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arXiv · 2007.03546

Self-dual varieties and networks in the lattice of varieties of completely regular semigroups

Norman R. Reilly
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Abstract

The kernel relation $K$ on the lattice $\mathcal{L}(\mathcal{CR})$ of varieties of completely regular semigroups has been a central component in many investigations into the structure of $\mathcal{L}(\mathcal{CR})$. However, apart from the $K$-class of the trivial variety, which is just the lattice of varieties of bands, the detailed structure of kernel classes has remained a mystery until recently. Kad'ourek [RK2] has shown that for two large classes of subvarieties of $\mathcal{CR}$ their kernel classes are singletons. Elsewhere (see [RK1], [RK2], [RK3]) we have provided a detailed analysis of the kernel classes of varieties of abelian groups. Here we study more general kernel classes. We begin with a careful development of the concept of duality in the lattice of varieties of completely regular semigroups and then show that the kernel classes of many varieties, including many self-dual varieties, of completely regular semigroups contain multiple copies of the lattice of varieties of bands as sublattices.

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Norman R. Reilly. 2020-07-07. Self-dual varieties and networks in the lattice of varieties of completely regular semigroups. https://arxiv.org/abs/2007.03546

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