Categorical Abstract Algebraic Logic: Subdirect Representation for Classes of Structure Systems
نویسنده
چکیده
The notion of subdirect irreducibility in the context of languages without equality, as presented by Elgueta, is extended in order to obtain subdirect representation theorems for abstract and reduced classes of structure systems. Structure systems serve as models of firstorder theories but, rather than having universal algebras as their algebraic reducts, they have algebraic systems in the sense of Categorical Abstract Algebraic Logic. The subdirect representation theory for partially ordered functors, presented in previous work by the author, becomes a special case of the theory presented here.
منابع مشابه
Categorical Abstract Algebraic Logic: Subdirect Representation of Pofunctors
Pałasińska and Pigozzi developed a theory of partially ordered varieties and quasi-varieties of algebras with the goal of addressing issues pertaining to the theory of algebraizability of logics involving an abstract form of the connective of logical implication. Following their lead, the author has abstracted the theory to cover the case of algebraic systems, systems that replace algebras in t...
متن کاملCategorical Abstract Algebraic Logic: Operators on Classes of Structure Systems
The study of structure systems, an abstraction of the concept of firstorder structures, is continued. Structure systems have algebraic systems, rather than universal algebras, as their algebraic reducts. Moreover, their relational component consists of a collection of relation systems on the underlying functors, rather than simply a system of relations on a single set. A variety of operators on...
متن کاملCategorical Abstract Algebraic Logic: Closure Operators on Classes of PoFunctors
Following work of Pa lasińska and Pigozzi on partially ordered varieties and quasi-varieties of universal algebras, the author recently introduced partially ordered systems (posystems) and partially ordered functors (pofunctors) to cover the case of the algebraic systems arising in categorical abstract algebraic logic. Analogs of the ordered homomorphism theorems of universal algebra were shown...
متن کاملCategorical Abstract Algebraic Logic: Leibniz Equality and Homomorphism Theorems
The study of structure systems, an abstraction of the concept of firstorder structures, is continued. Structure systems have algebraic systems rather than universal algebras as their algebraic reducts. Moreover, their relational component consists of a collection of relation systems on the underlying functors rather than simply a system of relations on a single set. Congruence systems of struct...
متن کاملAN ALGEBRAIC STRUCTURE FOR INTUITIONISTIC FUZZY LOGIC
In this paper we extend the notion of degrees of membership and non-membership of intuitionistic fuzzy sets to lattices and introduce a residuated lattice with appropriate operations to serve as semantics of intuitionistic fuzzy logic. It would be a step forward to find an algebraic counterpart for intuitionistic fuzzy logic. We give the main properties of the operations defined and prove som...
متن کامل