Automaton semigroups Alan J. Cain School of Mathematics & Statistics, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, United Kingdom Email: [email protected] Web page: www-groups.mcs.st-and.ac.uk/~alanc/ abstract The concept of an automaton group generalizes easily to semi- groups, and the systematic study of this area is beginning. This paper aims to contribute to that study. The basic theory of au- tomaton semigroups is briefly reviewed. Various natural semi- groups are shown to arise as automaton semigroups. The inter- action of certain semigroup constructions with the class of au- tomaton semigroups is studied. Semigroups arising from Cayley automata are investigated. Various open problems and areas for further research are suggested. Keywords: Automaton semigroup; constructions; Cayley au- tomata. MSC: 20M35; 68Q45. 1 introduction Automaton groups — groups of automorphisms of labelled rooted trees generated by actions of automata — were introduced in the early 1980s as interesting examples having ‘exotic’ properties. Grigorˇcuk’sinfinite peri- odic group [Gri80] was the first such example, and it inspired later ones such as the Gupta–Sidki group [GS83]. Since then, a substantial theory has de- veloped; see, for example, Nekrashevych’s monograph [Nek05] or one of the surveys by the school led by Bartholdi, Grigorchuk, Nekrashevych, and Šuni´c [BGN03, BGŠ03, GŠ07]. The notion of an automaton group naturally general- izes to semigroups, and the study of this generalization is beginning (see, for instance, Silva & Steinberg [SS05] or Grigorchuk, Nekrashevich & Sushchan- skii [GNS00, esp. §4 & §7.2]). This paper aims to contribute to this study. There are two main motivations for studying automaton semigroups. First, studies of other automata-theoretic ‘descriptions’ of groups have proved fruit- ful when generalized to semigroups: automatic structures in the sense of Ep- stein et al. [ECH+92, CRRT01]; automatic presentations [OT05, CORT09]; the context-free language view of hyperbolic structures [DG04]. Second, and more Acknowledgements: The author acknowledges much input by Victor Maltcev during the research that led to this paper. The author also thanks the two anonymous referees, whose comments greatly improved the paper. 1 importantly, studying the more general situation of automaton semigroups may shed light on the particular case of automaton groups: the ‘looser’ set- ting of semigroups may allow insights that can later be imported into the ‘stricter’ setting of groups. A historical parallel is the undecidability of the word problem. This was first established for finitely presented semigroups by Post [Pos47] and later proved for finitely presented groups by Boone [Boo57] and Novikov [Nov55]. Sections 2 and 3 introduce and develop the concept of an automaton semi- group without assuming any prior knowledge beyond elementary semigroup theory and automata theory. In particular, Section 3 reviews the basic theory of automaton semigroups. Section 4 then exhibits naturally occuring examples and classes of automa- ton semigroups. In particular, free semigroups and free commutative semi- groups are shown to be automaton semigroups, as are the free monoid of rank 1 and the [semigroup] free product of two trivial semigroups. The au- thor hopes that this section will win over any sceptics who perhaps suspect that the class of automaton semigroups might consist only of the automaton groups together with uninteresting, ‘artificial’, non-group semigroups. Semigroup constructions are considered next, in Section 5. This area is potentially broad and requires futher study, and this paper only takes the first steps into it and outlines a number of open problems. Section 6 is devoted to semigroups arising from Cayley automata — that is, automata obtained by viewing the Cayley graph of a finite semigroup as an automaton. The overmastering theme is to establish correspondences be- tween certain properties of finite semigroups and properties of the Cayley au- tomaton semigroups arising therefrom. A wider study of Cayley automaton semigroups has been undertaken by Maltcev [Mal09]. Various open questions are scattered throughout the paper in the appro- priate relevant contexts. [It should be emphasized that, despite their similar names, the notions of automaton groups and semigroups are entirely separate from the notions of automatic groups and semigroups whose study was begun by Epstein et al. [ECH+92] and Campbell et al. [CRRT01], respectively.] 2 automaton semigroups The present section is devoted to stating the various definitions required. These definitions will be illustrated by means of a running example, which will culminate in identifying the semigroup defined by a particular automaton. ∗ A basic point of notation: N denotes the natural numbers not including zero and N0 denotes the natural numbers including zero. 2.1 Automata, actions & semigroups An automaton A is formally a triple (Q; B; δ), where Q is a finite set of states, B is a finite alphabet of symbols, and δ is a transformation of the set Q × B. The automaton A is normally viewed as a directed labelled graph with vertex set Q and an edge from q to r labelled by x j y when (q; x)δ = (r; y): 2 x j y q r The interpretation of this is that if the automaton A is in state q and reads symbol x, then it changes to the state r and outputs the symbol y. Thus, starting in some state q0, the automaton can read a sequence of symbols α1α2 : : : αn and output a sequence β1β2 : : : βn, where (qi-1; αi)δ = (qi; βi) for all i = 1; : : : ; n. Such automata are more usually known in computer science as determinis- tic real-time (synchronous) transducers. In the established field of automaton groups, they are simply called ‘automata’ and this paper retains this terminol- ogy. The automaton A, which outputs exactly one symbol for every symbol it reads, is said to be synchronous. A more general type of automaton exists which can output zero or many letters on each input (in which case δ would ∗ be a function from Q × B to Q × B ). The present paper is solely concerned with synchronous automata. Example. Consider the following automaton A = (Q; B; δ): 0 j 0 1 j 1 a b 0 j 0 1 j 0 The state set Q of this automaton is fa; bg. Its alphabet B is f0; 1g. The function δ is formally defined by (a; 0) 7! (b; 0); (b; 0) 7! (b; 0); (a; 1) 7! (a; 1); (b; 1) 7! (a; 0): Suppose, for example, that A starts in state a and reads 0011. Then it outputs 0001. The sequence of states visited during this computation is a; b; b; a; a. ∗ Each state q 2 Q acts on B , the set of finite sequences of elements of B. The ∗ action of q 2 Q on B is defined as follows: α · q (the result of q acting on α) is defined to be the sequence the automaton outputs when it starts in the state q and reads the sequence α. That is, if α = α1α2 : : : αn (where αi 2 B), then α · q is the sequence β1β2 : : : βn (where βi 2 B), where (qi-1; αi)δ = (qi; βi) for all i = 1; : : : ; n, with q0 = q. Observe that, since A outputs one letter for each letter it reads, jα · qj = jαj. (The length of the sequence α is denoted jαj.) Example. Returning to the running example: ∗ The automaton A acts on the set of words B ; it has already been shown that 0011 · a = 0001. Notice that j0011j = j0001j = 4. ∗ As a further example, notice that for any i 2 N , 1i · a = 1i and 1i · b = 01i-1. ∗ The set B can be identified with an ordered regular tree of degree jBj. ∗ The vertices of this tree are labelled by the elements of B . The root vertex ∗ is labelled with the empty word ", and a vertex labelled α (where α 2 B ) has jBj children whose labels are αβ for each β 2 B. It is convenient not to distinguish between a vertex and its label, and thus one normally refers to ‘the vertex α’ rather than ‘the vertex labelled by α’. (Figure 1 illustrates the tree ∗ corresponding to f0; 1g .) 3 " 0 1 00 01 10 11 000 001 010 011 100 101 110 111 figure 1. The set f0; 1g∗ viewed as a rooted binary tree. ∗ The action of a state q on B can thus be viewed as a transformation of the corresponding tree, sending the vertex w to the vertex w · q. Notice that, 0 0 ∗ by the definition of the action of q, if αα · q = ββ (where α, β 2 B and 0 0 α ; β 2 B), then α · q = β. In terms of the transformation on the tree, this says 0 that if one vertex (α) is the parent of another (αα ), then their images under 0 the action by q are also parent (β) and child (ββ ) vertices. More concisely, the action of q on the tree preserves adjacency and is thus an endomorphism of the tree. Furthermore, the action’s preservation of lengths of sequences becomes a preservation of levels in the tree. The actions of states extends naturally to actions of words: w = w1 ··· wn ∗ (where wi 2 Q) acts on α 2 B by (··· ((α · w1) · w2) ··· wn-1) · wn: ∗ ∗ So there is a natural homomorphism ϕ : Q+ ! End B , where End B ∗ denotes the endomorphism semigroup of the tree B . The image of ϕ in ∗ End B , which is necessarily a semigroup, is denoted Σ(A). Definition 2.1. A semigroup S is called an automaton semigroup if there exists an automaton A such that S ' Σ(A). It is often more convenient to reason about the action of a state or word on a single sequence of infinite length than on sequences of some arbitrary fixed length.