The relative rank of the endomorphism monoid of a finite -set
Abstract
For a group acting on a set , let be the monoid of all -equivariant transformations, or -endomorphisms, of , and let be its group of units. After discussing few basic results in a general setting, we focus on the case when and are both finite in order to determine the smallest cardinality of a set such that generates ; this is known in semigroup theory as the relative rank of modulo .
Keywords: -set, equivariant transformation, -endomorphism monoid, -auto-morphism group, relative rank.
MSC 2020: 20B25, 20E22, 20M20.
1 Introduction
For any group , a -set is simply a set on which acts; this is, there exists a function such that , for all , and , for all , . In the context of semigroup theory, -sets are also known as -acts. A -equivariant transformation of , or a -endomorphism of , is a function such that , for all , . In general, -equivariant maps are the standard morphisms considered in the category of -sets ([4, p. 86]), and they are widely used in various areas of mathematics such as equivariant topology, topological dynamics, representation theory, and even in statistical inference, via the so-called equivariant estimators.
In this setting, two basic objects, which have been previously studied from a semigroup-theoretic perspective (e.g., see [5, 6, 12, 20]), are the monoid , consisting of all -equivariant transformations of equipped with composition, and its group of units , consisting of all bijective -equivariant transformations of . In particular, free -sets (i.e., -sets on which all point stabilizers are trivial) and their endomorphisms have gained special interest, as the former are examples of independence algebras (see [2, 10, 14, 15, 17]).
Our work in this paper is related with the concept of rank of a monoid , denoted by , which is the smallest cardinality of a generating set of . For any subset , define the relative rank of modulo by
Finding the relative rank of a finite monoid modulo its group of units is a natural and important question in semigroup theory (e.g., see [1, 3, 11, 16, 18, 19, 23]), as it is linked to the rank of via the formula
Our main theorem determines the relative rank of modulo when and are both finite. Before stating it, we introduce some notation: let be the conjugacy class of a subgroup , let be the set of conjugacy classes of point stabilizers in , let be the normalizer of in , and let be the set of all -orbits whose point stabilizer is conjugate to .
Theorem 1.
Let be a finite group acting on a finite set . Let be the list of different conjugacy classes of subgroups in . Let , and for any , consider the -conjugacy class . Then,
where and .
In order to prove Theorem 1, we begin by discussing few basic results that hold for an arbitrary -set and are probably known in the folklore of -equivariant maps, but we could not find them published anywhere. For example, we observe that when the action of on is transitive, if and only if a point stabilizer in is not properly contained in any of its conjugates. Hence, the situation always occurs for transitive actions where point stabilizers are finite, or finite-index, or normal subgroups of . Besides this, for a general -set, we use the Imprimitive Wreath Product Embedding Theorem [22, Theorem 5.5] to decompose as a direct product of wreath products:
One of the motivations for our study comes from the shift action of on , where is a set, and is the set of all functions . This action, which has fundamental importance in symbolic dynamics and the theory of cellular automata (see [9]), is defined as follows: for every and ,
When is considered as a topological space with the product topology of the discrete topology of , the -equivariant continuous transformations of turn out to be precisely the cellular automata over (see the Curtis-Hedlund Theorem in [9, Ch. 1]). Hence, the monoid , consisting of all cellular automata over , is a submonoid of . When and are both finite, it is clear that . This setting has been studied in [7], and the present paper generalizes and refines many of the results obtained there. In particular, Theorem 1 significantly generalizes Theorem 7 in [7], where the the relative rank of modulo was determined only when is a finite Dedekind group (i.e. all subgroups of are normal).
The structure of this paper is as follows. In Section 2, we provide all the basic results on -equivariant transformations that we need in order to prove Theorem 1; we believe some of these basic results may be of interest in their own right. In Section 3 we prove Theorem 1 in three steps. First, we show that the set , consisting of all transformations , with and , defined in (1), generates modulo (Lemma 10). Second, we show that a particular subset of also generates modulo (Lemma 11). Finally, we complete the proof by showing that any set that generates modulo must contain at least transformations (Theorem 17). The cardinality of is given by the formula of Theorem 1.
2 Basic results
For the rest of this paper, let be a group acting on a set . We assume that the reader is familiar with the basic results on group actions (e.g., see [22, Sec. 2]). Besides introducing notation, in this section we prove some required basic results on -sets that we may not find published anywhere else.
Denote by the symmetric group on . When is finite and , we may write instead of . Denote by and the -orbit and stabilizer of , respectively:
Two basic results are that the set of -orbits form a partition of , and that stabilizars are subgroups of .
We shall repeatedly use the conjugation action of on the set of its subgroups; this is defined, for any and any subgroup , by . The -orbits of this action, denoted by , are called conjugacy classes, while the stabilizers, denoted by , are called normalizers; in other words, for any ,
For a subset , define the setwise and elementwise stabilizers of as the subgroups and , respectively.
The main object of this paper is the monoid of -equivariant transformations of , which is denoted by
Let be the group of units of , i.e.
When is a subgroup of , i.e. is a permutation group, note that is equal to the centralizer of in .
It is clear that -equivariant transformations map -orbits to -orbits, as , for all , . Moreover, it is easy to see that if is bijective, then is also -equivariant, so .
Both and act naturally on by evaluation: for any and , this action is defined by .
A subset is -invariant if , for all . It is easy to prove that is -invariant if and only if is a union of -orbits. We may restrict the action of to , and consider the monoid and the group .
Lemma 2.
For any -invariant subset , the following hold:
-
1.
is isomorphic to a submonoid of , and is isomorphic to a subgroup of .
-
2.
If is also -invariant (i.e., , for all , ), then is isomorphic to a normal subgroup of .
-
3.
.
Proof.
The monoid is embedded in via the injective homomorphism given by
It is easy to check that indeed using the fact that if and only if . By restricting to , we show that is embedded in .
Now fix and , and assume that is also -invariant. Since , for all , then . Therefore , which shows that is normal in .
Finally, consider the restriction homomorphism given by , for all . This is a surjective homomorphism with kernel , so part (3) follows by the First Isomorphism Theorem. ∎
We shall introduce notation for three particular -equivariant transformations of . Let with .
-
1.
When , define by
(1) -
2.
When and there exists such that , define by
(2) -
3.
When and , define by
(3)
Observe that the -equivariant transformations given in (2) and (3) are bijections, and is defined if and only if , which is true if and only if .
For any , it is easy to check that for every . Using this simple fact combined with the previous constructions of -equivariant transformations, we obtain the following result.
Lemma 3.
Let .
-
1.
There exists such that if and only if .
-
2.
If , there exists such that if and only if .
Recall that the action of on is transitive if there exists a unique -orbit, so for any ; in other words, if for every there exists such that . If the action of on is not transitive, Lemma 3 shows that is properly contained in . On the other hand, when the action is transitive, the next result shows that if and only if, for any , the stabilizer is not properly contained in any of its conjugates (i.e., implies ). This last condition may seem unusual. In general, if is a finite, finite-index, or normal subgroup of , then cannot be properly contained in any of its conjugates; however, there are examples of infinite groups and infinite-index subgroups such that is properly contained in for some (see [21]).
Proposition 4.
Suppose that the action of on is transitive and let . The subgroup is not properly contained in any of its conjugates if and only if
Proof.
Assume that is not properly contained in any of its conjugates. Fix . It follows by transitivity and -equivariance that must be surjective. Suppose that , for some . By transitivity, there exists such that . Then , implies that . By the -equivarance of , we have , and again by transitivity there exists such that . Hence, by hypothesis, , so . Hence, , which shows that is injective.
Conversely, suppose that there exists and such that . Define as follows:
It is clear that , but we shall show that is not injective. Consider . Then . However, implies that
Therefore, is properly contained in . ∎
Example 5.
Let be an infinite group such that there exists a subgroup and with . In this situation, consider the shift action of on , where is a set with at least two elements. Without loss, assume that . Let be the indicator function of defined by
It is easy to check that , and so, by Proposition 4, is properly contained in .
2.1 Structure of
The next result is well-known (for example, see [24, Prop. 1.8]).
Lemma 6.
If the action of on is transitive, then
Proof.
Fix . The map from to , with defined as in Lemma 3, is a surjective homomorphism with kernel . ∎
Our next goal is to describe the structure of for intransitive actions.
First of all, we introduce some notation. Define
For each , define the set
Since , for every , , the set is -invariant. Moreover, by Lemma 3, is also -invariant, so, by Lemma 2, is isomorphic to a normal subgroup of . The number of -orbits inside is .
By the Orbit-Stabilizer Theorem, the cardinality of each -orbit inside is the index . Let be the set of -orbits contained in :
Example 7.
For the shift action of on , the values of may be calculated using the lattice of finite-index subgroups of and its corresponding Möbius function ; see [8] for details.
Example 8.
Consider the shift action of on , where we represent a function by the -tuple . In this case, . Figure 1 shows all the elements of partitioned by -orbits (depicted by ellipses) and by the sets (depicted by rectangles). In this case, we directly see that , and .

We briefly recall the definition of wreath product of groups. For a group , a set and a group acting on , the (unrestricted) wreath product of by , denoted by is the semidirect product , where is the direct product of indexed by and homomorphism given by , for all (see [22, p. 105] for more details). In our results, we shall always consider the case when and the action of on is the natural action by evaluation, so, in order to simplify notation, we write instead of .
For any set , let be the full transformation monoid of , i.e. the set of all self-maps of equipped with composition. When is finite, it is well-known that is generated by a generating set of together with a map with image size (e.g., see [13, Ch. 3]). Analogously to the case of groups, we may define a wreath product of monoids (see [1]).
The following result uses the Imprimitive Wreath Product Embedding Theorem, which, as stated in [22, Theorem 5.5], involves a partition with finitely many blocks; however, this assumption may be dropped and the same proof of the theorem works by using the axiom of choice.
Proposition 9.
Let be a group acting on a set ,
-
1.
For any ,
-
2.
-
3.
Suppose that is not properly contained in any of its conjugates. Then,
Proof.
-
1.
The set of -orbits inside forms a -invariant partition of , with . Using the axiom of choice, it is easy to see that the induced action of on is isomorphic to the whole symmetric group (pick representatives of -orbits in order to define corresponding -equivariant transformations). By [22, Theorem 5.5]) and Lemma 2, is permutationally isomorphic to a subgroup of , with . The kernel of the projection of to is isomorphic to the direct product . Therefore,
and the result follows by Lemma 6.
-
2.
For each , we define a function
by for every . The function is injective because is a partition of . To show that is surjective, observe that for any in the product, we may define by if and only if . It follows that is indeed -equivariant as, for every , is -equivariant and is -invariant. Finally, is a homomorphism since is -invariant so , for every . The result follows by Lemma 9.
-
3.
With the help of Proposition 4, this follows by a very similar argument as in the proof of the first point.
∎
3 The relative rank of modulo
For any monoid and , we say that a subset of generates modulo if .
For this section, assume that is a finite group acting on a finite set . We shall assume that is the list of different conjugacy classes of subgroups in . Define and . For , denote .
Lemma 10.
The set
generates modulo
Proof.
Suppose that the list of conjugacy classes is ordered such that
Fix and for each , define by
Each is indeed -equivariant as is -invariant. By Lemma 3, for all , so . This implies that we have the following factorization of
(4) |
which holds since is fixed by , for all , . Now fix and define
For , it is easy to see that is a -invariant subset of , so we may define by
It follows that . We finish the proof by showing that , for .
-
•
Case . We show that . In this case, is contained in a submonoid of isomorphic to . If , then , for , so assume that . By Proposition 9, , for any . The monoid is generated by together with any mapping with image size (see [19, Prop. 1.2]). For any with , the map induces on a mapping with image size . It follows that is generated by , and the result follows.
-
•
Case . We show that . Suppose that is a union of the following -orbits
for some . Write , for . As , then . Therefore, we have the factorizacion
∎
For any , define the -conjugacy class of by .
Lemma 11.
Let be a finite group acting on a finite set . For each , , and let be the list of -conjugacy classes such that and . We fix some elements of as follows:
-
•
For each , fix such that .
-
•
For each and , fix such that .
-
•
For each such that , fix such that and .
Then the set
generates modulo .
Proof.
By Lemma 10, it is enough to show that any , with , , , may be expressed as a product of elements of and . Without loss, assume that . Then, there exists such that ; since , we may assume that . There are two cases to consider.
-
•
Case . Note that . We have four subcases:
-
–
Case and . Here we have the factorization
-
–
Case and . There exists with . As , we may define as in the proof of Lemma 3. Then
which holds since for every ,
As , then
-
–
Case and . There exists with . Observe that
Therefore,
-
–
Case and . With the notation of the above cases,
-
–
-
•
Case . There exists such that , with . Thus, there exists such that
where . Now, because , and we may define . Hence,
This shows that if and only if , with . Again there are four subcases, which are done analogously as in the previous case. For example, for the subcase and , we obtain the factorization
∎
With the notation of the previous lemma, define
Corollary 12.
With the previous notation,
Example 13.
When is a finite group acting by the shift action on , with a finite set with at least two elements, it is not hard to show (see [7, Lemma 5]) that
Before being able to determine the exact relative rank of modulo , we need to introduce few definitions. For , define
This is an equivalence relation on , and it is linked with the Green’s relation in a semigroup of transformations (see [13, Theorem 4.5.1]).
Remark 14.
We state the following elementary properties of kernels of transformations:
-
1.
if and only if is a bijection.
-
2.
, we have .
-
3.
If , then , since if and only if .
Definition 15.
Let be such that . We say that is an elementary collapsing of type if there exist , with , such that:
-
1.
and .
-
2.
.
For example, when satisfy that and , then is an elementary collapsing of type .
Lemma 16.
-
1.
Suppose is an elementary collapsing of type and type . Then and .
-
2.
The number of possible types of elementary collapsings is precisely
Proof.
For the first part, there exist with and satisfying Definition 15. Then
This shows that and . The first equality implies that , so . Moreover, there exists such that , so
Now,
which implies that , and so
The second part of the lemma follows as an elementary collapsing of type exists if and only if . ∎
Theorem 17.
Let be a finite group acting on a finite set . Let be the list of different conjugacy classes of subgroups in . Then,
Proof.
Let such that . We shall show that must contain at least one elementary collapsing of each possible type. Suppose that satisfies that . Let be such that and . Since , there exist such that
Let be the smallest index such that is non-invertible. We must have .
Claim 18.
For all , we have:
-
1.
, and for every .
-
2.
If , then , and for every .
Proof.
Suppose there exists such that . Let . By Lemma 3, , and so . This implies that must map to an element of , which is impossible by Lemma 3. This proves that . If is properly contained in , then there must exist such that . However, this implies that and , with , satisfy that , which contradicts the definition of . Thus , and for every , follows by Lemma 3 and the finiteness of .
The second part of the claim is done analogously. ∎
We shall show that is an elementary collapsing of type by verifying the two properties of Definition 15.
- 1.
- 2.
Acknowledgments
The first and second author were supported by a CONACYT Basic Science Grant (No. A1-S-8013) and a PhD CONACYT National Scholarship, respectively. Both authors sincerely thank Csaba Schneider for his kind and enriching replies to our questions on the Imprimitive Wreath Product Embedding Theorem. We also thank the anonymous referee of this paper for his insightful comments.
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