2020/07/07 by Norman R. Reilly, Reilly, Norman R.
Biochemistry, Genetics and Molecular Biology · Computer Science · #20M05 #20M07 #20M10 (primary) #20M17 (secondary) #Advanced Algebra and Logic #Chemical Synthesis and Analysis #FOS: Mathematics #Rings and Algebras (math.RA) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2007.03546
openalex publication_date 2020/07/07 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
The kernel relation K on the lattice \L(\CR) of\nvarieties of completely regular semigroups has been a central component in many\ninvestigations into the structure of \L(\CR). However,\napart from the K-class of the trivial variety, which is just the lattice of\nvarieties of bands, the detailed structure of kernel classes has remained a\nmystery until recently. Kad'ourek [RK2] has shown that for two large classes of\nsubvarieties of \CR their kernel classes are singletons. Elsewhere\n(see [RK1], [RK2], [RK3]) we have provided a detailed analysis of the kernel\nclasses of varieties of abelian groups. Here we study more general kernel\nclasses. We begin with a careful development of the concept of duality in the\nlattice of varieties of completely regular semigroups and then show that the\nkernel classes of many varieties, including many self-dual varieties, of\ncompletely regular semigroups contain multiple copies of the lattice of\nvarieties of bands as sublattices.\n