Green index and finiteness conditions for semigroups

Abstract

Let S be a semigroup and let T be a subsemigroup of S. Then T acts on S by left and by right multiplication. If the complement S \ T has finitely many strong orbits by both these actions we say that T has finite Green index in S. This notion of finite index encompasses subgroups of finite index in groups, and also subsemigroups of finite Rees index (complement). Therefore, the question of S and T inheriting various finiteness conditions from each other arises. In this paper we consider and resolve this question for the following finiteness conditions: finiteness, residual finiteness, local finiteness, periodicity, having finitely many right ideals, and having finitely many idempotents.

Ends of semigroups

“…As K × E has finite Green index in G × E it must follow that K has finite Green index in G. It was shown in [6,Corollary 34] that if K is a subsemigroup of finite Green index in a group G then K is a subgroup of G with finite group index. ∈ S and a 1 , a 2 , . .…”

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.…”

“…There is a much better notion of index of a subsemigroup T in a semigroup S-the Green index, recently studied in [8], which generalizes both Rees and group indices. To define it we need the following relations on S. For x, y ∈ S we will say that xR T y if xT 1 = yT 1 , and xL T y if T 1 x = T 1 y.…”

Hopfian And Co-Hopfian Subsemigroups And Extensions

“…Each of these relations is an equivalence relation on S. When T = S, they coincide with the standard Green's relations on S. Furthermore, these relations respect T , in the sense that each R T -, L T -, and H T -class lies either wholly in T or wholly in S − T . Following [GR08], define the Green index of T in S to be one more than the number of H T -classes in S − T . If S and T are groups, then T has finite group index in S if and only if it has finite Green index in S [GR08, Proposition 8].…”

“…Rees index is more established, and many finiteness properties, such as finite generation and finite presentability, are known to be preserved on passing to or from subsemigroups of finite Rees index; see the brief summary in [Ruš98,§ 11] or the comprehensive survey [CM]. Green index is newer, but has the advantage that finite Green index is a common generalization of finite Rees index and finite group index, and some progress has been made in proving the preservation of finiteness properties on passing to or from subsemigroups of finite Green index; see [CGR12,GR08].…”