Overview

This monograph covers topics in the cohomology of monoids up through recent developments.  Jonathan Leech’s original monograph in the Memoirs of the American Mathematical Society dates back to 1975. This book is an organized, accessible, and self-contained account of this cohomology that includes more recent significant developments that were previously scattered among various publications, along with completely new material.  It summarizes the original Leech theory and provides a modern and thorough treatment of the cohomological classification of coextensions of both monoids and monoidal groupoids, including the case of monoids with operators. This cohomology is also compared to the classical Eilenberg-Mac Lane and Hochschild-Mitchell cohomologies. Connections are also established with the Lausch-Loganathan cohomology theory for inverse semigroups, the Gabriel-Zisman cohomology of simplicial sets, the Wells cohomology of small categories (also known as Baues-Wirschingcohomology), Grothendieck sheaf cohomology, and finally Beck’s triple cohomology.  It also establishes connections with Grillet’s cohomology theory for commutative semigroups.  The monograph is aimed at researchers in the theory of monoids, or even semigroups, and its interface with category theory, homological algebra, and related fields. However, it is also written to be accessible to graduate students in mathematics and mathematicians in general.

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9783031502576

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Antonio M. Cegarra earned his Ph.D. in Mathematical Sciences from the University of Granada in 1980. In 1989, he was appointed Professor of Algebra at the same university, where ever since he has dedicated himself to teaching and research. He has taught numerous algebra courses in both undergraduate and postgraduate programs, primarily spanning the range of Mathematics, Computer Science, and Physics. His research pursuits have predominantly revolved around category theory, homological algebra, and algebraic homotopy theory, driven by their applications in the realms of algebra, geometry, and topology.  Antonio has authored over eighty articles in esteemed international journals, many of which were co-authored with colleagues from various countries. He has also contributed to the Editorial Committees of renowned scientific journals, including the Journal of Homotopy and Related Structures as well as Homology, Homotopy and Applications. Additionally, he has lenthis expertise as a scientific advisor to diverse international agencies, such as the European Science Foundation and the Executive Agency for Higher Education, Research, Development and Innovation Funding.  Jonathan Leech earned his PhD from the University of California, Los Angeles, in 1969.  He has imparted mathematical knowledge at various institutions, including the University of Tennessee, Missouri Western State University, and ultimately at Westmont College in Santa Barbara, California. He has held positions as a Visiting Professor at Case Western Reserve University, the Universidad de Granada in Spain, and Universidade Mackenzie in Brazil. Additionally, he has served as a scholar in residence at both the University of Sydney and the University of Tasmania in Australia, as well as the University of Ljubljana in Slovenia, the University of Petru Maior in Romania and several others. Throughout his academic career, Professor Leech has made many research contributions. In addition to his seminal contributions to the cohomology of monoids and semigroups, he has delved deeply into algebraic structures associated with semigroups. Indeed, he along with colleagues from around the world have established the fundamental principles of modern noncommutative lattice theory through various publications, including his book Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond in the Famnit  Lectures series of the University of Primorska in Slovenia.

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