Structure (mathematical Logic)
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations, and relations that are defined on it.
Universal algebra studies structures that generalize the algebraic structures such as groups, rings, fields and vector spaces. The term universal algebra is used for structures with no relation symbols.
Model theory has a different scope that encompasses more arbitrary theories, including foundational structures such as models of set theory. From the model-theoretic point of view, structures are the objects used to define the semantics of first-order logic. For a given theory in model theory, a structure is called model, if it satisfies the defining axioms of that theory, although it is sometimes disambiguated as a semantic model when one discusses the notion in the more general setting of mathematical models. Logicians sometimes refer to structures as interpretations.
In database theory, structures with no functions are studied as models for relational databases, in the form of relational models.
Read more about Structure (mathematical Logic): Definition, Induced Substructures and Closed Subsets, Structures and First-order Logic, Many-sorted Structures
Famous quotes containing the word structure:
“Why does philosophy use concepts and why does faith use symbols if both try to express the same ultimate? The answer, of course, is that the relation to the ultimate is not the same in each case. The philosophical relation is in principle a detached description of the basic structure in which the ultimate manifests itself. The relation of faith is in principle an involved expression of concern about the meaning of the ultimate for the faithful.”
—Paul Tillich (1886–1965)