The variety of residuated lattices includes a vast proportion of the classes of algebras that are relevant for algebraic logic, e.g., \-groups, Heyting algebras, MV-algebras, or De Morgan monoids. Among the outliers, one counts orthomodular lattices and other varieties of quantum algebras. We suggest a common framework—pointed left-residuated \-groupoids—where residuated structures and quantum structures can all be accommodated. We investigate the lattice of subvarieties of pointed left-residuated \-groupoids, their ideals, and develop a theory of left nuclei. Finally, we extend some (...) parts of the theory of join-completions of residuated \-groupoids to the left-residuated case, giving a new proof of MacLaren’s theorem for orthomodular lattices. (shrink)

ABSTRACTIn this paper, we consider non-associative generalisations of Hájek's logics BL and psBL. As it was shown by Cignoli, Esteva, Godo, and Torrens, the former is the logic of continuous t-norms and their residua. Botur introduced logic naBL which is the logic of non-associative continuous t-norms and their residua. Thus, naBL can be viewed as a non-associative generalisation of BL. However, Botur has not presented axiomatization of naBL. We fill this gap by constructing an adequate Hilbert-style calculus for naBL. Although, (...) as was shown by Flondor, Georgescu, and Iorgulescu, there are no non-commutative continuous t-norms, Hájek's psBL can be viewed as BL's non-commutative generalisation. We present the logic psnaBL of psnaBL-algebras which can be viewed as naBL's non-commutative generalisation as well as psBL's non-associative generalisation and BL's both non-commutative and non-associative generalisation. (shrink)

This paper presents an abstract study of completeness properties of non-classical logics with respect to matricial semantics. Given a class of reduced matrix models we define three completeness properties of increasing strength and characterize them in several useful ways. Some of these characterizations hold in absolute generality and others are for logics with generalized implication or disjunction connectives, as considered in the previous papers. Finally, we consider completeness with respect to matrices with a linear dense order and characterize it in (...) terms of an extension property and a syntactical metarule. This is the final part of the investigation started and developed in the papers :417–446, 2010; Arch Math Logic 53:353–372, 2016). (shrink)