Algebraic structure

The definition of algebraic structure establishes a framework through which to compare otherwise very different abstract objects, and to study similarities and patterns made apparent by the definition.

1975, Ronald James Williams, Algebraic Structures Up to Homotopy, University of California, San Diego, page 16, In this chapter we propose a general means for describing algebraic structures, both strict and up to homotopy, and we apply the results of the preceding chapter to the study of these […] .

1995, George R. Kempf, Algebraic Structures, Bertelsmann (Vieweg), page 129, Thus the algebraic structures which we have been studying are part of category theory, but the important theorem is not generally categorical nonsense, although there are theorems in category theory which we will not study.

Any algebraic structure is defined by a non-empty set of elements, and a set of rules or conditions that the elements satisfy. Thus, even a set equipped with a relation, as described in Section 2.4, is an algebraic structure. This structure is sometimes called a setoid, a name that we won't have much use for.

A monoid is a purely algebraic structure consisting of an associative binary operation and an identity element for that operation.