Abstract
Ewald (J Symbolic Logic 51(1):166–179, 1986) considered tense operators \(G\), \(H\), \(F\) and \(P\) on intuitionistic propositional calculus and constructed an intuitionistic tense logic system called IKt. The aim of this paper is to give an algebraic axiomatization of the IKt system. We will also show that the algebraic axiomatization given by Chajda (Cent Eur J Math 9(5):1185–1191, 2011) of the tense operators \(P\) and \(F\) in intuitionistic logic is not in accordance with the Halmos definition of existential quantifiers. In this paper, we will study the IKt variety of IKt-algebras. First, we will introduce some examples and we will prove some properties. Next, we will prove that the IKt system has IKt-algebras as algebraic counterpart. We will also describe a discrete duality for IKt-algebras bearing in mind the results indicated by Orłowska and Rewitzky (Fundam Inform 81(1–3):275–295, 2007) for Heyting algebras. We will also get a general construction of tense operators on a complete Heyting algebra, which is a power lattice via the so-called Heyting frame. Finally, we will introduce the notion of tense deductive system which allowed us both to determine the congruence lattice in an IKt-algebra and to characterize simple and subdirectly irreducible algebras of the IKt variety.
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Notes
A unary operator \(\exists \) defined on a Heyting algebra \(\mathcal {A}\) is an existential quantifier if it satisfies the following conditions: \(\exists 0=0\), \(x\wedge \exists x=x\), \(\exists (x\wedge \exists y)=\exists x\wedge \exists y\) and \(\exists (x\vee y)=\exists x\vee \exists y\).
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We thank the anonymous referee for the very thorough reading and contributions to improve our presentation of the paper.
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Communicated by L. Spada.
The support of CONICET is gratefully acknowledged by Gustavo Pelaitay.
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Figallo, A.V., Pelaitay, G. An algebraic axiomatization of the Ewald’s intuitionistic tense logic. Soft Comput 18, 1873–1883 (2014). https://doi.org/10.1007/s00500-014-1317-6
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DOI: https://doi.org/10.1007/s00500-014-1317-6