The variety Sℋ of semi-Heyting algebras was introduced by H. P. Sankappanavar (in: Proceedings of the 9th "Dr. Antonio A. R. Monteiro" Congress, Universidad Nacional del Sur, Bahía Blanca, 2008) 13 ...as an abstraction of the variety of Heyting algebras. Semi-Heyting algebras are the algebraic models for a logic HsH, known as semi-intuitionistic logic, which is equivalent to the one defined by a Hilbert style calculus in Cornejo (Studia Logica 98(1-2): 9-25, 2011) 6. In this article we introduce a Gentzen style sequent calculus GsH for the semi-intuitionistic logic whose associated logic GsH is the same as HsH. The advantage of this presentation of the logic is that we can prove a cutelimination theorem for GsH that allows us to prove the decidability of the logic. As a direct consequence, we also obtain the decidability of the equational theory of semi-Heyting algebras.
Many intermediate logics, even extremely well-behaved ones such as IPC, lack the finite model property for admissible rules. We give conditions under which this failure holds. We show that frames ...which validate all admissible rules necessarily satisfy a certain closure condition, and we prove that this condition, in the finite case, ensures that the frame is of width 2. Finally, we indicate how this result is related to some classical results on finite, free Heyting algebras.
The IKt-algebras were introduced in the paper An algebraic axiomatization of the Ewald's intuitionistic tense logic by the first and third author. In this paper, our main interest is to investigate ...the principal and Boolean congruences on IKt-algebras. In order to do this we take into account a topological duality for these algebras obtained in Figallo et al. (Stud Log 105(4):673-701, 2017). Furthermore, we characterize Boolean and principal IKt-congruences and we show that Boolean IKt-congruence are principal IKt-congruences. Also, bearing in mind the above results, we obtain that Boolean IKt-congruences are commutative, regular and uniform. Finally, we characterize the principal IKt-congruences in the case that the IKt-algebra is linear and complete whose prime filters are complete and also the case that it is linear and finite. This allowed us to establish that the intersection of two principal IKt-congruences on these algebras is a principal one and also to determine necessary and sufficient conditions so that a principal IKt-congruence is a Boolean one on theses algebras.
So-called classical logic--the logic developed in the early
twentieth century by Gottlob Frege, Bertrand Russell, and
others--is computationally the simplest of the major logics, and it
is adequate ...for the needs of most mathematicians. But it is just
one of the many kinds of reasoning in everyday thought.
Consequently, when presented by itself--as in most introductory
texts on logic--it seems arbitrary and unnatural to students new to
the subject. In Classical and Nonclassical Logics , Eric
Schechter introduces classical logic alongside constructive,
relevant, comparative, and other nonclassical logics. Such logics
have been investigated for decades in research journals and
advanced books, but this is the first textbook to make this subject
accessible to beginners. While presenting an assortment of logics
separately, it also conveys the deeper ideas (such as derivations
and soundness) that apply to all logics. The book leads up to
proofs of the Disjunction Property of constructive logic and
completeness for several logics. The book begins with brief
introductions to informal set theory and general topology, and
avoids advanced algebra; thus it is self-contained and suitable for
readers with little background in mathematics. It is intended
primarily for undergraduate students with no previous experience of
formal logic, but advanced students as well as researchers will
also profit from this book.
The variety \(\mathbb{DHMSH}\) of dually hemimorphic semi-Heyting algebras was introduced in 2011 by the second author as an expansion of semi-Heyting algebras by a dual hemimorphism. In this paper, ...we focus on the variety \(\mathbb{DHMSH}\) from a logical point of view. The paper presents an extensive investigation of the logic corresponding to the variety of dually hemimorphic semi-Heyting algebras and of its axiomatic extensions, along with an equally extensive universal algebraic study of their corresponding algebraic semantics. Firstly, we present a Hilbert-style axiomatization of a new logic called "Dually hemimorphic semi-Heyting logic" (\(\mathcal{DHMSH}\), for short), as an expansion of semi-intuitionistic logic \(\mathcal{SI}\) (also called \(\mathcal{SH}\)) introduced by the first author by adding a weak negation (to be interpreted as a dual hemimorphism). We then prove that it is implicative in the sense of Rasiowa and that it is complete with respect to the variety \(\mathbb{DHMSH}\). It is deduced that the logic \(\mathcal{DHMSH}\) is algebraizable in the sense of Blok and Pigozzi, with the variety \(\mathbb{DHMSH}\) as its equivalent algebraic semantics and that the lattice of axiomatic extensions of \(\mathcal{DHMSH}\) is dually isomorphic to the lattice of subvarieties of \(\mathbb{DHMSH}\). A new axiomatization for Moisil's logic is also obtained. Secondly, we characterize the axiomatic extensions of \(\mathcal{DHMSH}\) in which the "Deduction Theorem" holds. Thirdly, we present several new logics, extending the logic \(\mathcal{DHMSH}\), corresponding to several important subvarieties of the variety \(\mathbb{DHMSH}\). These include logics corresponding to the varieties generated by two-element, three-element and some four-element dually quasi-De Morgan semi-Heyting algebras, as well as a new axiomatization for the 3-valued Łukasiewicz logic. Surprisingly, many of these logics turn out to be connexive logics, only a few of which are presented in this paper. Fourthly, we present axiomatizations for two infinite sequences of logics namely, De Morgan Gödel logics and dually pseudocomplemented Gödel logics. Fifthly, axiomatizations are also provided for logics corresponding to many subvarieties of regular dually quasi-De Morgan Stone semi-Heyting algebras, of regular De Morgan semi-Heyting algebras of level 1, and of JI-distributive semi-Heyting algebras of level 1. We conclude the paper with some open problems. Most of the logics considered in this paper are discriminator logics in the sense that they correspond to discriminator varieties. Some of them, just like the classical logic, are even primal in the sense that their corresponding varieties are generated by primal algebras.
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We characterise the intermediate logics which admit a cut-free hypersequent calculus of the form HLJ + R, where HLJ is the hypersequent counterpart of the sequent calculus LJ for propositional ...intuitionistic logic, and R is a set of so-called structural hypersequent rules, i.e., rules not involving any logical connectives. The characterisation of this class of intermediate logics is presented both in terms of the algebraic and the relational semantics for intermediate logics. We discuss various—positive as well as negative—consequences of this characterisation.
Algebraic Geometry over Heyting Algebras Nouri, Mahdiyeh
Journal of Siberian Federal University. Mathematics & Physics,
01/2020, Volume:
13, Issue:
4
Journal Article
Peer reviewed
Open access
In this article, we study the algebraic geometry over Heyting algebras and we investigate the properties of being equationally Noetherian and qω-compact over such algebras
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