Advances in the Theory of Varieties of Semigroups

​Edmond W. H. Lee is a professor of mathematics at Nova Southeastern University, USA. He holds a Ph.D. from Simon Fraser University, Canada, and a D.Sc. from National Research University Higher School of Economics, Russia. As a mathematician, Lee is primarily interested in the theory of varieties of semigroups—a topic that lies in the intersection of universal algebra and semigroup theory—and is best known for his work on equational theories of small semigroups. Lee is currently an active member of the editorial boards of Algebra Universalis and Semigroup Forum.

• Historical overview and main results.- Preliminaries. Part I - Semigroups.- Aperiodic Rees-Suschkewitsch varieties.- A problem of Pollak and Volkov on hereditarily finitely based identities.- Sufficient conditions for the non-finite basis property.- Semigroups without irredundant identity bases.- Part II - Involution Semigroups.- Involution semigroups with infinite irredundant identity bases.- Finitely based involution semigroups with non-finitely based reducts.- Counterintuitive examples of involution semigroups.- Equational theories of twisted involution semigroups.- Part III - Monoids.- Hereditarily finitely based varieties of monoids.- Varieties of aperiodic monoids with central idempotents.- Certain Cross varieties of aperiodic monoids with commuting idempotents.- Counterintuitive examples of monoids.

“The monograph is self-contained … .” (Peeter Normak, zbMATH 1542.20003, 2024)“This book can serve as a well-written introduction for mastering methods of a well-developed theory and can be recommended to beginning researchers as well as to experienced researchers.” (Grigory I. Mashevitzky, Mathematical Reviews, August, 2024)“The monograph by Edmond W. H. Lee is a significant contribution to the research of semigroup varieties and varieties of semigroups equipped with additional unary or 0-ary operations. It will undoubtedly be interesting and useful to all researchers involved in the theory of varieties of semigroups, the theory of semigroups as a whole and related branches of general algebra.” (Boris M. Vernikov, Semigroup Forum, Vol. 107 (2), 2023)