Generating Self-Map Monoids of Infinite Sets

Abstract

Let Ω be a countably infinite set, S = Sym(Ω) the group of permutations of Ω, and E = Self(Ω) the monoid of self-maps of Ω. Given two subgroups G 1 , G 2 ⊆ S, let us write G 1 ≈ S G 2 if there exists a finite subset U ⊆ S such that the groups generated by G 1 ∪ U and G 2 ∪ U are equal. Bergman and Shelah showed that the subgroups which are closed in the function topology on S fall into exactly four equivalence classes with respect to ≈ S . Letting ≈ denote the obvious analog of ≈ S for submonoids of E, we prove an analogous result for a certain class of submonoids of E, from which the theorem for groups can be recovered. Along the way, we show that given two subgroups G 1 , G 2 ⊆ S which are closed in the function topology on S, we have GLet Ω be an arbitrary infinite set, and set E = Self(Ω), the monoid of self-maps of Ω. Elements of E will be written to the right of their arguments. If U ⊆ E is a subset, then we will write U to denote the submonoid generated by U. The cardinality of a set Γ will be denoted by |Γ|. If Σ ⊆ Ω and U ⊆ E are subsets, letDefinition 1. Let M be a monoid that is not finitely generated. Then the cofinality c(M) of M is the least cardinal κ such that M can be expressed as the union of an increasing chain of κ proper submonoids.The main goal of this section is to show that c(E) > |Ω|, which will be needed later on.This section is modeled on Sections 1 and 2 of [2], where analogous statements are proved for the group of all permutations of Ω. Ring-theoretic analogs of these statements are proved in [10].Lemma 2. Let U ⊆ E and Σ ⊆ Ω be such that |Σ| = |Ω| and the set of self-maps of Σ induced by U {Σ} is all of Self(Σ). Then E = gUh, for some g, h ∈ E.Proof. Let g ∈ E be a map that takes Ω bijectively to Σ, and let h ∈ E be a map whose restriction to Σ is the right inverse of g. Then E = gU {Σ} h.We will say that Σ ⊆ Ω is a moiety if |Σ| = |Ω| = |Ω\Σ|. A moiety Σ ⊆ Ω is called full with respect to U ⊆ E if the set of self-maps of Σ induced by members of U {Σ} is all of Self(Σ). The following two results are modeled on group-theoretic results in [9].Lemma 3 (cf. [2, Lemma 3]). Let (U i ) i∈I be any family of subsets of E such that i∈I U i = E and |I| ≤ |Ω|. Then Ω contains a full moiety with respect to some U i .Proof. Since |Ω| is infinite and |I| ≤ |Ω|, we can write Ω as a union of disjoint moieties Σ i , i ∈ I. Suppose that there are no full moieties with respect to U i for any i ∈ I. Then in particular, Σ i is not full with respect to U i for any i ∈ I. Hence, for every i ∈ I there exists a map f i ∈ Self(Σ i ) which is not the restriction to Σ i of any member of (U i ) {Σ i } . Now, if we take f ∈ E to be the map whose restriction to each Σ i is f i , then f is not in U i for any i ∈ I, contradicting i∈I U i = E. Proposition 4. c(E) > |Ω|.Proof. Suppose that (M i ) i∈I is a chain of submonoids of E such that i∈I M i = E and |I| ≤ |Ω|. We will show that E = M i for some i ∈ I.By the preceding lemma, Ω contains a full moiety with respect to some M i . Thus, Lemma 2 implies that E = M i ∪ ...

The Bergman property for semigroups

“…Mesyan [19,Proposition 4] proved that cf(Self(Ω)) > ℵ 0 using an elementary diagonal argument, and an alternative proof of Theorem 4.1 can be obtained using a similar argument. In Galvin [11] it was shown that the symmetric group on an infinite set is strongly distorted.…”

Endomorphism rings generated using small numbers of elements

“…This equivalence relation and the results of this section are modeled on those in [2], where Bergman and Shelah define an analogous relation for groups and classify into equivalence classes the subgroups of the group of permutations of a countably infinite set that are closed in the function topology. Properties of such an equivalence relation defined for submonoids of the monoid of self-maps of an infinite set are investigated in [8].…”

The Bergman‐Shelah preorder on transformation semigroups

“…In [10] it was proved that contains at least two incomparable elements by constructing a subsemigroup S of N N such that S F 3 and F 3 S. In Section 4 we gave an example of a subsemigroup incomparable to all F n . The following theorem shows that there are anti-chains in of arbitrary finite length.…”