Hyperidentity and hypervariety results for varieties of semigroups
Shelly L. Wismath · Summit (Simon Fraser University) · 1988
A hypervariety is a class of varieties closed under the formation of equivalent, product, reduct, and sub-varieties.Hyperidentities are identities which define hypervarieties, in the same way that ordinary identities define varieties.This thesis explores the concepts of hypervariety and hyperidentity in relation to varieties of semigroups.Chapters 1 and 2 provide an introduction and background, describing the relationships between hyperidentities, hypervarieties, and varieties of clones.Chapter 3 gives the semigroup-theoretic results needed for later chapters; these are mainly of the form of an equational description of the joins of various equationallydefined varieties of semigroups.Chapter 4 begins the study of hyperidentities satisfied by various varieties of semigroups, and the properties of two operators, the hypervariety and the cIosure operators, on varieties of semigroups.For the lattice of all varieties of bands, we identify which varieties are closed, and produce hyperidentities to distinguish the corresponding hypervarieties, giving a countably infinite chain of hypervarieties.Similar results are obtained for the varieties of k-nilpotent semigroups, and for joins of these varieties with varieties of bands.The final two chapters consider the commutative varieties satisfying identities of the form xn = x " + ~, and other related varieties.We produce several families of hyperidentities satisfied by such varieties; in particular, we introduce a technique for producing a hyperidentity related to any given identity for semigroups.We also investigate length restrictions on what types of hyperidentities such varieties can satisfy.These results combine with the join results of Chapter 3 to provide information about the action of the hypervariety and closure operators on such varieties.this thesis, and for providing guidance and encouragement during its preparation. Thanks also to my husband Stephen