prégroup

A pregroup is a mathematical structure that generalizes the concept of a group. It is an algebraic structure consisting of a set with a binary operation, often called multiplication, and an inverse operation. However, unlike a group, a pregroup does not necessarily satisfy all the group axioms. Specifically, the associativity of the binary operation and the existence of an identity element might not be guaranteed.

The notion of a pregroup arose from attempts to capture the algebraic properties of certain operations that are not strictly associative or do not have a global identity. For instance, in some contexts, operations might be associative only on certain subsets or under specific conditions. Pregroups provide a framework to study such algebraic systems.

A common way to define a pregroup is through generators and relations, similar to how groups are often described. A pregroup can be presented by a set of generators and a set of relations between these generators. The elements of the pregroup are then interpreted as words formed from the generators, with the relations used to simplify or identify equivalent words.

Pregroups have found applications in various areas of mathematics, including abstract algebra, category theory, and theoretical computer science. They offer a more flexible algebraic setting than standard groups, allowing for the analysis of structures with weaker axiomatic properties. The study of pregroups often involves exploring the extent to which they can be embedded into or related to actual groups, or examining the properties that arise from their specific relational structure.