An algebraic structure or an algebraic system is a non-empty set together with one or more binary operations on that set. Algebraic structures whose binary operations satisfy particularly important properties are groupoids, semigroups, monoids, groups, rings, fields, modules and so on.
A groupoid is an algebraic structure consisting of a non-empty set G and a binary operation o on G. The pair (G, o) is called groupoid.
The set of real numbers with the binary operation of addition is a groupoid.
If (G, o) is a groupoid and if the associative rule (aob)oc = ao(boc) holds for all a, b, c ∈ G, then (G, o) is called a semigroup.
An element e of a groupoid (G, o) is called an identity element if eoa = aoe = a for all a ∈ G. If there is an identity element in a groupoid then it is unique.
A semigroup with identity element is called a monoid.
The set of all n x n matrices under the operation of matrix multiplication is a monoid. Here the identity element is the unit matrix of order n.
Let (G, o) be a monoid. An element a’ ∈ G is called an inverse of the element a ∈ G if aoa’ = a’oa = e (the identity element of G). The inverse of the element a ∈ G is denoted by a-1.
A monoid in which every element has an inverse is called group.
The set of all n x n matrices under the operation of matrix multiplication is not a group since not every n x n matrix has its multiplicative inverse, but if G is the set of all n x n nonsingular matrices, then G forms a group under the operation of matrix multiplication.