Singly Generated Quasivarieties and Residuated Structures

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985807%3A_____%2F20%3A00531245" target="_blank" >RIV/67985807:_____/20:00531245 - isvavai.cz</a>

Výsledek na webu

<a href="http://dx.doi.org/10.1002/malq.201900012" target="_blank" >http://dx.doi.org/10.1002/malq.201900012</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1002/malq.201900012" target="_blank" >10.1002/malq.201900012</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Singly Generated Quasivarieties and Residuated Structures

Popis výsledku v původním jazyce

A quasivariety K of algebras has the joint embedding property (JEP) if and only if it is generated by a single algebra A . It is structurally complete if and only if the free ℵ0‐generated algebra in K can serve as A . A consequence of this demand, called ‘passive structural completeness’ (PSC), is that the nontrivial members of K all satisfy the same existential positive sentences. We prove that if K is PSC then it still has the JEP, and if it has the JEP and its nontrivial members lack trivial subalgebras, then its relatively simple members all belong to the universal class generated by one of them. Under these conditions, if K is relatively semisimple then it is generated by one K ‐simple algebra. We also prove that a quasivariety of finite type, with a finite nontrivial member, is PSC if and only if its nontrivial members have a common retract. The theory is then applied to the variety of De Morgan monoids, where we isolate the sub(quasi)varieties that are PSC and those that have the JEP, while throwing fresh light on those that are structurally complete. The results illuminate the extension lattices of intuitionistic and relevance logics.

Název v anglickém jazyce

Singly Generated Quasivarieties and Residuated Structures

Popis výsledku anglicky

A quasivariety K of algebras has the joint embedding property (JEP) if and only if it is generated by a single algebra A . It is structurally complete if and only if the free ℵ0‐generated algebra in K can serve as A . A consequence of this demand, called ‘passive structural completeness’ (PSC), is that the nontrivial members of K all satisfy the same existential positive sentences. We prove that if K is PSC then it still has the JEP, and if it has the JEP and its nontrivial members lack trivial subalgebras, then its relatively simple members all belong to the universal class generated by one of them. Under these conditions, if K is relatively semisimple then it is generated by one K ‐simple algebra. We also prove that a quasivariety of finite type, with a finite nontrivial member, is PSC if and only if its nontrivial members have a common retract. The theory is then applied to the variety of De Morgan monoids, where we isolate the sub(quasi)varieties that are PSC and those that have the JEP, while throwing fresh light on those that are structurally complete. The results illuminate the extension lattices of intuitionistic and relevance logics.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/EF17_050%2F0008361" target="_blank" >EF17_050/0008361: Rozvoj lidských zdrojů pro výzkum v teoretické informatice</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2020

Kód důvěrnosti údajů

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

Název periodika

Mathematical Logic Quarterly

ISSN

0942-5616

e-ISSN

—

Svazek periodika

66

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

DE - Spolková republika Německo

Počet stran výsledku

23

Strana od-do

150-172

Kód UT WoS článku

000541456700001

EID výsledku v databázi Scopus

2-s2.0-85087178840