in this paper, countable compactness and the lindel¨of propertyare defined for l-fuzzy sets, where l is a complete de morgan algebra. theydon’t rely on the structure of the basis lattice l and no distributivity is requiredin l. a fuzzy compact l-set is countably compact and has the lindel¨ofproperty. an l-set having the lindel¨of property is countably compact if andonly if it is fuzzy compact. ...

Canonical extensions of lattice ordered algebras provide an algebraic formulation of what is otherwise treated via topological duality or relational methods. They were firstly introduced by Jónsson and Tarski for Boolean algebras with operators (see [8] and [9]) and generalized for distributive lattices with different operations in [5], [4] and [3]. If A = (A, {fi, i ∈ I}) is a distributive lat...

A distributive lattice L with 0 is finitary if every interval is finite. A function f : N0 ! N0 is a cover function for L if every element with n lower covers has f ðnÞ upper covers. All non-decreasing cover functions have been characterized by the author ([2]), settling a 1975 conjecture of Richard P. Stanley. In this paper, all finitary distributive lattices with cover functions are character...

It is well known that the set of all ideals(2) of a ring forms a complete modular lattice with respect to set inclusion. The same is true of the set of all right ideals. Our purpose in this paper is to consider the consequences of imposing certain additional restrictions on these ideal lattices. In particular, we discuss the case in which one or both of these lattices is complemented, and the c...

Let LM denote the coproduct of the bounded distributive lattices L and M. At the 1981 Bann Conference on Ordered Sets, the following question was posed: What is the largest class L of nite distributive lattices such that, for every non-trivial Boolean lattice B and every L 2 L, B L = B L 0 implies L = L 0 ? In this note, the problem is solved.

We prove strong completeness of a range of substructural logics with respect to their relational semantics by completeness-viacanonicity. 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 langu...

By a new partial ordering relation “ ≤ ” the set of convex sublattices CS(L) of a lattice L is again a lattice. In this paper we establish some results on the pseudocomplementation of (CS(L); ≤). We show that a lattice L with 0 is dense if and only if CS(L) is dense. Then we prove that a finite distributive lattice is a Stone lattice if and only if CS(L) is Stone. We also prove that an upper co...

Abstract The study of the sobriety Scott spaces has got a relatively long history in domain theory. Lawson and Hoffmann independently proved that space every continuous directed complete poset (usually called domain) is sober. Johnstone constructed first whose non-sober. Soon after, Isbell gave lattice with non-sober space. Based on Isbell’s example, Xu, Xi, Zhao showed there even Heyting algeb...