A structure obtained from a given structure by omitting some symbols of its language and retaining the same underlying domain and the interpretations of the remaining symbols. The reduct is the canonical result of the 'forgetting' operation that removes vocabulary without altering how the leftover symbols are interpreted on the same set.

Principle

Forgetting symbols is the organizing rule: removing symbols yields a new structure whose carrier set and interpretations for the remaining symbols are identical to those in the original, and there is a canonical projection from expanded structures to their reducts.

Demonstration

Given a structure A in language L′ = L ∪ {R}, the L-reduct of A is the structure with the same domain as A and with every symbol from L interpreted exactly as in A, but with R ignored. Concretely, the ordered field (R,+,·,0,1,<) has as reduct to the language of fields the field (R,+,·,0,1) obtained by forgetting the order relation.

Misapplication

Treating a reduct as if it adds information (for example, assuming the reduct can distinguish elements that only differed by the forgotten symbols) or confusing taking a reduct with taking a substructure or quotient; a reduct does not change the domain or collapse elements, it merely removes vocabulary.

Consequence

Reducts define a forgetful functor from structures in the richer language to structures in the smaller language; they preserve all truths expressible in the smaller language and provide canonical comparisons between expanded and reduced presentations of the same underlying set.

Reversal

Expansion: the inverse process is to add new symbols and specify their interpretations (possibly in many non-unique ways); an expansion of a reduct may be non-unique and can restore information lost by forgetting.

Boundary

Applies only to removal of symbols and their interpretations while keeping the same underlying set; it does not include forming substructures, quotients, or taking reducts that change domain elements. It excludes adding definitional abbreviations that implicitly carry new relations unless those are explicitly present in the reduced signature.

Semantic Tension

Reduct versus substructure: a reduct removes vocabulary but keeps the same domain, while a substructure keeps the language but restricts the domain; both reduce complexity but in orthogonal ways, which can lead to confusion in statements about containment and preservation.

Synthesis

A reduct is the canonical 'forgetful' image of a structure under deletion of vocabulary: it formalizes the idea of observing a structure through a smaller language, preserving the interpretations that remain while discarding the rest.