The Bergman property for semigroups

Abstract

In this article, we study the Bergman property for semigroups and the associated notions of cofinality and strong cofinality. A large part of the paper is devoted to determining when the Bergman property, and the values of the cofinality and strong cofinality, can be passed from semigroups to subsemigroups and vice versa.Numerous examples, including many important semigroups from the literature, are given throughout the paper. For example, it is shown that the semigroup of all mappings on an infinite set has the Bergman property but that its finitary power semigroup does not; the symmetric inverse semigroup on an infinite set and its finitary power semigroup have the Bergman property; the Baer-Levi semigroup does not have the Bergman property.

Generation of infinite factorizable inverse monoids

“…Recall from [15] that the Sierpiński rank of S, which we denote by SR(S), is the smallest integer n (if it exists) for which any countable subset of S is contained in an n-generated subsemigroup; if no such n exists, we say S has infinite Sierpiński rank and write SR(S) = ∞. Finally, following [21], we say that S has the semigroup Bergman property if the length function of S is bounded with respect to any generating set; that is, if for any generating set U of S we have S = U ∪ U 2 ∪ · · · ∪ U n for some n (which will typically vary, depending on the choice of U ). Clearly finite semigroups have finite Sierpiński rank (equal to the rank of the semigroup), and satisfy the semigroup Bergman property; it is in the setting of infinite semigroups that these notions become more interesting.…”

Katětov Functors

“…We prove that in the presence of a Katětov functor, additional mild assumptions ensure that the endomorphism monoid End(L) of the Fraïssé limit L is strongly distorted and its Sierpiński rank is at most five. Applying a result from [25], we conclude that if End(L) is not finitely generated, then it has the Bergman property. The constructions we present here are rather technical, and the reader might find it helpful to get acquainted with the constructions from [8] before going on.…”

“…Following [25], we say that a semigroup S is semigroup Cayley bounded with respect to a generating set U if S = U ∪ U 2 ∪ . .…”

Infinite partition monoids

“…Recall that a semigroup S has the semigroup Bergman property [34] if the length function of S is bounded with respect to any generating set for S. The property has this name since Bergman showed in [6] that an infinite symmetric group S X has the property. (Actually, Bergman showed that S X has this property with respect to group generating sets of S X , and the semigroup analogue was shown in [34,Corollary 2.5].…”

“…The property is so named because of the seminal paper of Bergman [6], in which it was shown that the infinite symmetric groups have this property; in fact, Bergman showed that infinite symmetric groups have the corresponding property with respect to group generating sets, and the semigroup analogue was proved in [34]. Further studies have investigated the semigroup Bergman property in the context of other transformation semigroups [10,34,36].…”