pregroups

Pregroups are algebraic structures studied within the field of algebra, particularly in the context of category theory and formal language theory. They serve as a generalization of groups and related structures, aiming to capture specific kinds of partial symmetry or partial invertibility.

A pregroup consists of a set equipped with a partial binary operation, an involution (a unary operation that is its own inverse), and certain axioms that generalize group properties. Unlike a group, where the binary operation is defined for all pairs of elements and every element has an inverse, a pregroup allows the operation to be only partially defined, reflecting the idea that not all elements can be combined.

The formal definition requires that the set with its partial operation and involution satisfy certain coherence conditions. Specifically, for every element, there is a "partial inverse" such that the composition of an element with its inverse behaves similarly to the identity wherever the operations are defined. Pregroups are useful in modeling systems where certain compositions are only valid under specific conditions, such as in linguistic structures, syntactic analysis, and certain algebraic systems.

Pregroups also serve as the foundational structures in the theory of pregroups in the sense of Lambek's algebraic approach to syntax, where they underpin the construction of free pregroups and their applications in categorial grammar. Their study provides insights into how partial symmetries and operations can be effectively modeled in both mathematical and computational contexts.