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This book is a unified treatment of the most important core developments in the theory of completely regular semigroup theory as it stands today. This volume focuses on the lattice of varieties of completely regular semigroups. Since any in-depth study of the lattice of varieties requires an understanding of free completely regular semigroups, the book begins by describing the free object on countably infinite sets and the properties of the lattice of fully invariant congruences on the free object. The authors introduce various associated relations and operators on the lattice of varieties of completely regular semigroups. Following that, the book covers the sublattice of varieties of bands with a focus on the influence of that sublattice on the structure of the whole lattice. The book concludes with the remarkable theorem due to Polák describing the whole lattice of varieties of completely regular as a subdirect product of lattices,some of which are well understood. The authors include recent advances, insights, results, and techniques throughout the book.
List of contents
Fully Invariant Relations.- Kernel and Trace Relations.- Bands.- Polák Theorem.
About the author
Mario Petrich received his Ph.D. from the University of Washington in 1961 and held positions in many universities across Austria, Canada, France, Germany, Italy, Portugal, Spain, the UK, and the US, including Pennsylvania State University, University of Western Ontario, University of Vienna, University of St Andrews, Simon Fraser University, and University of Montpellier. He was a founding editor of Semigroup Forum.
Norman Reilly received his Ph.D. from Glasgow University in 1965 and has held positions at Glasgow University, Tulane University, and Simon Fraser University. Dr. Reilly has published over 100 articles and two books. He is a long time editor of Semigroup Forum and the International Journal of Algebra and Computation.
Summary
This book is a unified treatment of the most important core developments in the theory of completely regular semigroup theory as it stands today. This volume focuses on the lattice of varieties of completely regular semigroups. Since any in-depth study of the lattice of varieties requires an understanding of free completely regular semigroups, the book begins by describing the free object on countably infinite sets and the properties of the lattice of fully invariant congruences on the free object. The authors introduce various associated relations and operators on the lattice of varieties of completely regular semigroups. Following that, the book covers the sublattice of varieties of bands with a focus on the influence of that sublattice on the structure of the whole lattice. The book concludes with the remarkable theorem due to Polák describing the whole lattice of varieties of completely regular as a subdirect product of lattices,some of which are well understood. The authors include recent advances, insights, results, and techniques throughout the book.
Additional text
“Current advancements with their insights, related recent results and their methodologies etc. are nicely and elaborately included in this book and some of which are enlighted with in-depth understanding and special mention of the whole lattice of varieties of completely regular as a subdirect product of lattices which is in fact a remarkable theorem due to Pola’k. The book consists of an insightful and resourceful bibliography at the end.” (Sanjib Kumar Datta, zbMATH 1545.20001, 2024)
Report
This review treats the two volumes as a single whole. One of the outstanding features consists of the Introduction and Review contained within each chapter and I can think of no better way to outline them than by quoting liberally from this material. The first author passed away before the publication of these two volumes. He will be greatly missed. (Peter R. Jones, Semigroup Forum, Vol. 111 (1), 2025)
Current advancements with their insights, related recent results and their methodologies etc. are nicely and elaborately included in this book and some of which are enlighted with in-depth understanding and special mention of the whole lattice of varieties of completely regular as a subdirect product of lattices which is in fact a remarkable theorem due to Pola k. The book consists of an insightful and resourceful bibliography at the end. (Sanjib Kumar Datta, zbMATH 1545.20001, 2024)
