Closed inverse subsemigroups of graph inverse semigroups

Abstract

As part of his study of representations of the polycylic monoids, M.V. Lawson described all the closed inverse submonoids of a polycyclic monoid Pn and classified them up to conjugacy. We show that Lawson's description can be extended to closed inverse subsemigroups of graph inverse semigroups. We then apply B. Schein's theory of cosets in inverse semigroups to the closed inverse subsemigroups of graph inverse semigroups: we give necessary and sufficient conditions for a closed inverse subsemigroup of a graph inverse semigroup to have finite index, and determine the value of the index when it is finite.

Congruences on graph inverse semigroups

“…The definition of graph inverse semigroups based on general directed graphs can be found in [7,8,12]. These inverse semigroups are related to various types of algebras, for instance, C * -algebras, Cohn path algebras, Leavitt path algebras and so on (see [1,2,[8][9][10][11]14]). Graph inverse semigroups have also been studied in their own right in recent years (see [2,5,7,8,12,13]).…”

Embedding of graph inverse semigroups into CLP-compact topological semigroups

“…Graph inverse semigroups play an important role in the study of rings and C * -algebras (see [1,4,17,28,35]). Algebraic theory of graph inverse semigroups is well developed (see [2,8,10,26,27,30,32]). Topological properties of graph inverse semigroups were investigated in [9,11,12,13,33].…”

“…(1) there exists cardinal k such that E = (⊔ α∈k E α ) ⊔ F where graph F is acyclic, for each vertex f ∈ F 0 the set I f is finite and for each α ∈ k the graph E α consists of one vertex and one loop; (2) if the graph F is non-empty, then the semigroup G(F ) is a compact subset of G(E); Consider statement (2). Observe that the set E(G(E)) is open and closed in S. By Lemma 2 the set E(G(E)) is d-compact.…”

An alternative look at the structure of graph inverse semigroups

“…Graph inverse semigroups play an important role in the study of rings and C * -algebras (see [1,3,12,18,25]). Algebraic theory of graph inverse semigroups is well developed (see [2,5,16,17,20,22]). Topological properties of graph inverse semigroups were investigated in [6,7,8,23].…”