具有完全正则平方的半群簇的格 II
The lattice of varieties of semigroups with completely regular square. II
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中文总结 AI 辅助
本文在模群意义下求解满足 $xy = (xy)^n$ 的自由半群词问题，证明簇 ${\cal {SCR}}_n$ 不等于 ${\cal{CR}}_n$ 与 ${\cal{SI}}$ 的并，并刻画 ${\cal{SO}}_n$ 的格结构。
中文摘要 AI 辅助
我们（在模群的意义下）求解了满足 $xy = (xy)^n$ 的自由半群的词问题。这使得我们能够证明由该恒等式定义的簇 ${\cal {SCR}}_n$ 不等于由恒等式 $x = x^n$ 和 $xy = (xy)^2$ 分别定义的簇 ${\cal{CR}}_n$ 与 ${\cal{SI}}$ 的并。因此，文献（this http URL, this http URL, 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）中关于格 $L({\cal {SCR}}_n)$ 是模格的结果并不能从先前关于 $L({\cal{CR}}_n)$ 和 $L({\cal{SI}})$ 的结果推出。然而，由 ${\cal {SCR}}_n$ 中所有幂等元构成子半群的半群组成的簇 ${\cal{SO}}_n$ 恰好等于所有正交群簇 ${\cal O}_n$ 与 ${\cal{SI}}$ 的并。我们由此推导出格 $L({\cal{SO}}_n)$ 在模群意义下的一个描述。
英文摘要
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 ${\cal{CR}}_n$ and ${\cal{SI}}$ 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 $L({\cal{CR}}_n)$ and $L({\cal{SI}})$. However the variety ${\cal{SO}}_n$ 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 ${\cal{SI}}$. We deduce from this a description of the lattice $L({\cal{SO}}_n)$ modulo groups.
