Properties of congruence lattices of graph inverse semigroups

Marina Anagnostopoulou-Merkouri; Zachary Mesyan; James Mitchell

doi:10.1142/S0218196724500139

Properties of congruence lattices of graph inverse semigroups

Marina Anagnostopoulou-Merkouri, Zachary Mesyan, James Mitchell^*

^*Corresponding author for this work

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Abstract

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

Original language	English
Article number	3
Pages (from-to)	371-396
Number of pages	26
Journal	International Journal of Algebra and Computation
Volume	34
Issue number	3
DOIs	https://doi.org/10.1142/S0218196724500139
Publication status	Published - 20 Apr 2024

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© 2024 World Scientific Publishing Company.

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title = "Properties of congruence lattices of graph inverse semigroups",

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

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T1 - Properties of congruence lattices of graph inverse semigroups

AU - Anagnostopoulou-Merkouri, Marina

AU - Mesyan, Zachary

AU - Mitchell, James

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Y1 - 2024/4/20

N2 - From any directed graph E one can construct the graph inverse semigroup G(E)𝐺(𝐸), whose elements, roughly speaking, correspond to paths in E. Wang and Luo showed that the congruence lattice L(G(E))𝐿(𝐺(𝐸)) of G(E)𝐺(𝐸) is upper-semimodular for every graph E, but can fail to be lower-semimodular for some E. We provide a simple characterization of the graphs E for which L(G(E))𝐿(𝐺(𝐸)) is lower-semimodular. We also describe those E such that L(G(E))𝐿(𝐺(𝐸)) is atomistic, and characterize the minimal generating sets for L(G(E))𝐿(𝐺(𝐸)) when E is finite and simple.

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

U2 - 10.1142/S0218196724500139

DO - 10.1142/S0218196724500139

M3 - Article (Academic Journal)

SN - 0218-1967

VL - 34

SP - 371

EP - 396

JO - International Journal of Algebra and Computation

JF - International Journal of Algebra and Computation

IS - 3

M1 - 3

ER -

Properties of congruence lattices of graph inverse semigroups

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