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Normality of marginal inverse subsemigroups in inverse semigroups
Abstract
For each pair of integers m≥ 1 and n≥ 2, we construct a finite inverse semigroup Sm,n⊆ Mm(ℂ) and explicitly determine its marginal inverse subsemigroup Mxn(Sm,n) associated with the word xn. We prove that the three conditions ∼xn=≡xn on Sm,n, Mxn(Sm,n)trianglelefteq Sm,n, and m≤ n are equivalent. As a consequence, we answer Problem 11 of Araújo, Araújo and Kinyon in [1]. Moreover, for a word ω in any inverse semigroup S, we prove that ∼ω=≡ω if and only if Mω(S)trianglelefteq S and ∼ω is invariant under inversion.
Related
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- Cartan inverse semigroups and twisted Steinberg algebras
- Covariant representations of actions of inverse semigroups: a new approach to the reduced and essential crossed-product C*-algebras
BibTeX
@misc{duan2026normality,
title = {{Normality of marginal inverse subsemigroups in inverse semigroups}},
author = {Bing Duan and Jizhi Zhang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05078},
eprint = {2611.05078},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05078},
}
RIS
TY - UNPB
TI - Normality of marginal inverse subsemigroups in inverse semigroups
AU - Bing Duan
AU - Jizhi Zhang
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05078
UR - https://arxiv.org/abs/2611.05078
AB - For each pair of integers m≥ 1 and n≥ 2, we construct a finite inverse semigroup S_m,n⊆ M_m(ℂ) and explicitly determine its marginal inverse subsemigroup M_x^n(S_m,n) associated with the word x^n. We prove that the three conditions ∼_x^n=≡_x^n on S_m,n, M_x^n(S_m,n)trianglelefteq S_m,n, and m≤ n are equivalent. As a consequence, we answer Problem 11 of Araújo, Araújo and Kinyon in [1]. Moreover, for a word ω in any inverse semigroup S, we prove that ∼_ω=≡_ω if and only if M_ω(S)trianglelefteq S and ∼_ω is invariant under inversion.
ER -