An rj-normal lattice is a distributive lattice with 0 such that each prime ideal contains at most n minimal prime ideals. A relatively ra-normal lattice is a distributive lattice such that each bounded closed interval is an /¡.-normal lattice. The main results of this paper are: (1) a distributive lattice L with 0 is 71,-normal if and only if for any %n, x,,*•', X e L such that x Ax. = 0 for an...

In the late 1930's Garrett Birkhoff [3] pioneered the theory of distributive lattices. An important component in this theory is the concept of exponentiation of lattices [4]: for a lattice L and a partially ordered set P let L denote the set of all order-preserving maps of P to L partially ordered b y / ^ g if and only if/(;c) ^ g(x) for each x e P (see Figure 1). Indeed, If is a lattice. This ...

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...

Let L be a bounded distributive lattice. For k 1, let Sk (L) be the lattice of k-ary functions on L with the congruence substitution property (Boolean functions); let S(L) be the lattice of all Boolean functions. The lattices that can arise as Sk (L) or S(L) for some bounded distributive lattice L are characterized in terms of their Priestley spaces of prime ideals. For bounded distributive lat...

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.

If L is a lattice, the automorphism group of L is denoted Aut(L). It is known that given a finite abstract group H, there exists a finite distributive lattice D such that Aut(D) £= H. It is also known that one cannot expect to find a finite orthocomplemented distributive (Boolean) lattice B such that Aut(B) s= H. In this paper it is shown that there does exist a finite orthomodular lattice L su...

Let L be a lattice and let L1, L2 be sublattices of L. Let be a congruence relation of L1. We extend to L by taking the smallest congruence of L containing . Then we restrict to L2, obtaining the congruence L2 of L2. Thus we have de ned a map ConL1 ! ConL2. Obviously, this is an isotone 0-preservingmap of the nite distributive lattice ConL1 into the nite distributive lattice ConL2. The main res...

Let L be a finite distributive lattice, and let J(L) denote the set of all join-irreducible elements of L. Set j (L) = IJ(Z)l. For each a C J(L), let u(a) denote the number of elements in the prime filter {x C L: x >~a}. Our main theorem is Theorem 1. For any finite distributive lattice L, 4 "(a) ~>j(L)41q ,'2. aEJ(L) The base 4 here can most likely be replaced by a smaller number, but it canno...

Let n be an ordinal number, let L be an add-associative right complementable right zeroed unital distributive non empty double loop structure, and let m1, m2 be monomials of n, L. Note that m1 ∗m2 is monomial-like. Let n be an ordinal number, let L be an add-associative right complementable right zeroed distributive non empty double loop structure, and let c1, c2 be constant polynomials of n, L...

In the present paper we introduce and study L-pre-T0-, L-pre-T1-, L-pre-T2 (L-pre-Hausdorff)-, L-pre-T3 (L-preregularity)-, L-pre-T4 (L-pre-normality)-, L-pre-strong-T3-, L-pre-strong-T4-, L-pre-R0-, L-pre-R1-separation axioms in (2, L)-topologies where L is a complete residuated lattice. Sometimes we need more conditions on L such as the completely distributive law or that the ”∧” is distribut...