Algebraic Structures

Definition: an algebraic structure consists of a collection of objects together with the operations that can be performed on those objects.

Many algebraic structures are familiar from traditional mathematics, including the natural numbers with arithmetic operations +, -, *, etc. Actually, this example is referred to as a homogeneous algebra – all its objects are of a single type.

Most often in computing we are interested in heterogeneous algebras – algebraic structures where there are several dissimilar kinds of objects, each with their own operations.

Again such algebras are familiar from traditional mathematics, including vector space over the real numbers. Here we have the real numbers and the arithmetic operations defined for them but not for vectors. In addition, we have vectors that can be added and multiplied ("dot product" of vectors yields a real number, vector product yields another vector).