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Singly generated quasivarieties and residuated structures
Author(s) -
Moraschini Tommaso,
Raftery James G.,
Wannenburg Johann J.
Publication year - 2020
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201900012
Subject(s) - mathematics , embedding , retract , variety (cybernetics) , class (philosophy) , simple (philosophy) , pure mathematics , extension (predicate logic) , algebra over a field , discrete mathematics , computer science , artificial intelligence , philosophy , statistics , epistemology , programming language
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 ℵ 0 ‐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.