Reducts

A reduct is a concept from model theory describing a structure that is obtained from another structure by discarding some of its language symbols. If M is a structure in a language L, and L′ is a sublanguage of L (i.e., L′ ⊆ L), then the L′-structure M|L′ is formed by keeping the same domain and interpreting exactly the symbols that remain in L′. In other words, all relations, functions, and constants in L′ are interpreted as they are in M, while the symbols not in L′ are simply forgotten.

Formally, if M = (D, interpret(S) for S ∈ L) is an L-structure, then M|L′ = (D, interpret(S) for S ∈ L′). The reduct preserves the domain and the meanings of the retained symbols, but it may lack the extra structure provided by the removed symbols. The opposite operation is expansion, where new symbols and interpretations are added to obtain a larger language.

Examples help illustrate the idea. Take a field F with the language {+, ×, 0, 1}. The reduct to {+, 0} gives the additive group structure (F, +, 0). The reduct to {×, 0, 1} yields the multiplicative monoid (F, ×, 0, 1). The reduct to the empty language yields the pure set (F, =). Reducts are often used to compare theories, to study definability, and to analyze how much of a structure’s behavior is captured by a smaller signature.

See also: expansion, definability, interpretability, interdefinability.