Many algebraic constructions can be viewed as algebra-valued functors. Using a category-theoretic formulation of universal algebra and first-order logic originated by F. W. Lawvere, we obtain algebraic and logical results concerning functors which correspond to important kinds of algebraic constructions--in particular, to Boolean powers and bounded Boolean powers.
The notion of an equational interpretation of an equational theory T' in an equational theory T is introduced and shown to be the syntactical counterpart to coalgebras. By means of equational interpretations T' (--->) T, the representable functors Mod(T) (--->) Mod(T') are shown to be obtainable as T'-algebras defined "within" the underlying-set functor U(,T): Mod(T) (--->) Set when U(,T) is treated as a T-algebra in the functor category Set('MOD(T)). The algebraic functors, i.e., the representable functors G: Mod(T) (--->) Mod(T') whose set-valued component U(,T),.G is monadic, are characterized similarly. The latter result is associated with a syntactical characterization of all the equational theories T' such that Mod(T') is equivalent as a category to Mod(T). Finitary and infinitary versions of the Boolean power constructions are described as algebraic functors which correspond to Post algebras in a functor category.
A functor-theoretic characterization of locally equational categories is given which is analogous to F. E. J. Linton's characterization of equational categories. The characterization of locally equational categories leads naturally to the notion of a locally algebraic functor. Bounded Boolean powers are described as locally algebraic functors, and a new proof of T. K. Hu's theorem characterizing the category of Boolean algebras as a locally equational category is sketched.
|School:||Simon Fraser University (Canada)|
|Source:||DAI-B 42/01, Dissertation Abstracts International|
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