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Properties of Congruence Lattices of Graph Inverse Semigroups

Marina Anagnostopoulou-Merkouri, Zak Mesyan, James D. Mitchell

Abstract

From any directed graph $E$ one can construct the graph inverse semigroup $G(E)$, whose elements, roughly speaking, correspond to paths in $E$. Wang and Luo showed that the congruence lattice $L(G(E))$ of $G(E)$ is upper-semimodular for every graph $E$, but can fail to be lower-semimodular for some $E$. We provide a simple characterisation of the graphs $E$ for which $L(G(E))$ is lower-semimodular. We also describe those $E$ such that $L(G(E))$ is atomistic, and characterise the minimal generating sets for $L(G(E))$ when $E$ is finite and simple.

Properties of Congruence Lattices of Graph Inverse Semigroups

Abstract

From any directed graph one can construct the graph inverse semigroup , whose elements, roughly speaking, correspond to paths in . Wang and Luo showed that the congruence lattice of is upper-semimodular for every graph , but can fail to be lower-semimodular for some . We provide a simple characterisation of the graphs for which is lower-semimodular. We also describe those such that is atomistic, and characterise the minimal generating sets for when is finite and simple.

Paper Structure

This paper contains 11 sections, 18 theorems, 89 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Definitions and statements of main results
  3. Graphs
  4. Inverse semigroups
  5. Lattices
  6. Congruence lattices on graph inverse semigroups
  7. Main results
  8. Meets, joins, and covers
  9. Modularity
  10. Atoms and atomistic congruence lattices
  11. Generating congruences

Key Result

Proposition 2.1

Let $E$ be a graph, let $G(E)$ be the graph inverse semigroup of $E$, and let $L(G(E))$ be the lattice of congruences on $G(E)$. The function $(H, W, f) \mapsto \varrho(H, W, f)$ is an order-isomorphism between the set of all Wang triples on $E$, ordered by $\leq$, and $L(G(E))$, ordered by containm

Figures (5)

  • Figure 1: Every connected simple graph $E$ with $4$ vertices, such that $L(G(E))$ is lower-semimodular, together with the corresponding lattice $L(G(E))$.
  • Figure 2: The diamond lattice $\mathfrak{M}_3$ and the pentagon lattice $\mathfrak{N}_5$.
  • Figure 3: A graph $E$, together with $L(G(E))$, which is not lower-semimodular. The vertices of $L(G(E))$ shown in orange are covered by their join, shown in blue, but they do not cover their meet, shown in purple.
  • Figure 4: An example of a graph satisfying the conditions of \ref{['thm-atomistic']}.
  • Figure 5: A graph $E$ with parallel edges, for which the conclusion of \ref{['thm-generators']} does not hold.

Theorems & Definitions (32)

  • Proposition 2.1: Proposition 1.2 in Luo2021aa
  • Proposition 2.2: Theorem 1.3 in Luo2021aa
  • Theorem 2.3
  • Corollary 2.4
  • Theorem 2.5
  • Corollary 2.6
  • Theorem 2.7
  • Proposition 3.1: Lemmas 2.7 and 2.8 in Luo2021aa
  • Proposition 3.2
  • proof
  • ...and 22 more