A Generalization of Groups

Abstract

For a nonempty set G, the authors define an operation * reckoned with closeness property (i.e.,* is an operation which is not a binary operation). Then they define the partial group as a generalisation of a group. A partial group G which is a non empty set satisfies following conditions hold for all a,b and c∈d:(PG1) If ab,(ab)c, bc and a (bc) is defined, then,(ab) c=a (bc)(PG2) . For every,a∈G there exists an e∈G such that ae and ea are defined and, ae=ea=a (PG3) . For every,a∈G there exists an a∈G such that aa and aa are defined and aa=aa=e. The * operation effects and changes the properties of group axioms. So that lots of group theoretic theorems and conclusions do not work in partial groups. Thus, this description gives us some fundemental and important properties and analogous to group theory. Also the authors have some differences from group theory.