The optimization problem over a distributive lattice

In this paper we give a necessary and sufficient condition for existence of minimal solution(s) of the linear system A * X ≥ b where A, b are fixed matrices and X is an unknown matrix over a lattice. Next, an algorithm which finds these minimal solutions over a distributive lattice is given. Finally, we find an optimal solution for the optimization problem min {Z = C * X | A * X ≥ b} where C is the given matrix of coefficients of objective function Z. This research was completed while the author was a visitor of the Center for Informatics and Applied Optimization, University of Ballarat, Ballarat, Australia.     

The dual of a generalized completely distributive lattice is a hypercontinuous lattice

In this short note, we prove that a complete lattice is a generalized completely distributive lattice if and only if its order dual is a hypercontinuous lattice.     

The Frattini sublattice of a finite distributive lattice

Frattini sublattices of finite distributive lattices are characterized and several applications are given thereof.Presented by J. Berman.The support of Consejo Nacional de Investigaciones Cientificas y Técnicas de la República Argentina is gratefully acknowledged.     

Indecomposable matrices over a distributive lattice

In this paper, the concepts of indecomposable matrices and fully indecomposable matrices over a distributive lattice L are introduced, and some algebraic properties of them are obtained. Also, some characterizations of the set F _n (L) of all n × n fully indecomposable matrices as a subsemigroup of the semigroup H _n (L) of all n × n Hall matrices over the lattice L are given.     

The lattice of idempotent distributive semiring varieties

A solution is given for the word problem for free idempotent distributive semirings. Using this solution the latticeL (ID) of subvarieties of the variety ID of idempotent distributive semirings is determined. It turns out thatL (ID) is isomorphic to the direct product of a four-element lattice and a lattice which is itself a subdirect product of four copies of the latticeL(B) of all band varieties. ThereforeL(ID) is countably infinite and distributive. Every subvariety of ID is finitely based. Project supported by the National Natural Science Foundation of China (Grant No. 19761004) and the Provincial Applied Fundamental Research Foundation of Yunnan (96a001z).     

About generalized ideals in a distributive lattice

Let L be a distributive lattice characterized by a ternary operation (, ,), where (a,b,c)=(ab)(bc)(ac)=(ab)(ac)(bc), a,b,cL. The note considers convex sublattices of L, called generalized ideals of L generated by the operation (, ,). Some remarks have been stated about the graph of a distributive lattice.     

The automorphism group of the universa distributive lattice

We describe the normal subgroup lattice of the automorphism groups of the countable universal homogeneous distributive lattice and of the countable atomless generalized Boolean lattice. Also, we show that subgroups of these automorphism groups of index less than lie between the pointwise and the setwise stabilizer of a finite set. Received March 8, 1999; accepted in final form September 16, 1999.     

On the length of the lattice of sublattices of a finite distributive lattice

Presented by I. Rival.