Directly decomposable ideals and congruence kernels of commutative semirings

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F61989592%3A15310%2F20%3A73603179" target="_blank" >RIV/61989592:15310/20:73603179 - isvavai.cz</a>

Result on the web

<a href="http://mat76.mat.uni-miskolc.hu/mnotes/article/2819" target="_blank" >http://mat76.mat.uni-miskolc.hu/mnotes/article/2819</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.18514/MMN.2020.2819" target="_blank" >10.18514/MMN.2020.2819</a>

Result language

angličtina

Original language name

Directly decomposable ideals and congruence kernels of commutative semirings

Original language description

As pointed out in the monographs [5, 6] on semirings, ideals play an important role despite the fact that they need not be congruence kernels as in the case of rings. Hence, having two commutative semirings S-1 and S-2, one can ask whether an ideal I of their direct product S = S-1 x S-2 can be expressed in the form I-1 x I-2 where I-j is an ideal of S-j for j = 1, 2. Of course, the converse is elementary, namely if I-j is an ideal of S-j for j = 1, 2 then I-1 x I-2 is an ideal of S-1 x S-2. Having a congruence Theta on a commutative semiring S, its 0-class is an ideal of S, but not every ideal is of this form. Hence, the lattice IdS of all ideals of S and the lattice KerS of all congruence kernels (i.e. 0-classes of congruences) of S need not be equal. Furthermore, we show that the mapping Theta bar right arrow [0]Theta need not be a homomorphism from ConS onto KerS. Moreover, the question arises when a congruence kernel of the direct product S-1 x S-2 of two commutative semirings can be expressed as a direct product of the corresponding kernels on the factors. In the paper we present necessary and sufficient conditions for such direct decompositions both for ideals and for congruence kernels of commutative semirings. We also provide sufficient conditions for varieties of commutative semirings to have directly decomposable kernels.

Czech name

—

Czech description

—

Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

OECD FORD branch

10101 - Pure mathematics

Project

<a href="/en/project/GF20-09869L" target="_blank" >GF20-09869L: The many facets of orthomodularity</a><br>

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2020

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Miskolc Mathematical Notes

ISSN

1787-2405

e-ISSN

—

Volume of the periodical

21

Issue of the periodical within the volume

1

Country of publishing house

HU - HUNGARY

Number of pages

13

Pages from-to

"113 "- 125

UT code for WoS article

000541509200008

EID of the result in the Scopus database

2-s2.0-85089540848