The lattice of varieties of semigroups with completely regular square. II
Abstract
We solve (modulo groups) the word problem for free semigroup satisfying xy = (xy)n. This enables us to show that the variety cal SCRn given by this identity is not equal to the join of the varieties calCRn and calSI defined by the identities x = xn and xy = (xy)2 respectively. Therefore the result of (M.V.Volkov, T.A.Ershova, The lattice of varieties of semigroup with completely regular square, Monash Conference on Semigroup Theory in Honour of G. B. Preston, World Scientific, Singapore, 1991, 306-322) that the lattice L(cal SCRn) is modular does not follow from the earlier results concerning L(calCRn) and L(calSI). However the variety calSOn consisting of all semigroups from cal SCRn whose idempotents form a subsemigroup turns out to be equal to the join of the varieties cal On of all orthogroups and calSI. We deduce from this a description of the lattice L(calSOn) modulo groups.
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BibTeX
@misc{ershova2026lattice,
title = {{The lattice of varieties of semigroups with completely regular square. II}},
author = {T. A. Ershova and M. V. Volkov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34588},
eprint = {2609.34588},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34588},
}
RIS
TY - UNPB
TI - The lattice of varieties of semigroups with completely regular square. II
AU - T. A. Ershova
AU - M. V. Volkov
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34588
UR - https://arxiv.org/abs/2609.34588
AB - We solve (modulo groups) the word problem for free semigroup satisfying xy = (xy)^n. This enables us to show that the variety cal SCR_n given by this identity is not equal to the join of the varieties calCR_n and calSI defined by the identities x = x^n and xy = (xy)^2 respectively. Therefore the result of (M.V.Volkov, T.A.Ershova, The lattice of varieties of semigroup with completely regular square, Monash Conference on Semigroup Theory in Honour of G. B. Preston, World Scientific, Singapore, 1991, 306-322) that the lattice L(cal SCR_n) is modular does not follow from the earlier results concerning L(calCR_n) and L(calSI). However the variety calSO_n consisting of all semigroups from cal SCR_n whose idempotents form a subsemigroup turns out to be equal to the join of the varieties cal O_n of all orthogroups and calSI. We deduce from this a description of the lattice L(calSO_n) modulo groups.
ER -