Partial groups, also known as partial semigroups or partial magma, refer to algebraic structures that possess some of the properties of a group or a semigroup but not necessarily all of them.
Formally, a partial group consists of a set G together with a partial binary operation *: G × G → G, denoted as (g,h) ↦ g * h or gh, where g and h are elements of G. The operation * is partial in the sense that it may not be defined for all pairs of elements in G, thus satisfying the axioms only for the defined pairs.
A partial group must satisfy four main axioms. Firstly, it must possess closure, which means that for any g, h in G such that gh is defined, then gh must also belong to G. Additionally, it must satisfy the existence of inverses, meaning that for every g in G, there must exist an element h in G such that both gh and hg are defined and equivalent to the "identity element" of the partial group. Thirdly, it must maintain the associativity property, implying that for every g, h, k in G such that gh and hk are defined, then (gh)k = g(hk). Lastly, it must satisfy partial cancellation, implying that if gh and gk are defined, then h = k.
Partial groups generalize the notion of groups and semigroups, allowing for the study of non-full operations and recognizing situations where not all elements possess an inverse. They are utilized in various branches of mathematics, such as algebraic topology, formal languages, and theoretical computer science.
