Research On Residual Lattice And Related Superstructure

Various logical algebras have been proposed as the semantical systems of non-classical logics. Among these logical algebras, residuated lattices are very im-portant algebraic structures, which provide algebraic semantics to the non-classical logical calculi known as substructural logics. In this paper, we mainly focus on in-vestigating state operators and derivations on residuated lattices and states on hyper MV-algebras. We aim to characterize some special types of residuated lattices and provide more general algebraic method for dealing with probability in substruc-tural logic. Also, we want to express an averaging of truth values of fuzzy events in fuzzy logic. The structure of this thesis is as follows:1. In the first chapter, we introduce state operators on residuated lattices and obtain state residuated lattices, which are a kind of more general logical algebras with internal states. Also, we give characterizations of Rl-monoids and Heyting algebras by state operators, and obtain that there is a bijection between compati-ble states on residuated lattices and states on the fixed point set of state operators. By studying state filters, we characterize state simple and state local residuated lat-tices. Moreover, we focus on algebraic structures of the set SF[L] of all state filters on state residuated lattices and obtain that SF[L] forms a coherent frame. Then, we introduce and study the relative co-annihilators by state filters. As applications of relative co-annihilators, we give the relative pseudocomplement in the pseudo-complemented lattice F[L]. In particular, we obtain that the set of all involutory filters relative a filter of residuated lattices forms a complete Boolean algebra.2. In the second chapter, we introduce some derivations in residuated lattices and give some characterizations of ideal derivations. Then, we obtain that the fixed point sets of ideal derivations in divisible residuated lattices and linearly divisible residuated lattices are a latticeal ideal and a prime latticeal ideal, respectively. And we prove that a given prime latticeal ideal of idempotent residuated lattices is the fixed point set of some ideal derivation. Also, we obtain that the fixed point set of good ideal derivations forms a residuated lattice. And we obtain that the set of all ideal derivation filters on residuated lattices can form a complete Heyting algebra. Moreover, the lattice of all ideal derivation filters is isomorphic to the lattice of all filters of the fixed point set. Finally, we obtain that the fixed point set of principal ideal derivations and that of their adjoint derivations are order isomorphic. In par-ticular, we obtain that a residuated lattice L is a (linearly ordered) Heyting algebra iff the fixed point set of a principal ideal derivation is a (prime) latticeal ideal of L.3. In the third chapter, we study states on hyper MV-algebras. In order to study hyperstructures of residuated lattices, we construct a hyperlattice by using fuzzy preordered lattices. Also, we obtain a lattice starting from a hyperlattice by strongly regular relations. Then, we introduce and study Riecan states and Bosbach states on a hyper MV-algebra. In particular, we obtain that Bosbach states are Riecan states, but the converse is not true in general. Also, we investigate and characterize regular Riecan states and regular Bosbach states. Moreover, we discuss some relationships between state morphisms and Bosbach states on hyper MV-algebras. And we obtain that state morphisms must be regular Bosbach states in hyper MV-algebras. Finally, using a Bosbach state on a hyper MV-algebra, we discuss a quotient hyper MV-algebra. And we obtain some equivalent conditions under which a quotient hyper MV-algebra becomes a Boolean algebra. Also, we establish isomorphic theorem on MV-algebras.