Singly Generated Quasivarieties and Residuated Structures

0531245 - ÚI 2021 RIV DE eng J - Journal Article
Moraschini, Tommaso - Raftery, J.G. - Wannenburg, J. J.
Singly Generated Quasivarieties and Residuated Structures.
Mathematical Logic Quarterly. Roč. 66, č. 2 (2020), s. 150-172. ISSN 0942-5616. E-ISSN 1521-3870
R&D Projects: GA MŠMT(CZ) EF17_050/0008361
EU Projects: European Commission(XE) 689176 - SYSMICS
Institutional support: RVO:67985807
Keywords : joint embedding property * passive structural completeness * relevance principle * abstract algebraic logic * relevance logic
OECD category: Pure mathematics
Impact factor: 0.240, year: 2020
Method of publishing: Limited access
http://dx.doi.org/10.1002/malq.201900012

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.
Permanent Link: http://hdl.handle.net/11104/0309943