Semigroups Arising from Asynchronous Automata
Abstract.
We introduce a new class of semigroups arising from a restricted class of asynchronous automata. We call these semigroups “expanding automaton semigroups.” We show that the class of synchronous automaton semigroups is strictly contained in the class of expanding automaton semigroups, and that the class of expanding automaton semigroups is strictly contained in the class of asynchronous automaton semigroups. We investigate the dynamics of expanding automaton semigroups acting on regular rooted trees, and show that undecidability arises in these actions. We show that this class is not closed under taking normal ideal extensions, but the class of asynchronous automaton semigroups is closed under taking these extensions. We construct every free partially commutative monoid as a synchronous automaton semigroup.
1. Introduction
Automaton groups were introduced in the 1980’s as examples of groups with fascinating properties. For example, Grigorchuk’s group is the first known group of intermediate growth and is also an infinite periodic group. Besides having interesting properties, many of these groups have deep connections with dynamical systems which were explored by Bartholdi and Nekrashevych in [3] and [9]. In particular, they use these groups to solve a longstanding problem in holomorphic dynamics (see [3]). For a general introduction to these groups, see [5] by Grigorchuk and Sunic or [9] by Nekrashevych.
Many generalizations of automaton groups have been studied. The most famous and wellstudied generalization is the class selfsimilar groups. A good introduction to these groups can be found in [5] or [9]. More recently, Slupik and Sushchansky study semigroups arising from partial invertible synchronous automata in [14]. Cain, Reznikov, Silva, Sushchanskii, and Steinberg investigate automaton semigroups, which are semigroups that arise from (not necessarily invertible) synchronous automata in [1], [10], and [13]. Grigorchuk, Nekrashevich, and Sushchanskii study groups arising from asynchronous automata in [4].
In all of the references listed above except for [4], the semigroups studied arose from synchronous automata. In [4], Grigorchuk et al. study groups arising from asynchronous automata. In particular, they give examples of automata generating Thompson groups and groups of shift automorphisms. This paper studies a class of semigroups that we call “expanding automaton semigroups.” These semigroups arise from a restricted class of asynchronous automata that we call “expanding automata,” and the class of expanding automata contains the class of synchronous automata. Thus the class of automaton semigroups is contained in the class of expanding automaton semigroups, and the class of expanding automaton semigroups is contained in the class of asynchronous automaton semigroups. As mentioned above, automaton semigroups and asynchronous automaton semigroups have been studied, but thus far a study of expanding automaton semigroups has not been done.
In Section 2 we give definitions of the different kinds of automata, and explain how the states of a given automaton act on a regular rooted tree. In particular, let be a finite set, and let denote the free monoid generated by . Then the states of a given automaton act on for some finite set . Thus we can consider the semigroup generated by the states of an automaton as a semigroup of functions from to . Given a free monoid , we associate a regular rooted tree with by letting the vertices of be and letting the edge set be for all and . The identity of is the root of the tree. The action of a semigroup associated with an asynchronous automaton on induces an action on the tree . Let denote the set of rightinfinite words over . Then is the boundary of the tree . The action of an asynchronous automaton semigroup on induces an action of the semigroup on , and so an asynchronous automaton semigroup acts on the boundary of a regular rooted tree.
Section 2 also contains examples of expanding automaton semigroups that are not automaton semigroups (Proposition 2.3), as well as asynchronous automaton semigroups that are not expanding automaton semigroups (Proposition 2.5). Thus Propositions 2.3 and 2.5 combine to show the following.
Proposition.
The class of automaton semigroups is strictly contained in the class of expanding automaton semigroups, and the class of expanding automaton semigroups is strictly contained in the class of asynchronous automaton semigroups.
We show the latter by proving that the bicyclic monoid (the monoid with monoid presentation ) is not a submonoid of any expanding automaton semigroup (Proposition 2.4), and then we demonstrate an asynchronous automaton semigroup that contains the bicyclic monoid as a submonoid (Proposition 2.5).
In Section 3 we investigate the dynamics of expanding automaton semigroups and asynchronous automaton semigroups on the trees on which they act. Example 3.2 gives an example of an expanding automaton semigroup acting on such that there are infinite words with and for all . Furthermore, if is not equal to or , then for all . Proposition 3.1 shows that automaton semigroups cannot have this kind of dynamical behavior when acting on the boundary of a tree. Thus the boundary dynamics of expanding automaton semigroups is richer than the boundary dynamics of automaton semigroups.
Section 3 also investigates several algorithmic problems regarding the actions of expanding automaton semigroups on a tree. Proposition 3.3 gives an algorithm that solves the uniform word problem for expanding automaton semigroups. This result is already known, as Grigorchuk et al. show in Theorem 2.15 of [4] that the uniform word problem is solvable for asynchronous automaton semigroups. We give an algorithm with our terminology for completeness. Proposition 3.8 gives an algorithm which decides whether a state of an automaton over induces an injective function from to .
Since the uniform word problem is decidable for these semigroups, there is an algorithm that takes as input an expanding automaton over an alphabet and states of the automaton and decides whether for all . On the other hand, Theorem 3.4 shows the following.
Theorem 3.4.

There is no algorithm which takes as input an expanding automaton
and states and decides whether or not there is a word with . 
There is no algorithm which takes as input an expanding automaton
and states and decides whether or not there is an infinite word such that .
The problem in part 1 of the above theorem is decidable for automaton semigroups: if is a synchronous automaton with , then (because and induce level producing functions ) there is a word such that if and only if there is a letter such that .
We close Section 3 by applying Theorem 3.4 to study the dynamics of asynchronous automaton semigroups. Theorem 3.6 shows that there is no algorithm which takes as input an asynchronous automaton over an alphabet , a subset , and a state of the automaton and decides whether there is a word such that . Thus we cannot decide if has a fixed point in . Furthermore, Theorem 3.6 also shows that there is no algorithm which takes as input an asynchronous automaton over an alphabet , a subset , and a state of the automaton and decides whether there is an infinite word such that . Thus undecidability arises when trying to understand the fixed points sets of asynchronous automaton semigroups on the boundary of a tree.
In Section 4 we give the basic algebraic theory of expanding automaton semigroups. Recall that a semigroup is residually finite if for all with then there is a finite semigroup and a homomorphism such that . Proposition 4.1 shows that expanding automaton semigroups are residually finite. It is already known that automaton groups are residually finite (see Proposition 2.2 of [5]) and automaton semigroups are residually finite (see Proposition 3.2 of [1]). Asynchronous automaton semigroups are not residually finite, as there is an asynchronous automaton generating Thompson’s group (see section 5.2 of [4]). This group is an infinite simple group, and so is not residually finite. Thus residual finiteness of expanding automaton semigroups also distinguishes this class from the class of asynchronous automaton semigroups. Recall that if is a semigroup, an element is said to be periodic if there are such that . Proposition 4.2 shows that the periodicity structure of expanding automaton semigroups is restricted. In particular, let denote the set of prime numbers that divide . If is an expanding automaton semigroup and is such that for some with , then the prime factorization of contains only primes from .
In Section 4.2 we provide information about subgroups of expanding automaton semigroups. Proposition 4.3 shows that an expanding automaton semigroup is a group if and only if is an automaton group. Proposition 3.1 of [1] shows that an automaton semigroup is a group if and only if is an automaton group; we use the idea of the proof of this proposition to obtain our result. Note that such a proposition does not apply to asynchronous automaton semigroups, as Thompson’s group can be realized with an asynchronous automaton. Proposition 4.4 shows that if is a subgroup of an expanding automaton semigroup, then there is a selfsimilar group such that is a subgroup of . Proposition 4.5 shows that if an expanding automaton semigroup has a unique maximal subgroup , then is selfsimilar. In particular, this proposition implies that if is the semigroup generated by the states of an invertible synchronous automaton, then the group of units of is selfsimilar (Corollary 4.6).
In Section 5.1 we study closure properties of expanding automaton semigroups. Let and be semigroups. The normal ideal extension of by is the disjoint union of and with multiplication defined by if or , if and , and if and . Note that if is a semigroup, then adjoining a zero to is an example of a normal ideal extension. Proposition 5.6 of [1] shows that the class of automaton semigroups is closed under normal ideal extensions. We show in Proposition 5.3 that the class of asynchronous automaton semigroups is closed under normal ideal extensions. On the other hand, we show in Proposition 5.2 that the free semigroup of rank 1 with a zero adjoined is not an expanding automaton semigroup. Example 3.2 shows that the free semigroup of rank 1 is an expanding automaton semigroup, and so we have that the class of expanding automaton semigroups is not closed under normal ideal extensions. Lastly, we show that the class of expanding automaton semigroups is closed under direct product (provided the direct product is finitely generated). In Proposition 5.5 of [1] Cain shows the same result for automaton semigroups, and our proof is similar.
Section 5.2 contains further constructions of expanding automaton semigroups. A free partially commutative monoid is a monoid generated by a set with relation set such that , i.e. a monoid in which the only relations are commuting relations between generators. We show the following.
Theorem 5.6.
Every free partially commutative monoid is an automaton semigroup.
2. Definitions and Examples
Given a set , let denote the free semigroup generated by . In the free monoid , let denote the identity. As defined in [4], an asynchronous automaton is a quadruple where is a finite set of states, is a finite alphabet of symbols, is a transition function, and is an output function. A synchronous automaton is defined analogously, the difference being that (the range of the output function is rather than ). In this paper, we study a restricted class of asynchronous automata.
An expanding automaton is a quadruple where is a finite set of states, is a finite alphabet of symbols, is a transition function, and is an output function. We view an expanding automaton as a directed labeled graph with vertex set and an edge from to labeled by if and only if and . Given an edge in the graph, we refer to as the input of the edge, and as the output of the edge. The interpretation of this graph is that if the automaton is in state and reads symbol , then it changes to state and outputs the word . Thus, if we fix , the automaton can read a sequence of symbols and output a sequence where and for .
Each state induces a function in the following way: acting on , denoted , is defined to be the sequence that the automaton outputs when the automaton starts in state and reads the sequence . We also insist that . This action of on induces an action of on . The state induces a function by if , and if is an edge in with endpoints and then where is the unique geodesic sequence of edges in connecting and . By abuse of notation, we identify with , as context should eliminate confusion. Considering the states of an automaton as functions leads to the following definition:
Definition 2.1.
Given an expanding automaton , we say that the expanding automaton semigroup (respectively monoid) corresponding to , denoted , is the semigroup (respectively monoid) generated by the states of .
An invertible synchronous automaton (or invertible automaton) is a quadruple where and, for any , the restricted function is a permutation of . The states of an invertible automaton induce bijections on . Furthermore, these functions are levelpreserving, i.e. for all and (where is the length function on ). Thus, given an invertible automaton , we define the automaton group associated with to be the group generated by the states of . An automaton semigroup is a semigroup generated by the states of a synchronous automaton. Thus the generators of an automaton semigroup over the alphabet induce levelpreserving functions on , but these functions are not necessarily bijective. Finally, an asynchronous automaton semigroup is a semigroup generated by the states of an asynchronous automaton.
A selfsimilar group is a group generated by the states of an invertible synchronous automaton with possibly infinitely states. A selfsimilar semigroup is defined analogously. Thus we define an expanding selfsimilar semigroup to be a semigroup generated by the states of an expanding automaton with possibly infinitely many states.
Note that if where is an expanding automaton semigroup acting on , then need not induce a levelpreserving function . Thus elements of expanding automaton semigroups are not necessarily graph morphisms. If is an expanding automaton, then the output function mapping into implies that for all , . We say that a function is prefixpreserving if is a prefix of in whenever is a prefix of in . We call a function lengthexpanding if for all and . If we topologize the tree by making each edge isometric to and imposing the path metric, then an element of an expanding automaton semigroup acting on will induce a prefixpreserving, lengthexpanding endomorphism of the tree. We call an expanding endomorphism if is prefixpreserving and lengthexpanding.
Let be an expanding endomorphsm of a tree , where . Then induces a function ; for the rest of the paper we denote this function by . Note that for any , the tree is isomorphic (as a graph or a metric space) to . Now for each , induces an expanding endomorphism ; for the rest of the paper we denote this induced endomorphism by . For any and , is characterized by the equation
The function is called the section of at . Inductively, given , there exists an expanding endomorphism such that for every . We call the section of at . To completely describe an expanding automorphism , we need only know the induced function and the sections . Thus, in keeping with the notation for automaton groups and semigroups in [1] and [9], any expanding endomorphism can be written as
where each is the section of at .
We denote a function by where . If and are expanding endomorphisms with and , then their composition (our functions act on the left) is given by the formula
(1) 
Let be an asynchronous automaton and . If , then is obtained by viewing the word as a path in starting at . The terminal vertex of this path is the section of at . Thus any section of a state of is itself a state of .
Let be an expanding automaton, and let be an element of . Equation (1) allows us to build an expanding automaton that contains as a state. Write . Using the original automaton , compute . If we iteratively use Equation (1) and , we can compute the section of at for any in terms of the sections of the ’s. Furthermore, a straightforward induction on word length in shows that if is a section of , then the word length of in is less than or equal to the word length of in . Thus we will compute an expanding automaton whose set of states has cardinality less than the cardinality of the set .
Before giving examples, we mention that we use the word “action” when describing the functions arising from these semigroups on regular rooted trees. In general, if one says that a monoid has an action on a set, one assumes that the identity of the monoid fixes each element of the set. In this case, however, we can have expanding automaton monoids (and indeed automaton monoids) in which the identity element of the monoid does not fix each vertex of the tree, so we do not include that assumption as part of the definition of “action”. Consider Example 2.2 below.
Example 2.2.
Consider the expanding automaton over the alphabet given by . See Figure 1 for the graphical representation of . We claim that is an identity element of even though does not fix every element of . To see this, first note that the range of is . Since fixes this set, . Now the range of is and fixes this set, so . Now let , and let denote the number of 0’s that occur in ; define analogously. Then , and therefore . Let be the word obtained from by deleting the first letter of . If 0 is the first letter of , then
Similarly, if starts with a 1 we have . Hence , and is an identity element. Thus the action of on includes the action of a semigroup identity that is not the identity function on .
We now show that there are semigroups which are expanding automaton semigroups but not automaton semigroups.
Proposition 2.3.
The class of automaton semigroups is strictly contained in the class of expanding automaton semigroups.
Proof.
Let , and let denote the
semigroup with semigroup presentation
. We show that is not an automaton semigroup
for any , but is an expanding automaton semigroup for
any .
Note that for any distinct with , the rewriting system defined by the rules and is terminating and confluent. Thus is a set of normal forms for , and so is not periodic in .
We begin by showing is not an automaton semigroup. Fix . Let be a synchronous automaton such that is generated by two elements and with and . We show that is periodic in . Note that and must both be states of as higher powers of and cannot multiply to obtain , and powers of cannot multiply to obtain . Let be such that there exists a minimal number with . Since the action of is lengthpreserving, there must exist such a . Let be the orbit of under the action of where for and .
First suppose that for each . If for some , then where , where , and so on. In this case, will have infinitely many sections, which cannot happen since is a state of a finite automaton. Thus for all . Note that if for some and , then the same logic implies that if for some then . Thus we see that if and the section of at is a power of for all , then the section of at is for all . Suppose that for all . Since the action of is lengthpreserving, there exist distinct such that . Then, as the only section of is , we have .
Suppose now that there is a letter such that there exists in the forward orbit of under the action of where . Since and is periodic, there exist distinct with such that for any . To see that this is true, let be the minimal number such that the orbit of under the action of is a cycle. Since the action of is lengthpreserving, there must exist such a . Suppose that there is a such that and the section of at is for some and . Then the relation implies that for any we have for some . Periodicity of then implies that there are as desired. Suppose, on the other hand, that the section of at is in for all . Let be the maximal number such that the section of at is not in and let . Then for some and the relation implies that . In this case we let and .
Let . By the preceding paragraph, for each choose such that . Since acts in a lengthpreserving fashion, there exist distinct such that for all . Thus we can choose distinct such that and for all and . We claim that . To see this, let . If , then the choice of and implies that . Fix . Then and , so the choice of and implies that . If , then
If then and , and so . Similarly, let and write . Suppose there is an such that and . Then
On the other hand, if then . Thus , and so is not .
Fix , and let be an alphabet. Let be the automaton with states and (which depend on ) defined by
where
Then in . Note also that the range of is , and fixes this set. So . Now fix such that . Then . Thus in if and only if and , and we have . ∎
Recall that the bicyclic monoid is the monoid with monoid presentation . Clifford and Preston show in Corollary 1.32 of [2] that in . Furthermore, the same corollary shows that if is a semigroup and such that , , , and , then the submonoid generated by ,, and is the bicyclic monoid if and only if .
Proposition 2.4.
Let be an expanding automaton semigroup. If is a submonoid of , then is not isomorphic to the bicyclic monoid.
Proof.
Let be an expanding automaton semigroup over an alphabet . Suppose such that , , , and . We show that .
If , let range denote . Since is idempotent, fixes range. The equations and imply that fixes range and range. Thus range, range range. So we see that must act injectively on range: if range and , then and and so . Furthermore, because cannot reduce word length, must act in a lengthpreserving fashion on range. Thus range=range. Now the equation implies that maps range onto range, and hence acts injectively and in a lengthpreserving fashion on range. Thus if range then . Suppose range. Then . Since range and acts injectively on range, . Thus ∎
We now distinguish the class of expanding automaton semigroups from the class of asynchronous automaton semigroups.
Proposition 2.5.
The class of expanding automaton semigroups is strictly contained in the class of asynchronous automaton semigroups.
Proof.
Let be the asynchronous automaton over the alphabet with four states defined by
Figure 2 gives the graphical representation of . Note that for all , so is an identity element of . Note also that by construction for any , but . Thus but in . So Corollary 1.32 of [2] implies that the submonoid generated by and is the bicyclic monoid, and Proposition 2.4 implies that is not an expanding automaton semigroup. ∎
3. Decision Properties and Dynamics
We begin this section by showing that expanding automaton semigroups have richer boundary dynamics than automaton semigroups. Proposition 3.1 restricts the kind of action that an automaton semigroup can have on the boundary of a tree, and Example 3.2 gives an expanding automaton semigroup which shows that this restriction does not extend to the dynamics of these semigroups. Example 3.2 also provides a realization of the free semigroup of rank 1 as an expanding automaton semigroup. Proposition 4.3 of [1] shows that the free semigroup of rank 1 is not an automaton semigroup, so Example 3.2 provides another example of an expanding automaton semigroup that is not an automaton semigroup. Let be a semigroup acting on a set , and . We say that is a fixed point of if
Proposition 3.1.
Let be an automaton semigroup with corresponding automaton . If every state of has at least two fixed points in , then every state of has infinitely many fixed points in .
Proof.
We begin with some terminology. We call a path in an inactive path if each edge on has the form for some .
Let . Since is a synchronous automaton, acts in a lengthpreserving fashion. Since has a fixed point in , in the finite automaton there must exist an inactive circuit accessible from via an inactive path . Let be a state on . As must also have two fixed points in , either there is another inactive circuit containing or there is another inactive circuit accessible from via an inactive path. In either case, has infinitely many fixed points in by “pumping” the two inactive circuits.∎
Example 3.2.
(ThueMorse Automaton): This example is constructed to model the substitution rules which give the ThueMorse sequence. This infinite binary sequence, denoted , is the limit of 0 under iterations of the substitution rules . The complement of the ThueMorse sequence, denoted , is the limit of under iterations of these substitution rules. For more information on these sequences, see Section 2.2 of [8] by Lothaire.
Consider the expanding automaton given by over the alphabet . First note that is the free semigroup of rank 1. To see this, by construction of we have for all , and thus for any .
Also by construction of , the action of has exactly two fixed points in : and . To see this, first notice that and are the fixed points of (see section 2.1 of [8]). Thus and are fixed points of for any . Furthermore, where maps 0 to the prefix of length of and maps 1 to the prefix of length of . Thus section 2.1 of [8] implies that has exactly two fixed points for all .
The following proposition gives an algorithm for solving the uniform word problem in the class of expanding automaton semigroups. This proposition is a special case of Theorem 2.15 of [4], which shows that asynchronous automaton semigroups have solvable uniform word problem. We include a proof for completeness.
Proposition 3.3.
Expanding automaton semigroups have solvable uniform word problem.
Proof.
Let be an expanding automaton, and let . Let and be elements of . If , then . If , then use Equation (1) to calculate and for all . If for some , then . If , then calculate for each with , and continue the process. Since and , this process stops in finite time.∎
We now turn to showing that undecidability arises in the dynamics of these semigroups.
Theorem 3.4.

There is no algorithm which takes as input an expanding automaton
and states and decides whether or not there is a word with . 
There is no algorithm which takes as input an expanding automaton
and states and decides whether or not there is an infinite word such that .
Proof.
We show undecidability by embedding the Post Correspondence Problem. Let be an alphabet, and let and be two lists of words over . Let and be alphabets such that . Undecidability of the Post Correspondence Problem implies that, in general, we cannot decide if there is a sequence of elements of such that .
We build an expanding automaton over the alphabet as follows. Let the state set of be , and let
Figure 3 shows where , , and .
Note that for any , does not contain the letter ; similarly, does not contain the letter . Now if contains a letter of , then we know since contains the letter and contains the letter . Thus if there is a word such that , then . By construction of , if and , then . Thus the expanding automaton simulates Post’s problem, and since we cannot decide the Post Correspondence Problem, we cannot decide if there is a word with . This proves part (1).
It is shown by Rouhonen in [12] that the infinite Post Correspondence Problem is undecidable. That is, there is no algorithm that takes as input two lists of words and over an alphabet and decides if there is an infinite sequence such that . Thus, using the same expanding automata and logic as above, (2) is proven. ∎
We now show that undecidability arises when trying to understand the fixed point sets of elements of asynchronous automaton semigroups. If for a set , let denote the prefix of of length .
Definition 3.5.
Let be a free monoid. A subset is a prefix code if

C is the basis of a free submonoid of

If , then for all
The prefix code Post correspondence problem is a stronger form of the Post Correspondence Problem. The input of the prefix code Post Correspondence Problem is two lists of words and over an alphabet such that and are prefix codes. A solution to the problem is a sequence of indices with such that . Rouhonen also shows in [12] that this form of Post’s problem is undecidable. We use the prefix code Post problem to prove the following:
Theorem 3.6.

There is no algorithm that takes as input an asynchronous automaton over an alphabet , a subset , and a state of and decides whether or not has a fixed point in , i.e. decides if there is a word such that .

There is no algorithm that takes as input an asynchronous automaton over an alphabet , a subset , and a state of and decides whether or not has a fixed point in , i.e. decides if there is an infinite word such that .
Proof.
Let be an alphabet, and let be prefix codes where and . Let be the expanding automaton with states that we constructed in the proof of Proposition 3.4. Then is an expanding automaton over the alphabet such that and