Inconsistency lemmas in algebraic logic

In this paper, the inconsistency lemmas of intuitionistic and classical propositional logic are formulated abstractly. We prove that, when a (finitary) deductive system is algebraized by a variety K, then has an inconsistency lemma—in the abstract sense—iff every algebra in K has a dually pseudo-complemented join semilattice of compact congruences. In this case, the following are shown to be equivalent: (1) has a classical inconsistency lemma; (2) has a greatest compact theory and K is filtral, i.e., semisimple with EDPC; (3) the compact congruences of any algebra in K form a Boolean lattice; (4) the compact congruences of any A ∈ K constitute a Boolean sublattice of the full congruence lattice of A. These results extend to quasivarieties and relative congruences. Except for (2), they extend even to protoalgebraic logics, with deductive filters in the role of congruences. A protoalgebraic system with a classical inconsistency lemma always has a deduction-detachment theorem (DDT), while a system with a DDT and a greatest compact theory has an inconsistency lemma. The converses are false.http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1521-3870hb201

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

Structural completeness in relevance logics

It is proved that the relevance logic R (without sentential constants) has no structurally complete consistent axiomatic extension, except for classical propositional logic. In fact, no other such extension is even passively structurally complete.http://link.springer.com/journal/112252017-06-30hb201

Idempotent residuated structures : some category equivalences and their applications

This paper concerns residuated lattice-ordered idempotent commutative monoids that are subdirect products of chains. An algebra of this kind is a generalized Sugihara monoid (GSM) if it is generated by the lower bounds of the monoid identity; it is a Sugihara monoid if it has a compatible involution :. Our main theorem establishes a category equivalence between GSMs and relative Stone algebras with a nucleus (i.e., a closure operator preserving the lattice operations). An analogous result is obtained for Sugihara monoids. Among other applications, it is shown that Sugihara monoids are strongly amalgamable, and that the relevance logic RMt has the projective Beth de nability property for deduction.http://www.ams.org//journals/tran/hb201

Relative congruence formulas and decompositions in quasivarieties

Quasivarietal analogues of uniform congruence schemes are discussed, and their relationship with the equational definability of principal relative congruences (EDPRC) is established, along with their significance for relative congruences on subalgebras of products. Generalizing the situation in varieties, we prove that a quasivariety is relatively ideal iff it has EDPRC; it is relatively filtral iff it is relatively semisimple with EDPRC. As an application, it is shown that a finitary sentential logic, algebraized by a quasivariety K, has a classical inconsistency lemma if and only if K is relatively filtral and the subalgebras of its nontrivial members are nontrivial. A concrete instance of this result is exhibited, in which K is not a variety. Finally, for quasivarieties M⊆K, we supply some conditions under which M is the restriction to K of a variety, assuming that K has EDPRC.The second author was supported in part by the National Research Foundation of South Africa (UID 85407).https://link.springer.com/journal/122482018-11-27hj2017Mathematics and Applied Mathematic

The algebraic significance of weak excluded middle laws

Please read abstract in the article.National Research Foundation of South Africa; Ministry of Science and Innovation of Spain; Agència de Gestió d'Ajuts Universitaris i de Recerca.https://onlinelibrary.wiley.com/journal/15213870hj2023Mathematics and Applied Mathematic

Varieties of De Morgan monoids : covers of atoms

The variety DMM of De Morgan monoids has just four minimal subvarieties. The join-irreducible covers of these atoms in the subvariety lattice of DMM are investigated. One of the two atoms consisting of idempotent algebras has no such cover; the other has just one. The remaining two atoms lack nontrivial idempotent members. They are generated, respectively, by 4{element De Morgan monoids C4 and D4, where C4 is the only nontrivial 0{generated algebra onto which nitely subdirectly irreducible De Morgan monoids may be mapped by non-injective homomorphisms. The homomorphic pre-images of C4 within DMM (together with the trivial De Morgan monoids) constitute a proper quasivariety, which is shown to have a largest subvariety U. The covers of the variety V(C4) within U are revealed here. There are just ten of them (all nitely generated). In exactly six of these ten varieties, all nontrivial members have C4 as a retract. In the varietal join of those six classes, every subquasivariety is a variety|in fact, every nite subdirectly irreducible algebra is projective. Beyond U, all covers of V(C4) [or of V(D4)] within DMM are discriminator varieties. Of these, we identify in nitely many that are nitely generated, and some that are not. We also prove that there are just 68 minimal quasivarieties of De Morgan monoids.The European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant, RVO 67985807 and by the CAS-ICS postdoctoral fellowship, the National Research Foundation of South Africa and DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa.https://www.cambridge.org/core/journals/review-of-symbolic-logic2021-06-01am2021Mathematics and Applied Mathematic

Epimorphisms, definability and cardinalities

We characterize, in syntactic terms, the ranges of epimorphisms in an arbitrary class of similar first-order structures (as opposed to an elementary class). This allows us to strengthen a result of Bacsich, as follows: in any prevariety having at most s non-logical symbols and an axiomatization requiring at most m variables, if the epimorphisms into structures with at most m+s+ℵ0 elements are surjective, then so are all of the epimorphisms. Using these facts, we formulate and prove manageable ‘bridge theorems’, matching the surjectivity of all epimorphisms in the algebraic counterpart of a logic ⊢ with suitable infinitary definability properties of ⊢, while not making the standard but awkward assumption that ⊢ comes furnished with a proper class of variables.The European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 689176 (project “Syntax Meets Semantics: Methods, Interactions, and Connections in Substructural logics”). The first author was also supported by the Project GA17-04630S of the Czech Science Foundation (GAČR). The second author was supported in part by the National Research Foundation of South Africa (UID 85407). The third author was supported by the DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa.http://link.springer.com/journal/112252020-02-07hj2019Mathematics and Applied Mathematic

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

Effects of edge-stiffened web openings on the behaviour of cold-formed steel channel sections under compression

The use of cold-formed steel (CFS) channel sections are becoming popular as the load-carrying members in building structures, and such channel sections often include web openings for the ease of installation of services. Traditional web openings are normally punched, and are unstiffened which can restrict the size and spacing of web openings. Recently, a new generation of CFS channel sections with edge-stiffened web openings has been developed, and is widely used in New Zealand. However, no experimental investigation has been reported in the literature for such channel sections under compression. In this paper, a total of 75 results comprising 26 axial compression tests and 49 finite element analysis results are reported on the compression resistance of CFS channel sections with both edge-stiffened and unstiffened web openings. For comparison, channel sections without web openings were also tested. For all specimens, initial imperfections were measured using a laser scanner. A nonlinear elasto-plastic finite element model was also developed, and the results showed good agreement with the test results. A parametric study was conducted using the validated finite element model to investigate the effect of opening spacing and column length on compression resistance of channel sections. It is shown that for the case of a channel section having seven edge-stiffened web openings, the compression resistance increased by as much as 22%, compared to a plain channel section. For comparison, the same section having unstiffened web openings had a 20% reduction in compression resistance, compared to a plain channel section