Inverse subsemigroups of finite index in finitely generated inverse semigroups

Abstract

Let S be an inverse semigroup with semilattice of idempotents E(S). Recall that the natural partial order on S is defined by s t ⇐⇒ there exists e ∈ E(S) such that s = et .A subset A ⊆ S is closed if, whenever a ∈ A and a s, then s ∈ A. The closure B ↑ of a subset B ⊆ S is defined asAn atlas in S is a subset A ⊆ S such that AA −1 A ⊆ A: that is, A is closed under the heap ternary operation a, b, c = ab −1 c (see [3]). Since, for all a ∈ A we have a, a, a = a, we see that A is an atlas if and only ifProposition 2.2. [18, Proposition 5.] A coset C that contains an idempotent e ∈ E(S) is an inverse subsemigroup of S, and in this case C = (CC −1 ) ↑ . Proof. If a, b ∈ C then ab aeb = a, e, b ∈ C and since C is closed, we have ab ∈ C. Furthermore, a −1 ea −1 e = e, a, e ∈ C and so a −1 ∈ C. Hence C is an inverse subsemigroup. Now ab −1 ∈ CC −1 and ab −1 ab −1 e = a, b, e ∈ C. Since C is closed we have (CC −1 ) ↑ ⊆ C. But if x ∈ C then x xe ∈ CC −1 and so x ∈ (CC −1 ) ↑ . Therefore C = (CC −1 ) ↑ .

Closed inverse subsemigroups of graph inverse semigroups

“…The closed inverse submonoids of free inverse monoids were completely described by Margolis and Meakin in [9]. For other related work on inverse subsemigroups of finite index, see [2] and the first author's PhD thesis [1].…”