structureswith

Structureswith is a term sometimes used in theoretical discussions to denote classes of mathematical structures that are equipped with additional components beyond a base signature. In this sense, a structure with extra features is described as a structure with or structureswith certain symbols, predicates, or relations added to its language. The idea is to study how properties of the original structure interact with the new, expanded features.

Formally, take a base logical signature L and a set Σ of new symbols. The expanded language L∪Σ allows one to define structures that interpret both the original symbols and the added ones. A structure A for L∪Σ is often referred to as a structurewith Σ, and the pair (A, interpretation of Σ) encodes the base structure along with its extra data. Researchers may then consider a property P that A must satisfy, such as a constraint on the added predicates or a particular relation between the base and the new components. The class of all L∪Σ-structures satisfying P is sometimes described as StructuresWith(P).

Examples include graphs with a distinguished subset of vertices marked by a unary predicate, groups with a specified subset of elements forming a substructure, or databases equipped with a designated set of active records. In model theory, expanding a language to form structureswith additional symbols can affect definability, categoricity, and interpretability. The concept is closely related to language expansion and to the study of how extra structure influences logical properties.