Comparing Numbers of Diagonal Subsemigroups and Congruences for Semigroups

📝 Abstract

Given a semigroup $S $, a diagonal subsemigroup $ρ$ is defined to be a reflexive and compatible relation on $S $, i.e. a subsemigroup of the direct square $S\times S$ containing the diagonal $\{ (s,s)\colon s\in S\} $. When $S$ is finite, we define the DSC coefficient $χ(S)$ to be the ratio of the number of congruences to the number of diagonal subsemigroups. In a previous work we observed that $χ(S) = 1$ if and only if $S$ is a group. Here we show that for any rational $α$ with $0 < α\leq 1 $, there exists a semigroup with $χ(S) = α $. We do this by utilizing the Rees matrix construction and adapting the congruence classification of such semigroups to describe their diagonal subsemigroups.

💡 Analysis

Given a semigroup $S $, a diagonal subsemigroup $ρ$ is defined to be a reflexive and compatible relation on $S $, i.e. a subsemigroup of the direct square $S\times S$ containing the diagonal $\{ (s,s)\colon s\in S\} $. When $S$ is finite, we define the DSC coefficient $χ(S)$ to be the ratio of the number of congruences to the number of diagonal subsemigroups. In a previous work we observed that $χ(S) = 1$ if and only if $S$ is a group. Here we show that for any rational $α$ with $0 < α\leq 1 $, there exists a semigroup with $χ(S) = α $. We do this by utilizing the Rees matrix construction and adapting the congruence classification of such semigroups to describe their diagonal subsemigroups.

📄 Content

One of the most fundamental concepts in algebra is that of a congruence. Given a semigroup S a congruence on S is an equivalence relation ρ that is compatible with the multiplication, i.e. (x, y), (z, t) ∈ ρ =⇒ (xz, yt) ∈ ρ. It is easy to see that every congruence is a subsemigroup of the direct square S × S. A more general type of subsemigroup of S × S is a diagonal subsemigroup, which is a reflexive relation that is compatible with the multiplication of the semigroup. Our previous paper on this topic [1] explored semigroups with the property that every diagonal subsemigroup is a congruence. We call such a semigroup DSC. The starting point in that paper is the following foundational observation: Theorem 1.1. Let S be a finite semigroup. Then S is DSC if and only if S is a group.

For infinite semigroups things are more interesting in several ways. Periodic groups are still DSC, but the additive group of integers is not DSC. We actually currently do not know whether there exists a DSC group that contains an element of infinite order. Furthermore, in [1] we showed that there exists DSC semigroups that are not groups.

We can re-phrase Theorem 1.1 as follows. Let Cong(S) be the set of congruences on S and Diag(S) the set of diagonal subsemigroups on S. When S is finite both of these sets are finite. We define the DSC coefficient of such S to be:

Clearly χ(S) ∈ Q ∩ (0, 1]. Then Theorem 1.1 asserts that:

One can think of the DSC coefficient as a measure of how close to being DSC a semigroup is and wonder what values χ(S) takes as S ranges over all finite semigroups. The main aim of this paper is to prove the following result:

Theorem 1.2. For any α ∈ Q∩(0, 1] there is a finite semigroup S with χ(S) = α.

To prove this result we will use the Rees matrix construction. This construction was originally introduced by Suschkewitsch who employed it to give a full structural description of completely simple semigroups [9]. Here we take a group, G, two index sets I and Λ and a Λ × I matrix P with entries from G. The Rees matrix semigroup M[G; I, Λ; P ] is the set I × G × Λ with multiplication:

(i, g, λ)(j, h, µ) = (i, gp λj h, µ).

The main tool we will use to prove Theorem 1.2 is characterizing the diagonal subsemigroups on a Rees matrix semigroup. There is a description of the congruences on a Rees matrix semigroup in terms of normal subgroups of the group and equivalence relations on the index sets, originally due to Preston [7]. It turns out that there is an analogous description for the diagonal subsemigroups on a Rees matrix semigroup, namely we can describe them using normal subgroups of the group and reflexive relations on the index sets, with the caveat that the group must be DSC.

The paper is structured as follows. In section 2 we give the characterization of the diagonal subsemigroups on a Rees matrix semigroup. In Section 3 we use this characterization to evaluate the DSC coefficient for a Rees matrix semigroup, and show that we can obtain any rational number between zero and one as the DSC coefficient of a Rees matrix semigroup. In the final section we make some observations about the DSC coefficient of Clifford semigroups and contrast this to the result for Rees matrix semigroups.

We will only require some basic concepts from semigroup theory which will be introduced as they are needed. For this paper we will use N to denote the set {1, 2, 3 . . . } and N 0 = N ∪ {0}. Given a set X we will let ∆ X = {(x, x) : x ∈ X} be the diagonal relation on X. For a subset ρ of X × Y , we will write ρ -1 for the set {(y, x) : (x, y) ∈ ρ}.

Let S be a Rees matrix semigroup M[G; I, Λ; P ] over the group G. We start this section by reviewing the congruence description for Rees matrix semigroups; for a more detailed exploration see [4,Section 3.5]. For any i, j ∈ I and λ, µ ∈ Λ we define an extract of the matrix P to be q λµij = p λi p -1 µi p µj p -1 λj . We will be interested in triples of the form (N, S, T ), where N is a normal subgroup of G, S is a relation on I and T is a relation on Λ. A triple is linked if for all i, j ∈ I and λ, µ ∈ Λ:

And a triple is an equivalence triple if the relations S, T are equivalence relations. Normally in the literature linked equivalence triples are simply called linked triples. However, as we will need another type of triple, where the relations are just reflexive, we will use the full term linked equivalence triple.

If we have a relation ρ on S, then we can get a triple (N ρ , ρ I , ρ Λ ) where:

If (N, S, T ) is a triple, we define a relation ρ N,S,T by: (i, g, λ) ρ N,S,T (j, h, µ) ⇐⇒ (i, j) ∈ S, (λ, µ) ∈ T ,

Then we have the following result due to Preston [7] that gives a direct correspondence between the congruences on a Rees matrix semigroup and the collection of linked equivalence triples:

Theorem 2.1. If ρ is a congruence then (N ρ , ρ I , ρ Λ ) is a linked equivalence triple and if (N, S, T ) is a linked equivalence triple then ρ N,S,T is a congruence. Furthermore th

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