Semigroup

Semigroup
Name	Semigroup
Type	Algebraic structure
Studied in	University of Cambridge, Princeton University, Massachusetts Institute of Technology, University of Oxford, Harvard University, Stanford University, University of Chicago, University of California, Berkeley, Yale University, Columbia University
Notable people	Emil Post, Alfred Tarski, John von Neumann, Andrey Kolmogorov, Israel Gelfand, Maurice Auslander, Paul Erdős, Saunders Mac Lane, Bertrand Russell, David Hilbert
Related subjects	Group (mathematics), Monoid, Category (mathematics), Ring (mathematics), Boolean algebra, Lattice (mathematics), Automaton theory, Formal language, Operator algebra

Semigroup A semigroup is an algebraic structure consisting of a set equipped with an associative binary operation, studied across many institutions and by numerous mathematicians. Rooted in early 20th-century work, it connects to structures investigated at University of Cambridge, Princeton University, and Massachusetts Institute of Technology and influences research topics pursued by figures such as Emil Post, Alfred Tarski, and John von Neumann.

Definition and Basic Examples

A semigroup is defined as a nonempty set S with a binary operation • satisfying associativity. Classical examples include the set of all functions on a set under composition considered in contexts like Harvard University research groups, the set of nonnegative integers under addition studied by Andrey Kolmogorov and colleagues, and matrices under multiplication as examined at Institute for Advanced Study and Princeton University. Other concrete instances arise from transformation semigroups on finite sets relevant to work at Bell Labs and AT&T, partial orders and endomorphism monoids appearing in seminars at University of Oxford and University of Cambridge, and languages under concatenation central to research at Stanford University and University of California, Berkeley.

Algebraic Properties and Elementary Results

Basic algebraic properties include notions of powers, idempotents, and cancellation; foundational theorems about structure and decomposition have been developed in schools such as University of Chicago and Columbia University. Important results parallel to group theory — existence of minimal ideals, Green’s relations, and Rees theorems — were studied by researchers affiliated with Yale University and University of Michigan. Semigroups also exhibit behavior explored in conferences at International Congress of Mathematicians and by authors publishing in journals associated with American Mathematical Society and London Mathematical Society.

Important Classes and Constructions

Important classes include inverse semigroups investigated by scholars linked to University of Edinburgh, commutative semigroups with connections to University of Bonn, regular semigroups studied at Universität Hamburg, and completely simple semigroups appearing in texts from University of Cambridge. Constructions such as free semigroups, Rees matrix semigroups, and direct/indirect limits are used across research programs at Max Planck Institute for Mathematics, Institut des Hautes Études Scientifiques, California Institute of Technology, and University of Tokyo; these constructions interact with notions introduced by mathematicians associated with École Normale Supérieure and Université Paris-Saclay.

Representations and Actions

Representations of semigroups as transformations or linear operators connect to work at Massachusetts Institute of Technology, Institute for Advanced Study, and Max Planck Institute for Mathematics. Actions on sets and modules, transformation semigroups, and representations by matrices relate to topics pursued at Imperial College London, National Academy of Sciences, and Russian Academy of Sciences. Operator semigroups used in functional analysis and partial differential equations are central to research at Courant Institute, University of California, Los Angeles, and Princeton University.

Homomorphisms, Ideals, and Congruences

Morphisms between semigroups, ideal theory, and congruence relations mirror categorical perspectives developed at University of Chicago and Harvard University. Studies of Green’s relations, minimal ideals, idempotent-generated subsemigroups, and Rees quotients have been advanced in collaborations involving European Mathematical Society and researchers at University of Manchester and University of Glasgow. Factorization by congruences and structure theorems are tools used in seminars at University of Illinois Urbana-Champaign and Ohio State University.

Applications and Connections to Other Areas

Semigroups interface with automata theory and formal languages, topics central to work at Bell Labs, MIT Computer Science and Artificial Intelligence Laboratory, and Carnegie Mellon University; they underpin models in theoretical computer science studied at Stanford University and Cornell University. Connections to operator theory, ergodic theory, and dynamical systems tie semigroup ideas to research at Steklov Institute of Mathematics, Institute for Advanced Study, and Princeton University. Applications in combinatorics, number theory, and coding theory have been explored by mathematicians at University of Cambridge, École Polytechnique, and University of Paris VI, while links to category theory and homological algebra relate to work at ETH Zurich and University of Bonn.

Category:Algebraic structures