# Semigroups and semirings

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A semigroup is an algebraic structure with a single associative binary operation; if it has an identity element it is called a 'monoid'.

Semigroups and monoids can be viewed as both "groups without inverses" and "rings without addition". Their study typically involves borrowing methods from group theory and ring theory, as well as ideas bespoke to the semigroup case. Their relatively unconstrained structure makes them ubiquitous and highly applicable - wherever there are non-invertible functions, there are usually semigroups to be found - but also hard to study in complete generality. As a result, much of the theory is specialised to particular classes of semigroups which arise frequently and are sufficiently constrained to admit a powerful theory.

Perhaps chief among these are the *inverse semigroups*: those in which each element x admits a unique "weak inverse'' x' such that x x' x = x and x' x x'=x'. Inverse semigroups model *partial symmetry* (symmetries between parts of a structure) in the same way that groups model total symmetry. Basic examples include the monoid of bijections between subsets of a given set (the *symmetric inverse monoid*) and the *bicyclic monoid* generated by two elements p and q such that pq=1 (but qp ≠ 1!). Other major areas of research include classes of semigroups with even weaker notions of inverses and/or cancellativity conditions, semigroups defined by (typically finitely many) generators and relations, linear representations of semigroups, and numerous interactions with theoretical computer science.

Semi*rings* and semi*fields* are structures which satisfy the axioms of rings and fields respectively, except for the invertibility of addition. They arise naturally in many areas, including semigroup theory where they are natural carriers for linear representations of semigroups which don't lend themselves to representation over fields. A prominent example is the *tropical semifield*, which is the algebraic structure formed by the real numbers under the operations of addition (playing the role of multiplication) and maximum (playing the role of addition), and which has a radically different but still rich algebra and geometry. *Tropical algebra* (also known as *max algebra* or *max-plus algebra*) is the tropical analogue of linear algebra, and is largely concerned with modules and matrices over the tropical semifield. *Tropical geometry* studies the (algebraic and convex) geometry of spaces over the tropical semiring, and what it can tell us about classical algebraic geometry over fields. Both have applications in a wide range of areas, ranging from combinatorial optimisation and scheduling to microprocessor design, and from biochemistry to statistics.

## Our research projects

Current topics of interest in Manchester include the geometry of finitely generated semigroups (including inverse semigroups), connections between inverse semigroups, étale groupoids and C*-algebras, tropical matrix semigroups and tropical representations of semigroups, and the combinatorics of tropical polytopes. Please see our individual researchers' pages for more details.