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...

An inequality between the number of coverings in the ordered set J(Con L) of join irreducible congruences on a lattice L and the size of L is given. Using this inequality it is shown that this ordered set can be computed in time O(n2 log2 n), where n = |L|. This paper is motivated by the problem of efficiently calculating and representing the congruence lattice Con L of a finite lattice L. Of c...

From a well-known decomposition theorem, we propose a tree representation for distributive and simplicial lattices. We show how this representation (called ideal tree) can be efficiently computed (linear time in the size of the lattice given by any graph whose transitive closure is the lattice) and compared with respect to time and space complexity. As far as time complexity is concerned, we si...

Abstract In the present paper, the interval-valued pP, P_ qq-fuzzy LI-ideal theory in lattice implication algebras is further studied. Some new properties of interval-valued pP, P_ qq-fuzzy LI-ideals are given. Representation theorem of interval-valued pP, P_ qq-fuzzy LI-ideal which is generated by an interval-valued fuzzy set is established. It is proved that the set consisting of all interval...

In recent years, the combinatorial properties of monomials ideals [7, 10, 14] and binomial ideals [9, 10] have been widely studied. In particular, combinatorial interpretations of minimal free resolutions have been given in both cases. In this present work, we will generalize existing techniques to obtain two new results. The first is S[Λ]-resolutions of Λ-invariant submodules of k[Z] where Λ i...

A calyx (multiplicative lattice) is a complete lattice endowed with the structure of a monoid such that multiplication by an element is a left adjoint functor of complete lattices (equivalently, a left adjoint functor which preserves colimits). Calyxes are a generalization of the set of ideals of a ring, which form a complete lattice under intersection and summation; in conjunction with the nat...

Since in 1936 Birkhoff and von Neumann regarded the lattice of all closed subspaces of a separable infinite-dimensional Hilbert space that is an orthomodular lattice as a proposition system for a quantum mechanical entity (Miklós, 1998), orthomodular lattices have been considered as a mathematical model for a calculus of quantum logic. With the development of the theory of quantum logics, ortho...