On Decompositions of Matrices over Distributive Lattices

Abstract

Let $L$ be a distributive lattice and ${\mathrm{M}}_{n,q}$ ($L)$ ${\mathrm{(M}}_n(L)$, resp.) the semigroup (semiring, resp.) of $n\times q$ ($n\times n$, resp.) matrices over $L$. In this paper, we show that if there is a subdirect embedding from distributive lattice $L$ to the direct product ${\prod }_{i=1}^m\mathrm{‍}{L}_i$ of distributive lattices $L_1,L_2,\mathrm{ }\dots ,L_m$, then there will be a corresponding subdirect embedding from the matrix semigroup ${\mathrm{M}}_{n,q}(L)$ (semiring ${\mathrm{M}}_n(L)$, resp.) to semigroup ${\prod }_{i=1}^m\mathrm{‍}{\mathrm{M}}_{n,q}(L_i)$ (semiring ${\prod }_{i=1}^m\mathrm{‍}{\mathrm{M}}_n(L_i)$, resp.). Further, it is proved that a matrix over a distributive lattice can be decomposed into the sum of matrices over some of its special subchains. This generalizes and extends the decomposition theorems of matrices over finite distributive lattices, chain semirings, fuzzy semirings, and so forth. Finally, as some applications, we present a method to calculate the indices and periods of the matrices over a distributive lattice and characterize the structures of idempotent and nilpotent matrices over it. We translate the characterizations of idempotent and nilpotent matrices over a distributive lattice into the corresponding ones of the binary Boolean cases, which also generalize the corresponding structures of idempotent and nilpotent matrices over general Boolean algebras, chain semirings, fuzzy semirings, and so forth.