On Decompositions of Matrices over Distributive Lattices

Yizhi Chen,Xianzhong Zhao

doi:10.1155/2014/202075

Yizhi Chen, Xianzhong Zhao

Open Access

https://doi.org/10.1155/2014/202075

Copy DOI

Abstract

LetLbe a distributive lattice andMn,q(L)(Mn(L), resp.) the semigroup (semiring, resp.) ofn×q(n×n, resp.) matrices overL. In this paper, we show that if there is a subdirect embedding from distributive latticeLto the direct product∏i=1m‍Liof distributive latticesL1,L2, …,Lm, then there will be a corresponding subdirect embedding from the matrix semigroupMn,q(L)(semiringMn(L), resp.) to semigroup∏i=1m‍Mn,q(Li)(semiring∏i=1m‍Mn(Li), 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.

Highlights

Introduction and PreliminariesA semiring is an algebra (R, +, ⋅) with two binary operations + and ⋅ such that both (R, +) and (R, ⋅) are semigroups and such that the distributive laws x (y + z) ≈ xy + xz, (x + y) z ≈ xz + yz (1)are satisfied
A partially ordered semiring means a semiring R equipped with a compatible ordering ≤; that is, ≤ is a partial order on R satisfying the following condition: a ≤ b, c ≤ d 󳨐⇒ a + c ≤ b + d, ac ≤ bd (2)
A distributive lattice L is a lattice which satisfies either of the distributive laws and whose addition + and the multiplication ⋅ on L are as follows: a + b = a ∨ b, ab = a ∧ b

Summary

Introduction and Preliminaries

A semiring is an algebra (R, +, ⋅) with two binary operations + and ⋅ such that both (R, +) and (R, ⋅) are semigroups and such that the distributive laws x (y + z) ≈ xy + xz, (x + y) z ≈ xz + yz (1). We will introduce several kinds of distributive lattices which will often occur: general Boolean algebras (including binary Boolean algebras), chain semirings (including chains), and fuzzy semirings. 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, including the corresponding results in [6, 8, 30]. We translate the characterizations of idempotent and nilpotent matrices over a distributive lattice into the corresponding ones of the binary Boolean cases, which generalize and extend the corresponding structures of idempotent and nilpotent matrices over Boolean algebras, chain semirings, fuzzy semirings, and so forth. For notations and terminologies that occurred but are not mentioned in this paper, readers are referred to [32,33,34]

Decompositions of Matrices over a Distributive Lattice

Some Applications