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A band is an idempotent semigroup. Every commutative band is a semilattice and uniquely determined by the left and right identity sets of its elements or equivalently by the left and right zero sets of its elements. We generalize this notion by defining a groupoid or semigroup to be stabilized with respect to binary relations, in particular the binary relations defined by the one-sided identity and zero sets of its elements, if and only if for any groupoid or semigroup on the same set with the same binary relations, their binary operations are identical. We prove every right group with maximal subgroup size $2$ is a stabilized semigroup with respect to the one-sided identity [zero] sets of its elements. We define a commutative-rectangular band to be a band in which every pair of elements either commutes or are generalized inverses of each other, and we prove a commutative-rectangular band is a stabilized semigroup with respect to the one-sided identity [zero] sets of its elements.
| Subjects: | Group Theory (math.GR) |
| MSC classes: | 20M10 |
| Cite as: | arXiv:2410.23473 [math.GR] |
| (or arXiv:2410.23473v5 [math.GR] for this version) | |
| https://doi.org/10.48550/arXiv.2410.23473 arXiv-issued DOI via DataCite |
From: Julia Maddox [view email]
[v1]
Wed, 30 Oct 2024 21:36:45 UTC (9 KB)
[v2]
Wed, 5 Mar 2025 04:18:13 UTC (9 KB)
[v3]
Tue, 31 Mar 2026 01:03:34 UTC (15 KB)
[v4]
Fri, 17 Apr 2026 05:36:47 UTC (15 KB)
[v5]
Mon, 25 May 2026 15:31:43 UTC (16 KB)
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