On lattice of congruences on graph inverse semigroups

Abstract

Congruences on a graph inverse semigroup were recently described in terms of the underline graph. Based on such descriptions, we show that the lattice of congruences on a graph inverse semigroup is upper semimodular but not lower semimodular.

Properties of Congruence Lattices of Graph Inverse Semigroups

“…Proposition 3.1 (cf. Lemma 2.18 in [10] and Proposition 1.2 in [6]). Let E be a finite acyclic digraph, and let ρ(H 1 , W 1 ) and ρ(H 2 , W 2 ) be any congruences on G(E).…”

“…In this section we present a number of results from [6] and [10], and some refinements of our own, that are central to the current article.…”

“…There have been a number of more recent papers specifically about graph inverse semigroups also; see, for example, [7]. Of particular, relevance here are the papers of Wang [10], and Luo and Wang [6]. In the first paper [10], Wang completely characterises the congruences of a graph inverse semigroup G(E) in terms of E. This characterisation is used to show that the lattice L(G(E)) of every graph inverse semigroup G(E) is Noetherian, i.e.…”

“…G(E) does not have any infinite strictly ascending chains of congruences. In the second paper [6], Luo and Wang show that the lattice of congruences of a graph inverse semigroup is always upper-semimodular (Proposition 3.2), but not lower-semimodular in general. Luo and Wang's result in [6] follows in a long tradition of determining the lattice theoretic properties of lattices naturally associated to various algebraic objects.…”

“…In the second paper [6], Luo and Wang show that the lattice of congruences of a graph inverse semigroup is always upper-semimodular (Proposition 3.2), but not lower-semimodular in general. Luo and Wang's result in [6] follows in a long tradition of determining the lattice theoretic properties of lattices naturally associated to various algebraic objects. For example, among many others, the lattice of normal subgroups of a group is modular but not distributive in general.…”