The Canonical Structure as a Minimum Structure

In this paper a structure of a syslem is defined as a mathematical structure [formula omitted], where [formula omitted]is a first-order logic language and Σ is a set of sentences of the given first-order logic. It is shown that a canonical structure determined by I, which is similar to those used in proving the Gödel's completeness theorem, satisfies a universality in the sense of category theory when homomorphisms are used as morphisms, and a freeness in the sense of universal algebra when Σ-morphisms, which preserve Σ, are used. The universality and the freeness give the minimality of the canonical structure.As an example, a structure of a stationary system is defined as a pair [formula omitted] Its canonical structure is actually constructed. In a sense this canonical structure accords with models constructed by Nerode realization.