Stonean residuated lattices are closely related to Stone algebras since the bounded lattice reduct of a distributive Stonean residuated lattice is a Stone algebra. In the present work we follow the ideas presented by Chen and Grätzer and try to apply them for the case of Stonean residuated lattices. Given a Stonean residuated lattice, we consider the triple formed by its Boolean skeleton, its a...

a heyting algebra is a distributive lattice with implication and a dual $bck$-algebra is an algebraic system having as models logical systems equipped with implication. the aim of this paper is to investigate the relation of heyting algebras between dual $bck$-algebras. we define notions of $i$-invariant and $m$-invariant on dual $bck$-semilattices and prove that a heyting semilattice is equiva...

The aim of this paper is to extend the truth value table oflattice-valued convergence spaces to a more general case andthen to use it to introduce and study the quantale-valued fuzzy Scotttopology in fuzzy domain theory. Let $(L,*,varepsilon)$ be acommutative unital quantale and let $otimes$ be a binary operationon $L$ which is distributive over nonempty subsets. The quadruple$(L,*,otimes,varep...

We prove strong completeness of a range of substructural logics with respect to their relational semantics by completeness-via-canonicity. Specifically, we use the topological theory of canonical (in) equations in distributive lattice expansions to show that distributive substructural logics are strongly complete with respect to their relational semantics. By formalizing the problem in the lang...

Algebraic properties of lattices of quotients of nite posets are considered. Using the known duality between the category of all nite posets together with all order-preserving maps and the category of all nite distributive (0; 1)-lattices together with all (0; 1)-lattice ho-momorphisms, algebraic and arithmetic properties of maximal proper sublattices and, in particular, Frattini sublattices of...

According to Chajda and Eigenthaler ([1]), a d-lattice is a bounded lattice L satisfying for all a, c ∈ L the implications (i) (a, 1) ∈ θ(0, c) → a ∨ c = 1; (ii) (a, 0) ∈ θ(1, c) → a ∧ c = 0; where θ(x, y) denotes the least congruence on L containing the pair (x, y). Every bounded distributive lattice is a d-lattice. The 5-element nonmodular lattice N 5 is a d-lattice. Theorem 1 A bounded latti...

A complemented modular lattice L is coordinatizable, if it is isomorphic to the lattice L(R) of principal right ideals of some von Neumann regular ring R. All known sufficient conditions for coordinatizability, due first to J. von Neumann, then to B. Jónsson, are first-order. Nevertheless, we prove that coordinatizability of complemented modular lattices is not firstorder, even for countable 2-...