A note on dual prehomomorphisms from a group into the Margolis–Meakin expansion of a group

We give a category-free order theoretic variant of a key result in Auinger and Szendrei (J Pure Appl Algebra 204(3):493–506, 2006) and illustrate how it might be used to compute whether a finite X-generated group H admits a canonical dual prehomomorphism into the Margolis–Meakin expansion M(G) of a finite X-generated group G. We show that for G the Klein four-group a suitable H must be of exponent 6 at least and recapture a result of Szakács.


Introduction
The following note considers canonical, i.e. generator preserving dual prehomomorphisms from an X-generated group H into the Margolis-Meakin expansion M(G) of an X-generated group G. It was shown by Auinger and Szendrei [1] that such mappings play an important role in constructing (finite) F-inverse covers for (finite) inverse monoids. We give a necessary and sufficient order theoretic condition for M(G) to admit a canonical dual prehomomorphism from an X-generated group H. It can be seen as a variant of the key statement Lemma 3.1 in [1] and might be applicable on a computer. The idea is to represent the elements of both M(G) and H as congruence classes of words in the free monoid with involution (X ∪ X −1 ) * . This enables us to handle the elements of H in relation to M(G) by systematically going through the words in (X ∪ X −1 ) * . We use the slightly different view of M(G), introduced in [2], to show how already known positive examples fit into the picture. Further, for G the Klein four-group, we prove that a suitable group H must be of exponent 6 at least and recapture a result of Szakács [6]. It should be noted that in our construction the groups H, we consider as possible candidates for admitting a canonical dual prehomorphism into M(G), may be arbitrarily X-generated extensions by G. This is in contrast to [1], where H is assumed to be an X-generated subgroup of a semidirect product of a relatively free group by G.

Preliminaries and notations
For all undefined notions and notations, the reader is referred to [3,5]. Let X be a nonempty set and let G be an X-generated group with respect to an injection G ∶ X → G ⧵ {1 G } . Note that the mapping G can be uniquely extended to a homomorphism G ∶ (X ∪ X −1 ) * → G , where (X ∪ X −1 ) * is the free monoid with involution on X. For w ∈ (X ∪ X −1 ) * we denote w G by w . By the Cayley graph Γ(G) with respect to G , we mean the directed graph whose vertex set V(Γ(G)) is G and whose edge set E(Γ(G)) is G × X , where for each g ∈ G, x ∈ X, (g, x) denotes an edge with initial vertex g and terminal vertex gx . Put There is a natural action of G on the semilattice of all subgraphs of Γ(G) with operation the set theoretic union, defined as follows: Put g� = � , and for each nonempty subgraph Γ of Γ(G) and g ∈ G , let gΓ be the subgraph of Γ(G) with The graphs we consider do not have isolated vertices, whence they are solely determined by their edge sets, and we conveniently may regard them as (possibly empty) subsets of X × G.
The following theorem was essentially proved in [4].
We often represent the elements of M(G) by their corresponding images ⟨w⟩ in (X ∪ X −1 ) * ∕ ker M(G) , where M(G) denotes the unique extension of M(G) to a homomorphism from (X ∪ X −1 ) * onto M(G). Then obviously ⟨∅⟩ corresponds to (�, 1 G ) .
A note on dual prehomomorphisms from a group into the Margolis-…

Canonical dual prehomomorphisms into M(G)
In M(G) the natural partial order is given by ⟨v⟩ ≤ ⟨w⟩ if and only if v = w and Γ(⟨w⟩) ⊆ Γ(⟨v⟩) . The following order theoretic statements are straightforward.
Note that the greatest lower bound ∧ i∈I ⟨w i ⟩ exists in M(G) for each finite set I if and only if all w i are equal to a given w , say, in which case ∧ i∈I ⟨w i ⟩= (∪ i∈I Γ(⟨w i ⟩), w) . Note further that ∨ i∈I ⟨w i ⟩ exists if and only if the set {⟨w i ⟩, i ∈ I} has an upper bound in M(G).
Let H be an X-generated group via an injection H ∶ X → H ⧵ {1 H } . Like with M(G) we may represent the elements of H by their corresponding images [w] in (X ∪ X −1 ) * ∕ ker H , where H denotes the unique extension of H to a homo- [5]. According to [1], we call canonical if ([x]) = ⟨x⟩ for all x ∈ X . Note that a canonical dual prehomomorphism ∶ H → M(G) always induces a generator respecting homomorphism from H onto G, given by [w] ↦ w , which follows from the fact that in M(G) we have that (Γ(⟨v⟩), v) ≤ (Γ(⟨w⟩), w) implies v = w and respects generators. Thus H necessarily must be an extension by G.
In what follows we give a necessary and sufficient condition for M(G) to admit a canonical dual prehomomorphism ∶ H → M(G) . Our condition is of an order theoretic form.

Theorem 3.2 Let G and H be groups as defined above. Then H admits a canonical dual prehomomorphism ∶ H → M(G) if and only if the following sequence of least upper bounds exists for each
. Continuing this process we see that all P n ([w]), n ∈ ℕ 0 exist. Sufficiency: Let the condition in the assumption of Theorem 3.2 be satisfied. Note that {P n ([w])} n∈ℕ 0 is increasing and will be constant after a finite number of steps, for each [w] ∈ H , since all occurring graphs are finite.
This fact holds in any inverse semigroup S and easily follows from Note that the above defined mapping P is the least possible canonical dual prehomomorphism with respect to the pointwise order of mappings, since in the necessity proof of Theorem 3.
Proof Under the assumptions we obtain for arbitrary In what follows we describe a way of doing that for finite H and G which might be implemented on a computer. We start to determine a finite subset T of (X ∪ X −1 ) * satisfying the following property: For each w ∈ (X ∪ X −1 ) * there is v ∈ T such that [w] = [v] and Γ(⟨v⟩) ⊆ Γ(⟨w⟩) . To compute such a set T we describe a simple algorithm which directly implements the defining property of T.
(1) For T, constructed so far, construct a superset T ′ of T in the following way: Put all elements of T into T ′ . List the elements of T × X × {−1, 1} and check . If the answer for a given (w, x, ) is yes, go to the next triple in the list. If the answer is no, put wx into T ′ and go to the next triple in the list. (2) If T is a proper subset of T ′ , as constructed in (1), take T ′ as new T and start (1) again. If T = T � the algorithm stops.
Note that since H and M(G) are finite, the computation stops after a finite number of steps. To see that in the end T has the required property, we note that if a word w ′ is dropped in (1) of the above algorithm because , where uv is in T or has been dropped earlier in (1) Even for a small finite noncyclic X-generated group G, an X-generated group H admitting a canonical dual prehomomorphism ∶ H → M(G) might be large. The following theorem points into this direction.
Since the intersection of both graphs does not contain a connected subgraph having at least one edge and vertex 1 G , we conclude that Γ([u] ) = � , whence ([u]) = 1 M(G) . We infer and on the other hand ([x 2 y −1 x]) ≥ ⟨x 2 y −1 x⟩ which means with contradiction, since the intersection on the right hand side does not contain a connected subgraph with vertices 1 G and x 2 y −1 x = gh .
It is an open question whether the finite group B(2; 6) admits a canonical dual prehomomorphism into M(G) with G the Klein four-group, or a contradiction can be achieved following the pattern in the proof of Theorem 3.4. It is also an open question whether the group G U , as defined in [1], with U the variety of all groups of exponent n = 3 , respectively n = 4 , admits a suitable mapping ∶ G U → M(G) in this case. In our setting G U may be represented by FG({x, y}) ∕≡ , where ≡ is the congruence on the free group FG({x, y}) generated by the relators w 3 = 1 , respectively FG({x, y}) . Since, by construction in [1], G U is a subgroup of a semidirect product of the finite groups B(8; 3), respectively B(8; 4) by G, it is finite. Obviously B(2; 4) is a homomorphic image of G U in case n = 4 . However B(2; 4) itself is not of the form G V for some group variety V, since the only possible choice of such V would be the variety of elementary Abelian 2-groups. Only if V has exponent 2, the group G V has exponent 2 ⋅ 2 = 4 . But in this case G V is a subgroup of a semidirect product of the free elementary Abelian 2-group of rank 8 by G whence |G V | < 2 8 ⋅ 2 2 = 2 10 < 2 12 = |B(2;4)| . Note in particular that G U has exponent 6 in case n = 3 , and exponent 8 in case n = 4 . Anyway it follows from [1], Proposition 4.4., referring to a remark of V. Guba, that ∶ G U → M(G) exists if U is the variety of all groups of sufficiently large odd exponent n.
We continue our considerations with a theorem which also follows from a result of Szakács [6]. For sake of completeness we give an elementary direct proof.

Theorem 3.5 Let G be an X-generated noncyclic group, and let H be a generator respecting X-generated extension by G such that the homomorphism H → G , defined by [w] ↦ w has a nontrivial Abelian kernel K. Then there is no canonical dual prehomomorphism ∶ H → M(G).
Proof We show first that under the assumptions Γ(G) contains a subgraph consisting of two disjoint cycles connected by a path, of the form where u 1 , u 2 , v, z 1 , z 2 are nonempty words in (X ∪ X −1 ) * , labeling the respective paths. Assume first that there is y ∈ X such that y has finite order m ≥ 2 . Since G is noncyclic there is x ∈ X such that x ≠ y n , for all n ∈ ℕ . Consequently, by use of the  (y, x) . Since A is a subgroup of G and B = yxA , with yx ∉ A by assumption, we obtain A ∩ B = �. Assume now that there is x ∈ X , such that x has infinite order. Since K is nontrivial there is a nonempty reduced word w = y 1 … y m , m ≥ 2 , with y i ∈ X ∪ X −1 , 1 ≤ i ≤ m , such that w = 1 G , and Γ(⟨w⟩) forms a cycle. Let u 1 = y 1 = z 1 , u 2 = (y 2 ⋯ y m ) −1 , z 2 = y 2 ⋯ y m , and v � = x n , where n is such that the equality y 1 x k a = b can only hold for at most one k ∈ ℕ by the assumption that x has infinite order, and the set A, whence A × A , is finite. We may use u 1 , u 2 , v ′ , z 1 , z 2 to define a graph which consists of the two disjoint cycles with vertex sets A = {1 G , y 1 , … , y 1 ⋯ y m−1 } and B = {y 1 x n , y 1 x n y 1 , … , y 1 x n y 1 ⋯ y m−1 } , connected by the path with initial vertex y 1 labeled by v � = x n . Let p, q ∈ {1, … , n} such that p is the least element with y 1 x p ∉ A for all k, p ≤ k ≤ n , and such that q is the least element with y 1 x q ∈ B . Then the path with initial vertex y 1 x p−1 and label v = x q−p+1 connects the cycles with vertex sets A and B precisely as shown in the graph above. We conclude On the other hand we obtain Hence for any canonical dual prehomomorphism ∶ H → M(G) we get It is shown in [2], see also Example 3.2, that for any {x}-generated cyclic group G of order n there is a canonical dual prehomomorphism from the {x}-generated cyclic group H of order 2n into M(G). Clearly the homomorphism [w] ↦ w has Abelian kernel. If we regard, however, G as an e.g. {x, y}-generated group where y = x 2 , say, n ≥ 3 , then the assertion of Theorem 3.5 remains true, although G is cyclic, as the following example shows for n = 3.

Example 3.3
Let G = {1 G , g, g 2 } be the three element cyclic group generated by X = {x, y} with respect to x = g, y = g 2 . The Cayley graph Γ(G) looks as follows:  In particular Γ(⟨x⟩) is a breaking path in her terminology.
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