Structure (mathematical logic)

In mathematical logic, the notion of a structure generalizes mathematical objects such as groups, rings, fields, lattices or ordered sets. A structure is a set equipped with any number of named constants, operations and relations. For example the ordered group of integers can be regarded as a structure consisting of the set of integers $$\mathbb Z = \{\dots, -2, -1, 0, 1, 2, \dots\}$$ together with the constant 0, the binary operation $$+$$ (addition), the unary function $$-$$ (which maps each integer to its inverse), and the binary relation $$<$$. This structure is often denoted by $$(\mathbb Z, 0, +, -, <)$$.

Structures are studied in model theory, where the term model is often used as a synonym. Structures without relations are studied in universal algebra, and a structure with only constants and operations is often referred to as an algebra or, to avoid confusion with algebras over a field, as a universal algebra.