substructures

Substructures are subsets that inherit the structure’s operations and relations. In algebra, a substructure of a structure A is a subset B of A with the same interpretation of the operation symbols, closed under those operations, and containing any constant symbols.

In practice, substructures include subgroups of a group, subrings of a ring, submodules of a module, subspaces of a vector space, and subfields of a field, as well as induced subgraphs of a graph. In model theory, a substructure is required to preserve the interpretation of function and relation symbols; an elementary substructure additionally preserves all first-order statements.

Key properties include that the intersection of substructures is a substructure, and the union of a chain of substructures is a substructure. Proper substructures are those strictly smaller than the whole structure. The concept generalizes to subobjects in category theory as a way to study internal structure.

In other disciplines, the term substructure is used more loosely. In chemistry and materials science, it denotes a fragment of a molecule or a structural component within a crystal, often relevant to synthesis, characterization, or defect analysis. A substructure thus encompasses both precise mathematical notions of inheritance within a formal system and broader, domain-specific fragments used to analyze complex objects.