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Register a new userSelf-dual varieties and networks in the lattice of varieties of completely regular semigroups
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Norman Reilly Professor Emeritus
Publication date
2020
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English
Authors
Norman R. Reilly
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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. Kadourek [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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Several complete congruences on the lattice L(CR) of varieties of completely regular semi- groups have been fundamental to studies of the structure of L(CR). These are the kernel relation K , the left trace relation Tl , the right trace relation Tr and their intersections KcapTl,Kcap Tr . However, with the exception of the lattice of all band varieties which happens to coincide with the kernel class of the trivial variety, almost nothing is known about the internal structure of individual K-classes beyond the fact that they are intervals in L(CR). Here we present a number of general results that are pertinent to the study of K -classes. This includes a variation of the renowned Polak Theorem and its relationship to the complete retraction V -> V cap B , where B denotes the variety of bands. These results are then applied, here and in a sequel, to the detailed analysis of certain families of K -classes. The paper concludes with results hinting at the complexity of K -classes in general, such as that the classes of relation K/Kl may have the cardinality of the continuum.
Given a subdirectly irreducible *-regular ring R, we show that R is a homomorphic image of a regular *-subring of an ultraproduct of the (simple) eRe, e in the minimal ideal of R; moreover, R (with unit) is directly finite if all eRe are unit-regular. Finally, unit-regularity is shown for every member of the variety generated by artinian *-regular rings (endowed with unit and pseudo-inversion).
This paper is a contribution to the theory of finite semigroups and their classification in pseudovarieties, which is motivated by its connections with computer science. The question addressed is what role can play the consideration of an order compatible with the semigroup operation. In the case of unions of groups, so-called completely regular semigroups, the problem of which new pseudovarieties appear in the ordered context is solved. As applications, it is shown that the lattice of pseudovarieties of ordered completely regular semigroups is modular and that taking the intersection with the pseudovariety of bands defines a complete endomorphism of the lattice of all pseudovarieties of ordered semigroups.
There are continuum many clones on a three-element set even if they are considered up to emph{homomorphic equivalence}. The clones we use to prove this fact are clones consisting of emph{self-dual operations}, i.e., operations that preserve the relation ${(0,1),(1,2),(2,0)}$. However, there are only countably many such clones when considered up to equivalence with respect to emph{minor-preserving maps} instead of clone homomorphisms. We give a full description of the set of clones of self-dual operations, ordered by the existence of minor-preserving maps. Our result can also be phrased as a statement about structures on a three-element set, ordered by primitive positive constructability, because there is a minor-preserving map from the polymorphism clone of a finite structure $mathfrak A$ to the polymorphism clone of a finite structure $mathfrak B$ if and only if there is a primitive positive construction of $mathfrak B$ in $mathfrak A$.
This paper enriches the list of properties of the congruence sequences starting from the universal relation and successively performing the operations of lower $t$ and lower $k$. Three classes of completely regular semigroups, namely semigroups for which $ker{sigma}$ is a cryptogroup, semigroups for which $ker{ u}$ is a cryptogroup and semigroups for which $kappa$ is over rectangular bands, are studied. $((omega_t)_k)_t$, $((mathcal{D}_t)_k)_t$ and $((omega_k)_t)_k$ are found to be the least congruences on $S$ such that the quotient semigroups are semigroups for which $ker{sigma}$ is a cryptogroup, $ker{ u}$ is a cryptogroup and $kappa$ is over rectangular bands, respectively. The results obtained present a response to three problems in Petrich and Reillys textbook textquotelefttextquoteleft Completely Regular Semigroupstextquoterighttextquoteright.
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