Ideals and Finiteness Conditions for Subsemigroups

Abstract

In this paper we consider a number of finiteness conditions for semigroups related to their ideal structure, and ask whether such conditions are preserved by sub-or supersemigroups with finite Rees or Green index. Specific properties under consideration include stability, D = J and minimal conditions on ideals.2010 Mathematics Subject Classification. 20M05, 20M12.

On hopfian cofinite subsemigroups

“…The related question of the existence of finite complete rewriting system has been settled only very recently, see [16]. Some related cohomological finiteness conditions are considered in [11] and [15], residual finiteness is treated in [13], and some surprising behaviour in relation to the ideal structure is exhibited in [3].…”

Ends for subsemigroups of finite index

“…Moreover, subsemigroups of finite Green index preserve finiteness and finite generation (the question of finite presentability is still open). For more details about Green index see [3,8,9]. The analog of Theorem 4.1 for Green index does not hold for arbitrary semigroups, as can be seen from the following example: Proof That T has Green index 2 in S is obvious.…”

Growths of Endomorphisms of Finitely Generated Semigroups

“…In this subsection, we show that the Green index serves as a better analogy of the group index [20] and gives us equality, directly generalizing the result for groups. The Green index, which was introduced in [20], has proven to be a very useful generalization of both the grouptheoretic notion of index and the more established Rees index for semigroups, and has yielded many Reidemeister-Schreier-type theorems about the inheritance of various finiteness properties by subsemigroups or extensions of finite index; see, for example, [7,10,19,20,24,28]. We recall the definition here: let T be a subsemigroup of a semigroup S .…”