On Subdirect Decompositions of Finite Distributive Lattices

Yizhi Chen,Zhongzhu Liu,Jing Tian

doi:10.1155/2017/6490903

Yizhi Chen, Zhongzhu Liu + Show 1 more

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https://doi.org/10.1155/2017/6490903

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Abstract

Subdirect decomposition of algebra is one of its quite general and important constructions. In this paper, some subdirect decompositions (including subdirect irreducible decompositions) of finite distributive lattices and finite chains are studied, and some general results are obtained.

Highlights

Introduction and PreliminariesAs for as semirings concerned, there are several ways of approaching subdirect decompositions of semirings
Introduction and PreliminariesA semiring R is an algebraic structure (R, +, ⋅) consisting of a nonempty set R together with two binary operations + and ⋅ on R such that (R, +) and (R, ⋅) are semigroups connected by distributivity, that is, a(b + c) = ab + ac and (b + c)a = ba+ca, for all a, b, c ∈ R [1, 2]
There is a third way of approaching subdirect decompositions which is based on another Birkhoff theorem verified in [4], which, in terms of semirings, says that a semiring R is a subdirect product of a family of semirings {Ri}i∈I if and only if there exists a family of factor congruences {ρi}i∈I on R such that ⋂i∈I ρi = εR and R/ρi = Ri for each i ∈ I; here, εR is the identity congruence on R

Summary

Introduction and Preliminaries

As for as semirings concerned, there are several ways of approaching subdirect decompositions of semirings In most cases they can be obtained from various semirings theoretical constructions. There is a third way of approaching subdirect decompositions which is based on another Birkhoff theorem verified in [4], which, in terms of semirings, says that a semiring R is a subdirect product of a family of semirings {Ri}i∈I if and only if there exists a family of factor congruences {ρi}i∈I on R such that ⋂i∈I ρi = εR and R/ρi = Ri for each i ∈ I; here, εR is the identity congruence on R. Some subdirect decompositions of a finite distributive lattice are discussed in [8], and the subdirect decompositions of a finite chain are studied in [2], the results in this paper will be more general. For notations and terminologies occurred but not mentioned in this paper, the readers are referred to [1, 4]

Subdirect Decompositions of Finite Distributive Lattices

Findings

Conclusion