Singly generated quasivarieties and residuated structures

A quasivariety K of algebras has the joint embedding property (JEP) iff it is generated by a single algebra A. It is structurally complete iff the free countably 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. It is a minimal quasivariety if, moreover, it is PSC but fails to unify some finite set of equations. We also prove that a quasivariety of finite type, with a finite nontrivial member, is PSC iff 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

Singly generated quasivarieties and residuated structures

Please read abstract in the article.H2020 Marie Skłodowska-Curie Actions; DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa and National Research Foundation of South Africa.https://onlinelibrary.wiley.com/journal/15213870hj2021Mathematics and Applied Mathematic

Admissible rules and the Leibniz hierarchy

This paper provides a semantic analysis of admissible rules and associated completeness conditions for arbitrary deductive systems, using the framework of abstract algebraic logic. Algebraizability is not assumed, so the meaning and signi cance of the principal notions vary with the level of the Leibniz hierarchy at which they are presented. As a case study of the resulting theory, the non-algebraizable fragments of relevance logic are considered.This work is based on research supported in part by the National Research Foundation of South Africa (UID 85407).https://www.dukeupress.edu/notre-dame-journal-of-formal-logichb2016Mathematics and Applied Mathematic