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Varieties of de Morgan monoids: Covers of atoms

T. Moraschini, J. G. Raftery & J. J. Wannenburg

Review of Symbolic Logic 13 (2):338-374 (2020)

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The algebraic significance of weak excluded middle laws.Tomáš Lávička, Tommaso Moraschini & James G. Raftery - 2022 - Mathematical Logic Quarterly 68 (1):79-94.details For (finitary) deductive systems, we formulate a signature‐independent abstraction of the weak excluded middle law (WEML), which strengthens the existing general notion of an inconsistency lemma (IL). Of special interest is the case where a quasivariety algebraizes a deductive system ⊢. We prove that, in this case, if ⊢ has a WEML (in the general sense) then every relatively subdirectly irreducible member of has a greatest proper ‐congruence; the converse holds if ⊢ has an inconsistency lemma. The result extends, in (...) Download Export citation Bookmark 2 citations
Singly generated quasivarieties and residuated structures.Tommaso Moraschini, James G. Raftery & Johann J. Wannenburg - 2020 - Mathematical Logic Quarterly 66 (2):150-172.details A quasivariety 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 can serve as A. A consequence of this demand, called ‘passive structural completeness’ (PSC), is that the nontrivial members of all satisfy the same existential positive sentences. We prove that if is PSC then it still has the JEP, and if it has the JEP and (...) Download Export citation Bookmark 6 citations
Two Maximality Results for the Lattice of Extensions of $$\vdash _{\mathbf {RM}}$$.Krzysztof A. Krawczyk - 2022 - Studia Logica 110 (5):1243-1253.details We use an algebraic argument to prove that there are exactly two premaximal extensions of \’s consequence. We also show that one of these extensions is the minimal structurally complete extension of the unique maximal paraconsistent extension of \. Precisely, we show that there are exactly two covers of the variety of Boolean algebras in the lattice of quasivarieties of Sugihara algebras and that there is a unique minimal paraconsistent quasivariety in that lattice. We also obtain a corollary stating that (...) Download Export citation Bookmark
On Pretabular Extensions of Relevance Logic.Asadollah Fallahi & James Gordon Raftery - forthcoming - Studia Logica:1-19.details We exhibit infinitely many semisimple varieties of semilinear De Morgan monoids (and likewise relevant algebras) that are not tabular, but which have only tabular proper subvarieties. Thus, the extension of relevance logic by the axiom $$(p\rightarrow q)\vee (q\rightarrow p)$$ ( p → q ) ∨ ( q → p ) has infinitely many pretabular axiomatic extensions, regardless of the presence or absence of Ackermann constants. Download Export citation Bookmark

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