Inverse semigroup shifts over countable alphabets
Abstract
In this work we characterize shift spaces over infinite countable alphabets that can be endowed with an inverse semigroup operation. We give sufficient conditions under which zerodimensional inverse semigroups can be recoded as shift spaces whose correspondent inverse semigroup operation is a 1block operation, that is, it arises from a group operation on the alphabet. Motivated by this, we go on to study block operations on shift spaces and, in the end, we prove our main theorem, which states that Markovian shift spaces, which can be endowed with a 1block inverse semigroup operation, are conjugate to the product of a full shift with a fractal shift.
This is a precopyediting, authorproduced PDF of an article accepted for publication in Semigroup Forum, following peer review.
1 Introduction
In this work we are concerned with semigroup operations defined on shift spaces over infinite countable alphabets.
Let be a nonempty countable set (called an alphabet) with the product topology. We define the sets , and , as the sets of all onesided infinite sequences over elements of , and all twosided infinite sequences over elements of , respectively. That is,
and
where denotes the set of the nonnegative integers, and denotes the set of the integers. Whenever a definition, or result, works for both and we will use the symbol meaning “either or ”. For now we consider with the product topology. On the map , which shifts every entry of a given sequence one to the left, is called the shift map. A shift space is any subset of which is closed and invariant under (that is, ). Note that if the alphabet is a finite group then becomes a topological group when given the operation of entrywise multiplication. In the twosided case, the shift map is an expansive group automorphism on the zero dimensional group .
A major inspiration for what follows is the seminal paper by Kitchens [kitchens], which concerns the converse of the above.
Theorem 1.1.
[kitchens, Theorem 1] Suppose is a compact, zero dimensional topological group, and that is an expansive group automorphism. Then

is topologically conjugate to via a group isomorphism, where is a onestep shift of finite type and is a finite group;

is topologically conjugate to via a group isomorphism, where and are finite groups, is a group automorphism, and the group operation on is given as an extension of by .
A shift space in which carries a topological group operation which commutes with the shift is called a topological group shift, and the above shows that topological group shifts are modelled by group operations on the alphabet (after a possible recoding). In [sobottka2007], similar results were obtained for topological quasigroup shifts over finite alphabets.
Here, we are interested in what one can say when the alphabet is countably infinite. In this case we will consider the compactification of proposed in [Ott_et_Al2014], which is such that the elements introduced when compactifying can be seen as finite words in together with an empty word . The twosided analogue of this construction is the space , which was considered in [GoncalvesSobottkaStarling2015_2].
If is a group, then producing a binary operation on from the one on poses the immediate question of how to multiply sequences of different lengths. The natural thing to do is when multiplying two words and with the length of less than the length of , that one would truncate to the length of and then multiply entrywise as normal. The result here of course is a word the same length as . Under this operation, is not a group – indeed, for every element ! It is however, an inverse semigroup, and it is for this reason that this paper concerns shift spaces with inverse semigroup operations on them – “inverse semigroup shifts” – and studies to what extent we can obtain results akin to Theorem 1.1 in the infinite alphabet case (see Section 2.2 for the details of the construction of ).
After setting notation and background in Section 2, we prove a result similar to Theorem 1.1.i above for the onesided case, Proposition 3.5. To do this, we first define the notion of an expansive partition, and go on to show that if a dynamical system is an inverse semigroup and admits such a partition, then it is conjugate to an inverse semigroup shift. Section 4 is then dedicated to inverse semigroup shifts whose operation is induced from a group action on the alphabet. In Section 4.1 we see that a group operation over an infinite alphabet always induces a 1block inverse semigroup operation on a full shift which uses that alphabet (but not necessarily a topological operation). Section 4.2 is dedicated to showing that any inverse semigroup satisfying certain conditions is semigroup isomorphic to an inverse semigroup shift, while Section 4.3 shows general properties of socalled 1block operations. Here we show that, in contrast to the finite alphabet case, follower sets and predecessor sets may have different cardinalities. In Section 5, we define a class of fractal shift spaces and then find an analogue to Theorem 1.1.ii, by showing that any twosided Markovian 1block inverse semigroup shift is isomorphic and topologically conjugate to the product of a fractal shift and a twosided full shift, see Theorem 5.18.
2 Background
In this section we will set notation and recall the background necessary for the paper. For detailed references on the topics covered in this section, see [La98] for inverse semigroups, [LindMarcus] and [Ott_et_Al2014] for one sided shift spaces, [GoncalvesSobottkaStarling2015_2] for two sided shift spaces over infinite alphabets, and [GoncalvesSobottkaStarling2015] for sliding block codes between infinite alphabet shift spaces.
2.1 Semigroups and dynamical systems
Recall that a semigroup is a pair where is a set and is an associative binary operation. If is a topological space and is continuous with respect to the topology of , we say that is a topological semigroup. An inverse semigroup is a semigroup such that, for all , there exists a unique such that and .
We call an element an idempotent if , and the set of all such elements will be denoted . It is true that, for all and we have that , , , and . For all , the elements and are idempotents.
If is an inverse semigroup with an identity (that is an element such that for all ) then it is called an inverse monoid. An element in is called a zero (and denoted by ) if for all , and if , and then we say that and are divisors of zero. If they exist, the zero element and identity element are both seen to be unique.
Given an element , we have the Green’s relations and : if and only if ; if and only if . Given its class is denoted , and its class is denoted .
If and are two inverse semigroups, then a map is said to be a semigroup homomorphism if for all we have that . For such a , for all , we have that and for all . If a semigroup homomorphism is bijective then we say that it is an isomorphism.
Any inverse semigroup carries a natural partial order given by saying that if and only if there exists some such that .
A dynamical system (DS) is a pair , where is a locally compact space and is a map, and if in addition is continuous then we will say that is a topological dynamical system (TDS). Two dynamical systems and are said to be conjugate if there exists a bijective map such that . If, in addition, and are topological dynamical systems and is a homeomorphism we say that and are topologically conjugate and the map is called a (topological) conjugacy for the dynamical systems.
We say that a dynamical system is expansive if there exists a partition of such that, for all in , there exists such that and are in different elements of (where if is not invertible, and otherwise). In such a case, is called an expansive partition for .
Lastly, recall that a locally compact Hausdorff space is said to be zero dimensional or totally disconnected if it has a basis consisting of clopen sets.
2.2 Compactified shift spaces over countable alphabets
Let be a countable set with , which we will call an alphabet and whose elements will be called symbols or letters. Define a new symbol , which we will call the empty letter and define the extended alphabet . We will consider the set of all infinite sequences over , and define by
and
We will denote by the constant sequence whose entries are all the empty letter , that is, where for all .
Definition 2.1 (Onesided full shift).
The onesided full shift over is the set
Definition 2.2 (Twosided full shift).
The twosided full shift over is the set
Definition 2.3.
For , we define
and call the length of .
We will refer to sequences in as finite sequences, sequences in as infinite sequences and we will refer to as the empty sequence. To simplify the notation, giving a sequence we will frequently omit the entries with the empty letter when denoting it, that is, we will denote , where .
Given and a finite set , we define
(1) 
In the special case that , then we denote simply by . Furthermore, given we define
(2) 
In the case , given , where each is a finite word of length 1, we define .
The sets of the form (1) and (2) are called generalized cylinders and they form the basis of the topology that we consider in . For this topology we have that is zero dimensional (generalized cylinders are clopen sets), Hausdorff and compact. Furthermore, when the topology is metrizable (see [Ott_et_Al2014], Section 2), while for the topology is not first countable (there is no countable neighborhood basis for  see [GoncalvesSobottkaStarling2015_2, Proposition 2.7]). We refer the reader to [GoncalvesSobottkaStarling2015_2] and [Ott_et_Al2014] for more details about this topology.
Note that if then coincides with an usual cylinder of the product topology in . In particular if , and we allow to be finite, then with the topology generated by the generalized cylinders is a classical shift space over (see [Ott_et_Al2014, Remark 2.24]).
As mentioned in the introduction, the shift map is defined as the map given by for all . Note that and, for all , , we have that . Furthermore, when is infinite and we have that is continuous everywhere but at . In any other situation (either under and finite , or under and countable ) we have that is a continuous map (see [GoncalvesSobottkaStarling2015_2, Proposition 2.12]).
Given , let and be the set of finite sequences and the set of infinite sequences in , respectively. We consider on the topology induced from , that is, the topology whose basis are the sets of the form and .
For each , let
be the set of all blocks of length in .
Furthermore, define
The language of is
Also, define the letters of to be – this is the set of elements of used in sequences of .
Given , the follower set of in is the set
(3) 
while the predecessor set of in is
(4) 
We say that a subset satisfies the infinite extension property if, for all ,
Notice that when , if, and only if, , while if then we have that if, and only if, .
Definition 2.4.
A set is said to be a shift space over if the following three properties hold:

is closed with respect to the topology of ;

is invariant under the shift map, that is, ;

satisfies the infinite extension property.
We notice that condition 2 could be relaxed when to . However in this work we will just consider, even when , shift spaces such that . We remark that satisfies the infinite extension property if and only if is dense in (see [Ott_et_Al2014, Proposition 3.8] and [GoncalvesSobottkaStarling2015_2, Lemma 2.18]).
Given such that we can define the shift space generated by as the smallest shift space such that . It follows that , where stands for the closure of .
Shift spaces can be characterized in terms of forbidden words:
Definition 2.5.
Let be a subset of whose elements do not use the empty letter. We define as the subset of such that: (i) its infinite sequences are exactly those that do not contain a block belonging to ; (ii) its finite sequences satisfy the ‘infinite extension property’; (iii) (where this last assumption is only necessary to assure that, when , condition 2 of Definition 2.4 is satisfied).
From [Ott_et_Al2014, Theorem 3.16] and [GoncalvesSobottkaStarling2015_2, Proposition 2.25] we have that is a shift space if, and only if, for some whose elements do not use the empty letter.
Definition 2.6.
Given a shift space , the projection of onto the nonnegative coordinates is the map given by .
The projection establishes a relationship between twosided shift spaces and onesided shift spaces. We refer the reader to sections 2 and 4 of [GoncalvesSobottkaStarling2015_2] for more details about it.
We now summarize some important types of shift spaces. If is a shift space, then we say that is a:

shift of finite type (SFT): if for some finite ;

finitestep shift: if for some . In the case that , we will say that is an step shift (step shifts will also be referred as Markov (or Markovian) shifts);

infinitestep shift: if it is not a finitestep shift. Note that just twosided shift spaces can contain proper infinitestep shifts;

edge shift: if , where and is a directed graph with no sources and sinks.

rowfinite shift: if for all we have that is a finite set;

columnfinite shift: if for all we have that is a finite set.
2.3 Sliding block codes
Roughly speaking, a slinding block code is a code which encodes a sequence of a shift space as a new sequence (possibly in other shift space) in a continuous way and which is invariant by translations.
A pseudo cylinder of a shift space is a set of the form
where and . We say that the pseudo cylinder has memory and anticipation . Note that if is a onesided shift space, then any pseudocylinder of has memory equal to zero. We adopt the convention that the empty set is a pseudo cylinder of whose memory and anticipation are zero.
Given , we will say that is finitely defined in if there exist two collections of pseudo cylinders of , namely and , such that
In this case we say that has memory and anticipation .
Definition 2.7.
Let and be two countably infinite alphabets, and let and be two shift spaces. Suppose is a pairwise disjoint partition of , such that:
 1.

for each the set is finitely defined in ;
 2.

is shift invariant (that is, ).
A map is called a sliding block code if
where is the characteristic function of the set and stands for the symbolic sum.
In this case, we say that has memory and anticipation , where and stand for the memory and the anticipation of each , respectively, If we will say that is a block, while if or , we will say that has unbounded memory or anticipation, respectively.
Intuitively, is a sliding block code if it has a local rule, that is, if for all and , there exist which depends on the configuration of around , such that we only need to know to determine . In the particular case that is a sliding block code with memory and anticipation , then its local rule can be written as a function such that for all and , it follows that .
We refer the reader to [GoncalvesSobottkaStarling2015], [GoncalvesSobottkaStarling2015_2] and [SG] for more details about sliding block codes between shift spaces over infinitely countable alphabets.
2.4 Shift spaces as dynamical systems and semigroups
If is a shift space, then is a dynamical system. If , [GoncalvesSobottkaStarling2015] ensures that is a topological dynamical system if, and only if, is a columnfinite shift. On the other hand, if , then [GoncalvesSobottkaStarling2015_2] ensures that is always a topological dynamical system.
Definition 2.8.
Two shift spaces and are said to be (topologically) conjugate if the dynamical systems and are (topologically) conjugate.
We remark that a sliding block code is always a conjugacy between shift spaces (see [GoncalvesSobottkaStarling2015, Proposition 3.12] and [GoncalvesSobottkaStarling2015_2, Proposition 3.10]). Furthermore, in [GoncalvesSobottkaStarling2015] and [GoncalvesSobottkaStarling2015_2] conditions are given for a sliding block code to be a topological conjugacy.
We now seek to incorporate a semigroup operation to the shift space structures defined so far.
Definition 2.9.
We will say that a shift space is a semigroup shift if there exists a binary operation defined on such that is a semigroup and the shift map is a semigroup homomorphism. If, in addition, the operation is continuous, then we will say that is a topological semigroup shift.
A particular type of semigroup shifts that we will study are the block semigroup shifts.
Definition 2.10.
Let be an alphabet, be a subshift, and be a binary operation on . Suppose that a pairwise disjoint partition of , such that:
 1.

for each the set is finitely defined;
 2.

is shift invariant.
We say that is a block operation if
(5) 
where is the characteristic function of the set , and stands for the symbolic sum.
Denoting by and the memory and the anticipation of each , respectively, let and . If and are finite, we will say that is a block operation with memory and anticipation (in the case we always have ).
Once again it is best to think of a block operation as an operation with a local rule. In the particular case of being a block operation with memory and anticipation , its local rule can be written as , such that given and ,
One notices that the above definition of a block operation is similar to the definition of sliding block codes (see [GoncalvesSobottkaStarling2015, GoncalvesSobottkaStarling2015_2, SG]). However, since is not a shift space, the map is not a sliding block code and we cannot use the results established for them. At any rate, block operations share several properties with sliding block codes, and, for the particular case of 1block semigroups (see Subsection 4.1), such an operation will be a sliding block code defined from the shift space to (see Section 2.5 for the definition of ).
2.5 The Product of Shift Spaces
Given two shift spaces and , the Cartesian product is not a shift space on the alphabet (if either or is infinite). In light of this, we define product of shift spaces as follows:
Definition 2.11.
Given two shift spaces and , the product of and is the shift space generated by , that is, , where we are identifying an element with .
We remark that is the smallest shift space whose set of infinite sequences is . Alternatively, we can consider the map , defined, for all and , by:
(6) 
and it follows that , , and .
Proposition 2.12.
Let and be two shift spaces, consider with the product topology and with the topology generated by generalized cylinders. Then the map defined by (6) is continuous.
Proof.
Suppose that and are two shift spaces. Let
be a generalized cylinder of , where , and is a finite set, say .
Let and , where for all (notice that and are finite sequences and not sets). It follows that if , then
where , , with for and and with for and .
Since for each , the set is a clopen of , then its complement is also a clopen of . Hence is a finite intersection of clopen sets and thus it is a clopen set of .
On the other hand, given we have that
By the first part of the proof we have that each is a clopen set and hence we conclude that is also a clopen set. ∎
3 Expansive dynamics and inverse semigroup shifts
In [kitchens], it is proven that if is a compact zerodimensional group and is an expansive and transitive group automorphism of , then is conjugate to the full shift over a finite group. In our situation, the full shift over a countably infinite group, when given the operation of entrywise multiplication, has the structure of an inverse semigroup with the empty sequence being the zero element. The shift map fixes the empty sequence, and in general the empty sequence is the only possible point of discontinuity of the shift map.
In this section we prove a result similar to Kitchens’, that is under some conditions, an expansive automorphism over a zerodimensional inverse semigroup can be modeled by the shift map over , for a suitable infinite alphabet .
Lemma 3.1.
Let be a compact space and let be a function. Suppose that there exisits such that and that is continuous on . Then, for all and all , the set is open.
Proof.
Take to be open. Note that will not contain for any , because . Hence we only need to show that is open.
If , then . Since is continuous at , there exists an open set with such that , that is, . Hence, is open.
∎
Proposition 3.2.
Let be a compact Hausdorff space, and let be a function. Suppose that we have such that and that is continuous on . Further, suppose that there exists an infinite partition of clopen sets of , such that is an expansive partition for . Then there exists an infinite alphabet and a continuous injection , which is a homeomorphism onto its image, such that . If we assume further that
(7) 
is dense in , then the image of is a subshift of .
Proof.
Let be a clopen partition of , and let be an expansive partition for . Consider the full shift over the index set .
Let be defined by
Since is an expansive partition, it is easy to see that is injective. We claim that is a homeomorphism onto its image, and that its image is a subshift of .
We first prove that is continuous. Suppose that in . We have to prove that in . If , then is an infinite sequence in . Take and let
Note that is open (by Lemma 3.1) and nonempty, and that . Since , there exists such that for all , . Thus, for all , the first entries of the sequence are equal to that of . Hence, .
Suppose now that with . Let be finite, say . We have to find such that implies that , for . and .
Let
(8) 
We note that , since . In addition, we have that . Hence, is continuous at all points in . We also note that does not contain , because it is contained in . Hence, is continuous on .
We claim that is open. Let . Then , an open set. Since is continuous at , there exists an open set containing such that , that is, . Hence
is open, is contained in , and contains . Hence, is open.
Now, since , there exists such that whenever , . This implies that whenever , we have , for , and . Hence we have proven that converges to .
Finally, suppose . Then . Let be finite, say , and take as in (8). Since , and is an open set containing , there exists such that for all we have . This means that for all , . Since was arbitrary, this means that .
Now that we have that is continuous and injective, since is compact and is Hausdorff, is automatically a homeomorphism onto its image.
We now prove our final statement. We have that is closed and is clearly closed under the shift, and so it remains to show that it has the infinite extension property. Suppose that and that with . Then with . As before, let be finite, form , and let
which is an open neighborhood of . Again, is continuous on . Because the set in (7) is dense in , there exists in . Now, the first entries of match . Furthermore, the element must be different from any element of because of the form of . Since was arbitrary, we are forced to conclude that the set
is infinite. Hence, has the infinite extension property.
∎
Remark 3.3.
From now on we suppose that we have a compact Hausdorff space , a map which satisfies the hypotheses of Proposition 3.2 and denote the subshift of given by the image of the map by .
Corollary 3.4.
If then is a rowfinite subshift of .
Proof.
It is clear that if then for all and so the only finite word in is the zero word and hence, by Proposition 3.21 in [Ott_et_Al2014], the result follows.
∎
Proposition 3.5.
Suppose that