Directly decomposable ideals and congruence kernels of commutative semirings

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

Directly decomposable ideals and congruence kernels of commutative semirings

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

Directly decomposable ideals and congruence kernels of commutative semirings

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GF20-09869L" target="_blank" >GF20-09869L: Ortomodularita z různých pohledů</a><br>

Návaznosti

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

Rok uplatnění

2020

Kód důvěrnosti údajů

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

Název periodika

Miskolc Mathematical Notes

ISSN

1787-2405

e-ISSN

—

Svazek periodika

21

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

HU - Maďarsko

Počet stran výsledku

13

Strana od-do

"113 "- 125

Kód UT WoS článku

000541509200008

EID výsledku v databázi Scopus

2-s2.0-85089540848