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.