Paragroups

Paragroups are a concept in mathematics, specifically in abstract algebra, that generalize the idea of a group. A paragroup can be thought of as a collection of elements with an associative binary operation, an identity element, and an inverse for each element, but with a relaxation of certain group axioms. Typically, this relaxation involves the group operation not necessarily being defined for all pairs of elements, or the inverse property holding only under specific conditions. The term "paragroup" is not as standardized as "group" and can sometimes refer to slightly different structures depending on the context and the author.

One common interpretation of a paragroup involves a structure where the operation is partial rather than total. In a total operation, like in a standard group, the operation must be defined for every pair of elements. In a partial operation, it is only defined for certain pairs. Another perspective on paragroups can involve a weakening of the associative property or the existence of inverses. For instance, a structure might satisfy associativity for all defined operations but not necessarily have an inverse for every element.

Paragroups have found applications in various areas of mathematics, including the study of algebraic structures, category theory, and theoretical computer science. They allow mathematicians to explore algebraic phenomena that do not strictly adhere to the rigorous definitions of groups, rings, or other established algebraic objects. This broader scope enables the modeling and analysis of systems with more complex or restricted interaction rules. The development and study of paragroups contribute to a deeper understanding of algebraic structures and their properties.