Cayley Graphs of Finite Semigroups Daniel Glasson Supervised by Dr. Graham Clarke Royal Melbourne Institute of Technology Vacation Research Scholarships are funded jointly by the Department of Education and Training and the Australian Mathematical Sciences Institute. This project will seek to find interesting and relevant properties of the Cayley graphs of finite semigroups. We will investigate how the algebraic properties of semigroups relate to their graph theoretic properties. We will start with small cases, primarily a survey of semigroups of sizes 2 and 3 and then selected examples from semigroups of size 4-6. Where possible we will extend these results to arbitrarily large or even infinitely large semigroups. 1 Background 1.1 Semigroup Theory A semigroup is a set S with an operation ∗ such that (i) For all a; b 2 S; a ∗ b 2 S. (Closure) (ii) For all a; b; c 2 S; a ∗ (b ∗ c) = (a ∗ b) ∗ c. (Associativity) For convenience we will omit ∗ so that a ∗ b will be written ab. We will introduce some important concepts in semigroup theory here. However, some definitions and basic results will be provided when discussing the findings of the project. If for every a; b 2 S; ab = ba then we say S is commutative. If there exists an element a 2 S such as = a[sa = a] for all s 2 S then we call a a left [right] zero. An element that is both a left and right zero is called a zero. If every element is a left [right] zero then S is called a left [right] zero semigroup. An element e of a semigroup S is a left [right] identity if for all s 2 S, es = s [se = e] and a (two-sided) identity if es = se = s for all s 2 S. We will make use of the notation S1 to mean a semigroup with an adjoined identity if it does not already have one. We also commonly use the letter e to denote an idempotent; that is an element whose square is itself (e2 = e). Unlike in group theory where every group has one and only one idempotent, namely the identity, a semigroup can have as many idempotents as elements (or none at all, however a finite semigroup must have at least one idempotent). A semigroup where every element is idempotent is called a band. An element a of a semigroup S is said to be left [right] cancellable if, for any x; y 2 S, ax = ay [xa = ya] ) x = y. If every element in S is left [right] cancellable, then we say S is left [right] cancellative. If for an element a of a semigroup S there exists x 2 S such that axa = a then we say that a is regular. If every element of S is regular we say S is a regular semigroup. Two elements a; b 2 S are 1 said to be inverses of each other if aba = a and bab = b. In fact if a is regular in S and axa = a then a has at least one inverse, namely xax (this is easily verified). S is called an inverse semigroup if every element of S has a unique inverse. If I is a non-empty subset of a semigroup S and SI ⊆ I [IS ⊆ I], then I is called a left [right] ideal of S. An ideal of S is a non-empty subset that is both a left and right ideal of S. For an element a 2 S, we call S1a [aS1] the left [right] principal ideal generated by a, and S1aS1 the principal two-sided ideal. This brings us onto an important concept in semigroup theory, that of Green's relations: • L -relation: aL b () S1a=S1b i.e. 9u; v 2 S1 such that ua = b and vb = a. • R-relation: aRb () aS1=bS1 i.e. 9u; v 2 S1 such that au = b and bv = a. • D-relation: aDb () 9c 2 S such that aL c and cRb. • J -relation: aJ b () S1aS1=S1bS1 i.e. 9s; t; u; v 2 S1 such that sau = b and tbv = a; and finally, • H -relation: aH b () aL b and aRb. i.e. H = R \ L . For our purposes, it is important to note that: Proposition 1. In a finite semigroup S, J = D. Proof. Before we proceed we mention that for any element x in a finite semigroup S there exists m n m; n 2 N (m 6= n) such that x = x since otherwise we would have infinite distinct powers of x. m+r m m m+1 m+r−1 Then for some r 2 N, x = x and the subset fx ; x ; ··· ; x g is a cyclic subgroup of S and so S contains an idempotent. We only need to show that J ⊆ D since by definition D ⊆ J . Suppose aJ b in S, then there exists s; t; u; v 2 S1 such that sau = b and tbv = a. By repeated substitutions we have a = (ts)ma(uv)m m m m m m m for some m 2 N such that (ts) and (uv) are idempotent. Then a = (ts) (ts) a(uv) = (ts) a. We have b = [s]au and au = (ts)mau = (ts)m−1tsau = [(ts)m−1t]b ) bL au. Similarly, a = a(uv)m so a = au[v(uv)m−1] and au = a[u] ) auRa. Hence aDb and D = J . In fact this proof shows D = J in any semigroup where every element has finite order (obvioulsy this is a necessary condition for a semigroup of finite cardinality). It is often useful to imagine the D-class structure of a semigroup by using an egg box diagram. Imagine a grid, where each row is an R-class, each column an L -class and each cell a H -class (see 2 figure 1). We often use the notation Ra to represent the R-class which a is in. Obviously if b 2 Ra then Ra = Rb. Similarly we use La;Ha;Da;Ja for the L ; H ; D; J -classes. We usually denote an idempotent element by an asterix. Lb ∗ Ra a b Rc c d Figure 1: An example of a D-class. Each row is an R-class, each column an L -class and each cell a H -class. 1.2 Cayley Graphs We denote a directed graph G by G(V; E) where V is the set of vertices and E the set of ordered pairs such that if a; b 2 V and (a; b) 2 E then there is an edge from a to b. A left Cayley graph of a semigroup S, Cay(T;S), is a graph whose vertex set consists of the elements S and whose edge set consists of ordered pairs (a; b) if and only if a 6= b and sa = b for some s 2 T ⊆ S. Note that we do not include (a; a) in the edge set. We can define a right Cayley graph analogously, however we will only be investigating the left Cayley graphs of semigroups. If not specifically mentioned, the generating set, T , will be the whole semigroup itself. For more information on Cayley graphs see Keralev [1] and Cain [2]. We will make use of some graph theoretic terms throughout this paper, however we will introduce any definitions and required background when an unfamiliar term is presented. 3 b ab 0 a ba Figure 2: The Cayley graph of the 5-element Brandt Semigroup B with generating relation B2 = fa; b j a2 = b2 = 0; aba = a; bab = bg For example, take the Cayley graph Cay(B2; B2) of the 5-element Brandt Semigroup B2 = fa; b j 2 2 a = b = 0; aba = a; bab = bg in figure 2. We can see that (s; 0) 2 E for all s 2 B2, 0s = 0. Similarly (a; ba); (ba; a) 2 E since aRba i.e. b[a] = ba and a = a[ba]. 2 Results 2.1 A survey of semigroups of size 2 and 3 When counting the number of semigroups, it is customary to exclude not only isomorphic but anti- isomorphic semigroups. Semigroups S,T are anti-isomorphic if there exists a bijection φ from S to T such that (ab)φ = bφaφ (a; b 2 S). There are 4,18,126,1160 semigroups of sizes 2,3,4 and 5 respectively [3]. However, the Cayley graphs of a semigroup and its anti-isomorphic partner are not necessarily graph isomorphic. Take for example, a left-zero semigroup S and its anti-isomorphism copy T , a right-zero semigroup. Cay(S; E) is a complete graph (every vertex has an edge to every other vertex) while Cay(T;E) is a null graph (it has no edges). We will exclude the study of self-dual semigroups, those who are isomorphic to their anti-isomorphic partners. For example, all commutative semigroups are self-dual by the homomorphism property. Where necessary we will include the Cayley table of a semigroup. A Cayley table gives all products of a finite semigroup in a way similar to a multiplication table and is read as such (Figure 3). 4 a b ab ba 0 a 0 ab 0 a 0 b ba 0 b 0 0 ab a 0 ab 0 0 ba 0 b 0 ba 0 0 0 0 0 0 0 Figure 3: The Cayley Table for the Brandt Semigroup of Order 5 2.1.1 Semigroups of Order 2 We have 4 distinct non-[anti]isomorphic semigroups of order 2.