Completely Regular Semigroup Varieties
Applications and Advanced Techniques
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Beschreibung
This book presents further developments and applications in the area of completely regular semigroup theory, beginning with applications of Polák’s theorem to obtain detailed descriptions of various kernel classes including the K-class covers of the kernel class of all bands. The important property of modularity of the lattice of varieties of completely regular semigroups is then employed to analyse various principal sublattices. This is followed by a study of certain important complete congruences on the lattice; the group, local and core relations. The next chapter is devoted to a further treatment of certain free objects and related word problems. There are many constructions in the theory of semigroups. Those that have played an important role in the theory of varieties of completely regular semigroups are presented as they apply in this context. The mapping that takes each variety to its intersection with the variety of bands isa complete retraction of the lattice of varieties of completely regular semigroups onto the lattice of band varieties and so induces a complete congruence for which every class has a greatest member. The sublattice generated by these greatest members is then investigated with the help of many applications of Polák’s theorem. The book closes with a fascinating conjecture regarding the structure of this sublattice.
This book presents further developments and applications in the area of completely regular semigroup theory, beginning with applications of Polák’s theorem to obtain detailed descriptions of various kernel classes including the K-class covers of the kernel class of all bands. The important property of modularity of the lattice of varieties of completely regular semigroups is then employed to analyse various principal sublattices. This is followed by a study of certain important complete congruences on the lattice; the group, local and core relations. The next chapter is devoted to a further treatment of certain free objects and related word problems. There are many constructions in the theory of semigroups. Those that have played an important role in the theory of varieties of completely regular semigroups are presented as they apply in this context. The mapping that takes each variety to its intersection with the variety of bands is a complete retraction of the lattice of varieties of completely regular semigroups onto the lattice of band varieties and so induces a complete congruence for which every class has a greatest member. The sublattice generated by these greatest members is then investigated with the help of many applications of Polák’s theorem. The book closes with a fascinating conjecture regarding the structure of this sublattice.
Expands readers’ knowledge of the lattice of varieties of completely regular semigroups Includes recent advances, insights, results, and techniques in the area of completely regular semigroup theory Utilizes the Polák’s ladder technique from the previous volume to develop explicit descriptions of noteworthy sublattices
Autor*in
Mario Petrich
Themen in »Completely Regular Semigroup Varieties«
Word Problems Semidirect Products Wreath Products Rhodes Expansion Varieties of Completely Regular Semigroups Applications of Polak’s Theorem Free Completely Regular Semigroups
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“The book is well equipped with powerful results with detailed methodology and illustrative examples and also a new fascinating conjecture regarding the structure of the aforesaid sublattice. The book has an updated and insightful reference list at the end.” (Sanjib Kumar Datt, zbMATH 1567.20003, 2025)
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