Subdirect products and propagating equations with an application to the Moufang Theorem

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F24%3A10509656" target="_blank" >RIV/00216208:11320/24:10509656 - isvavai.cz</a>

Výsledek na webu

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=i4meQKls93" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=i4meQKls93</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.26493/2590-9770.1715.3ef" target="_blank" >10.26493/2590-9770.1715.3ef</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Subdirect products and propagating equations with an application to the Moufang Theorem

Popis výsledku v původním jazyce

We introduce the concept of propagating equations and focus on the case of associativity propagating in varieties of loops.An equation ε propagates in an algebra X if ε(-RIGHTWARDS ARROWy ) holds whenever ε(-RIGHTWARDS ARROWx ) holds and the elements of -RIGHTWARDS ARROWy are contained in the subalgebra of X generated by -RIGHTWARDS ARROWx . If ε propagates in X then it propagates in all subalgebras and products of X but not necessarily in all homomorphic images of X. If V is a variety, the propagating core V[ε] = {X ELEMENT OF V :ε propagates in X} is a quasivariety but not necessarily a variety. We prove by elementary means Goursat&apos;s Lemma for loops and describe all subdirect products of Xk and all finitely generated loops in HSP(X) for a nonabelian simple loopX. If V is a variety of loops in which associativity propagates, X is a finite loop in whichassociativity propagates and every subloop of X is either nonabelian simple or contained in V, then associativity propagates in HSP(X) LOGICAL OR V. We study the propagating core S[x(yz)=(xy)z] of Steiner loops with respect to associativity. While this is not a variety, we exhibit many varieties contained in S[x(yz)=(xy)z], each providing a solution to Rajah&apos;s problem, i.e., a variety of loops not contained in Moufang loops in which the Moufang Theorem holds.

Název v anglickém jazyce

Subdirect products and propagating equations with an application to the Moufang Theorem

Popis výsledku anglicky

We introduce the concept of propagating equations and focus on the case of associativity propagating in varieties of loops.An equation ε propagates in an algebra X if ε(-RIGHTWARDS ARROWy ) holds whenever ε(-RIGHTWARDS ARROWx ) holds and the elements of -RIGHTWARDS ARROWy are contained in the subalgebra of X generated by -RIGHTWARDS ARROWx . If ε propagates in X then it propagates in all subalgebras and products of X but not necessarily in all homomorphic images of X. If V is a variety, the propagating core V[ε] = {X ELEMENT OF V :ε propagates in X} is a quasivariety but not necessarily a variety. We prove by elementary means Goursat&apos;s Lemma for loops and describe all subdirect products of Xk and all finitely generated loops in HSP(X) for a nonabelian simple loopX. If V is a variety of loops in which associativity propagates, X is a finite loop in whichassociativity propagates and every subloop of X is either nonabelian simple or contained in V, then associativity propagates in HSP(X) LOGICAL OR V. We study the propagating core S[x(yz)=(xy)z] of Steiner loops with respect to associativity. While this is not a variety, we exhibit many varieties contained in S[x(yz)=(xy)z], each providing a solution to Rajah&apos;s problem, i.e., a variety of loops not contained in Moufang loops in which the Moufang Theorem holds.

Druh

J_SC - Článek v periodiku v databázi SCOPUS

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/LTAUSA19070" target="_blank" >LTAUSA19070: Komutátory, kvazigrupy a Yang-Baxterova rovnice</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)<br>I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2024

Kód důvěrnosti údajů

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

Název periodika

The Art of Discrete and Applied Mathematics

ISSN

2590-9770

e-ISSN

2590-9770

Svazek periodika

7

Číslo periodika v rámci svazku

3

Stát vydavatele periodika

SI - Slovinská republika

Počet stran výsledku

20

Strana od-do

nestránkováno

Kód UT WoS článku

—

EID výsledku v databázi Scopus

2-s2.0-85213223251