Summand absorbing submodules of a module over a semiring

Zur Izhakian; Manfred Knebusch; Louis Rowen

doi:10.1016/j.jpaa.2018.11.001

Summand absorbing submodules of a module over a semiring

Zur Izhakian^* (Corresponding Author), Manfred Knebusch^* (Corresponding Author), Louis Rowen^* (Corresponding Author)

^*Corresponding author for this work

Mathematical Science

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

14 Downloads (Pure)

Abstract

An R-module V over a semiring R lacks zero sums (LZS) if x+y=0⇒x=y=0. More generally, a submodule W of V is summand absorbing in V if ∀x,y∈V: x+y∈W⇒x∈W,y∈W. These arise in tropical algebra and modules over idempotent semirings. We explore the lattice of summand absorbing submodules of a given LZS module, especially those that are finitely generated, in terms of the lattice-theoretic Krull dimension, and describe their explicit generation.

Original language	English
Pages (from-to)	3262-3294
Number of pages	33
Journal	Journal of Pure and Applied Algebra
Volume	223
Issue number	8
Early online date	9 Nov 2018
DOIs	https://doi.org/10.1016/j.jpaa.2018.11.001
Publication status	Published - 31 Aug 2019

Keywords

Semiring
lacking zero sums
direct sum decomposition
free (semi)module
projective (semi)module
indecomposable
semidirect complement
upper bound monoid
weak complement
Upper bound monoid
Lacking zero sums
Direct sum decomposition
Indecomposable
Projective (semi)module

Access to Document

10.1016/j.jpaa.2018.11.001Licence: Unspecified

summand absorbing submodules of a module over a semiringSubmitted manuscript, 381 KBLicence: Unspecified

http://arxiv.org/abs/1705.10089v1Licence: Unspecified

Cite this

@article{0829e385792441788ba2587c10b0b9ef,

title = "Summand absorbing submodules of a module over a semiring",

abstract = "An R-module V over a semiring R lacks zero sums (LZS) if x+y=0⇒x=y=0. More generally, a submodule W of V is summand absorbing in V if ∀x,y∈V: x+y∈W⇒x∈W,y∈W. These arise in tropical algebra and modules over idempotent semirings. We explore the lattice of summand absorbing submodules of a given LZS module, especially those that are finitely generated, in terms of the lattice-theoretic Krull dimension, and describe their explicit generation.",

keywords = "Semiring, lacking zero sums, direct sum decomposition, free (semi)module, projective (semi)module, indecomposable, semidirect complement, upper bound monoid, weak complement, Upper bound monoid, Lacking zero sums, Direct sum decomposition, Indecomposable, Projective (semi)module",

author = "Zur Izhakian and Manfred Knebusch and Louis Rowen",

year = "2019",

month = aug,

day = "31",

doi = "10.1016/j.jpaa.2018.11.001",

language = "English",

volume = "223",

pages = "3262--3294",

journal = "Journal of Pure and Applied Algebra",

issn = "0022-4049",

publisher = "Elsevier",

number = "8",

}

TY - JOUR

T1 - Summand absorbing submodules of a module over a semiring

AU - Izhakian, Zur

AU - Knebusch, Manfred

AU - Rowen, Louis

PY - 2019/8/31

Y1 - 2019/8/31

N2 - An R-module V over a semiring R lacks zero sums (LZS) if x+y=0⇒x=y=0. More generally, a submodule W of V is summand absorbing in V if ∀x,y∈V: x+y∈W⇒x∈W,y∈W. These arise in tropical algebra and modules over idempotent semirings. We explore the lattice of summand absorbing submodules of a given LZS module, especially those that are finitely generated, in terms of the lattice-theoretic Krull dimension, and describe their explicit generation.

AB - An R-module V over a semiring R lacks zero sums (LZS) if x+y=0⇒x=y=0. More generally, a submodule W of V is summand absorbing in V if ∀x,y∈V: x+y∈W⇒x∈W,y∈W. These arise in tropical algebra and modules over idempotent semirings. We explore the lattice of summand absorbing submodules of a given LZS module, especially those that are finitely generated, in terms of the lattice-theoretic Krull dimension, and describe their explicit generation.

KW - Semiring

KW - lacking zero sums

KW - direct sum decomposition

KW - free (semi)module

KW - projective (semi)module

KW - indecomposable

KW - semidirect complement

KW - upper bound monoid

KW - weak complement

KW - Upper bound monoid

KW - Lacking zero sums

KW - Direct sum decomposition

KW - Indecomposable

KW - Projective (semi)module

UR - http://www.scopus.com/inward/record.url?scp=85056831608&partnerID=8YFLogxK

UR - http://www.mendeley.com/research/summand-absorbing-submodules-module-semiring

U2 - 10.1016/j.jpaa.2018.11.001

DO - 10.1016/j.jpaa.2018.11.001

M3 - Article

SN - 0022-4049

VL - 223

SP - 3262

EP - 3294

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

IS - 8

ER -

Summand absorbing submodules of a module over a semiring

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

Cite this