subgroups : a ~ 1} , if and only if is always denoted by I given by 8 given by or ab~l_ € N . A/N . Similarly, for a ring there is a one-one correspondence between congruences ideals N ~ and two-sided A I = a ~ b and the quotient i4/~ {a € A : a ~ 0} , if and only if is always written or a - b i I , A/I . In semigroups no such device is available and we as such. must study congruences So semigroups consititute the simplest, most manageable and most natural class of algebras to which the methods of universal algebra must be applied. Applied Mathematical reasons 3. Let me now turn to less exalted reasons for studying semigroups. One of the striking aspects of semigroup conferences these days is that many of the participants, between a third and a half, at a guess, come from departments of computer science. The reason is that semigroups have found significant applications in the theory of automata, languages and codes. If A is a non-empty set (an alphabet, as we often want to call it) then the set of all finite words in the alphabet position. word 1 A is a semigroup if we define multiplication by juxta Denote the length of (with jl| = 0 ) u by This is the free monoid, generated by A* is usually denoted by Now let f j If we include the empty 4+ . A . The set of non-empty words in A subset of 4* is called a language . we normally write Q . The function f(.q,a) f simply as qa can be extended to and think of Q * A* A q1 = q(q € Q) q(wa) = (qw)a = (Q,f) is an {q € Q , w € A* , A*-automation. 9 A by defining (inductively) We say that A* . be a finite non-empty set and suppose that we have a map : Q * A —+ Q ; as 'acting' on ju| . then we obtain a monoid, which we denote by a € A) . (In the terminology of Eilenberg [4] this is a complete deterministic automaton.) We may think of it as a very rudimentary machine whose states (the elements of be altered by various input (the elements of Suppose now that among the elements of Q there is an element we call the initial state and that there is a subset .set of terminal states. A language L recognizing Example. T of. Q = {u € i4* ; which iw € T} . is the language recognized by the automaton A . is called recognizable if there exists an automaton We can picture an automaton via its state graph. lb = 1 , i called the A L . A = {a b] , Let L can Let L Then we say that Q) A) . Q = {0,1,2,3} , 2b = 3 , i = 1 , Qa = Ob = 0 , = 1 , T = {1,2,3} . la = 2a = 2 , If 3a = 0 , then we can draw the picture It is easy to see that 0 is a 'sink' state from which no escape is possible, and that 1 aba In fact L , words in A* If L , by L . Let 2 aba = 0 . the language recognized by this automaton, consists of all not containing wz (. L2 } . = L2 c A* If Lei* F then then aba as a segment. .L2 < L> is defined as ^w xw z ' ■ w\ ^ denotes the submonoid of be the set of all finite 10 subsets of A* . A* > generated Then the set Rat A* from of rational subsets of F A* is the set of subsets of by means of the operations of U (finite union), A* . obtained and < >. This leads to an important characterization of recognizable subsets of Theorem 3. (Kleene [7]). A language L A* : is recognizable if and only if it is rational. Another characterization of recognizable languages is more algebraic in character. of L If L is any subset of is the relation ~ on w 1 ~ w2 A* A* then the syntactic congruence ~ defined by iff v € L iff uu2 y € L] . (7) (I have not gone into the mathematical theory of grammar at all, but one can perhaps dimly see that this is saying that and w2 are mutually interchangeable in the set of 'meaningful' sentences that constitutes Thus (very roughly speaking) cat ~ dog but if cat ^ black . u = (the), A* . Then is usually denoted by Theorem 4. L L . then So 'cat' and 'dog' are syntactically equiva lent but 'cat' and 'black' are not.) congruence on u = (sat on the mat) 4*/~ It is easy to verify that ~ is a is called the syntactic monoid of L and M(L) . is recognizable if and only if its syntactic monoid M(L) is finite. This is by no means as deep as the Kleene theorem. round it is virtually obvious. of A on M{L) A = (M{L) ,f) = L = {u recognizes 4 z € L , w ~ z . i.e. (<; *) w a into an initial state and z M(L) Indeed one way is finite we can define an action by f(w,a) making If = wa [w € M(L) A*-automation. : w € L) But then if and only if lul € L A and hence = (M{L) ,f) If we then take 1 as as the set of terminal states then lz f I , if and only if there exists that the automation = 4*/~) w in A* z = lzl € L recognizes 11 i.e. L . A if and only if such that by (7) . w € L and We conclude I mention this proof (or rather half-proof) because it emphasised the very close links between automata and monoids. A further connection is provided by the theory of codes. It is not the case that every submonoid of a free monoid is free. example, in {a,b}* consider the submonoid 14* = {a 1 : i > 2} . The base of indecomposable elements of freely generated by M) M (i.e. are codes. = M* 'J {1} , the set is {a2 ,a3} , {a2 ,a3} . but M = [a2 ,aba , ab 2 ,b}, C2 M . For where Af*-\ [M*)2 of is certainly not We say that a subset if it is the base of a free submonoid of Cj M C of A* is a code For example, = {a4 , b ,ba 2 ,ab ,aba2 } This means, for example, that any word in four letters - say x 3 x 1 x 1+x 2 x 3 - can be encoded unambiguously (in Cx , say) as ab2a2baba2b2 . is in fact an example of a prefix code: any word in be decoded without hesitation by reading from the left. C 1 fl C 1 A+ = 0 . <C-1> = C* can This is because By contrast if one tries to decode aba^ba2 using C2 the answer (unique since might first have tried x^xj ? or C2 is a code) is x 5XjX 2 ? , but one before reaching the correct solution. That the theory of codes is intimately bound up with the theory of monoids is illustrated by. Theorem 5. base. Then Let C M be a submonoid of a free monoid is a prefix code if and only if u , {M M ux € M =» x € M . . is called left unitary.) 12 A* and let satisfies C be its This has been a very sketchy introduction to a rich and rapidly growing area. Those whose appetites have been whetted should turn to Eilenberg [4,5] Lallement [8 ] , Berstel and Perrin [l] , and Lothaire [9] . REFERENCES 1. J. Berstel and D. Perrin, Theory of codes, Academic Press, 1984 . 2. A.H. Clifford and G.B. Preston, The algebraic theory of semigroups, American Math. Soc., 1961 . 3. P.M. Cohn, Universal Algebra, Harper and Row, 1965 . 4. S. Eilenberg, Automata, Languages and Machines, vol. A , Academic Press 1974 . 5. S. Eilenberg, Automata, Languages and Machines, vol. B , Academic Press 1976 . 6 . 7. J. M. Howie, An introduction to semigroup theory, Academic Press, 1976 . S.C. Kleene, Representation of events in nerve nets and finite auotmata, Automata Studies, pp. 3-42 (Princeton University Press, 1956) . 8. G. Lallement, Semigroups and combinatorial applications, Wiley, 1979 . 9. M. Lothaire, Combinatorics on Words, Addison-Wesley, 1983 . 10. D.B. McAlister, Embedding inverse semigroups in coset semigroups, Semigroup Forum, 20 (1980), 255-267 . 11. M. Petrich, Inverse semigroups, Wiley, 1984 . 12. G.B. Preston, Inverse semi-groups, J. London Math. Soc. 29 (1954), 396-403 . 13. G.B. Preston, Inverse semi-groups with minimal right ideals, J. London Math. Soc. 29 (1954), 404-411 . 14. G.B. Preston, Representations of inverse semi-groups, J. London Math. Soc. 29 (1954), 411-419 . 15. B.M. Schein, Semigroups of strong subsets, Volzskii Matem. Sbomik 4 (1966), 180-186 . (Russian). 13 16. V.V. Vagner, Generalised groups, Doklady Akad. Nauk SSSR 84 (1952), 1119-1122 . 17. (Russian). V.V. Vagner, Theory of generalised heaps and generalised groups, Matem. Sobomik (N.S.) 32 (1953), 545-632 . University of St. Andrews, SCOTLAND. 14 (Russian).

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