In 1973, Katriňák proved that regular double
-algebras can be regarded as (regular) double Heyting algebras by ingeniously constructing binary terms for the Heyting implication and its dual in terms ...of pseudocomplement and its dual. In this paper, we prove a converse to Katriňák’s theorem, in the sense that in the variety
of regular dually pseudocomplemented Heyting algebras, the implication operation → satisfies Katriňák’s formula. As applications of this result together with the above-mentioned Katriňák’s theorem, we show that the varieties
,
,
and
of regular double
-algebras, regular dually pseudocomplemented Heyting algebras, regular pseudocomplemented dual Heyting algebras, and regular double Heyting algebras, respectively, are term-equivalent to each other and also that the varieties
,
,
,
of regular De Morgan
-algebras, regular De Morgan Heyting algebras, regular De Morgan double Heyting algebras, and regular De Morgan double
-algebras, respectively, are also term-equivalent to each other. From these results and recent results of Adams, Sankappanavar and Vaz de Carvalho on varieties of regular double
-algebras and regular pseudocomplemented De Morgan algebras, we deduce that the lattices of subvarieties of all these varieties have cardinality
. We then define new logics,
,
and
, and show that they are algebraizable with
,
and
, respectively, as their equivalent algebraic semantics. It is also deduced that the lattices of extensions of all of the above mentioned logics have cardinality
T-rough Heyting algebras were introduced by Eric SanJuan in 2008 as an algebraic formalism for reasoning on finite increasing sequences over Boolean algebras in general and on generalizations of ...rough set concepts in particular. In this paper, we introduce the variety of algebras, which we call T-rough symmetric Heyting algebras. These algebras constitute an extension of T-rough Heyting algebras and a generalization of symmetric Heyting algebras of order n. Our main interest is the representation theory of tense operators on T-rough symmetric Heyting algebras. In order to do this, a discrete-style duality for these algebras is developed.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Inspired by the definition of tense operators on distributive lattices presented by Chajda and Paseka in 2015, in this paper, we introduce and study the variety of tense distributive lattices with ...implication and we prove that these are categorically equivalent to a full subcategory of the category of tense centered Kleene algebras with implication. Moreover, we apply such an equivalence to describe the congruences of the algebras of each variety by means of tense 1-filters and tense centered deductive systems, respectively.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
We show that if a subgroup of the automorphism group of the Fraïssé limit of finite Heyting algebras has a countable index, then it lies between the pointwise and setwise stabilizer of some finite ...set.
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Recent research on algebraic models of
quasi-Nelson logic
has brought new attention to a number of classes of algebras which result from enriching (subreducts of) Heyting algebras with a special ...modal operator, known in the literature as a
nucleus
. Among these various algebraic structures, for which we employ the umbrella term
intuitionistic modal algebras
, some have been studied since at least the 1970s, usually within the framework of topology and sheaf theory. Others may seem more exotic, for their primitive operations arise from algebraic terms of the intuitionistic modal language which have not been previously considered. We shall for instance investigate the variety of
weak implicative semilattices
, whose members are (non-necessarily distributive) meet semilattices endowed with a nucleus and an implication operation which is not a relative pseudo-complement but satisfies the postulates of Celani and Jansana’s strict implication. For each of these new classes of algebras we establish a representation and a topological duality which generalize the known ones for Heyting algebras enriched with a nucleus.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
8.
AN ALGEBRAIC APPROACH TO INQUISITIVE AND -LOGICS BEZHANISHVILI, NICK; GRILLETTI, GIANLUCA; QUADRELLARO, DAVIDE EMILIO
The review of symbolic logic,
12/2022, Volume:
15, Issue:
4
Journal Article
Peer reviewed
Open access
Abstract This article provides an algebraic study of the propositional system $\mathtt {InqB}$ of inquisitive logic. We also investigate the wider class of $\mathtt {DNA}$ -logics, which are negative ...variants of intermediate logics, and the corresponding algebraic structures, $\mathtt {DNA}$ -varieties. We prove that the lattice of $\mathtt {DNA}$ -logics is dually isomorphic to the lattice of $\mathtt {DNA}$ -varieties. We characterise maximal and minimal intermediate logics with the same negative variant, and we prove a suitable version of Birkhoff’s classic variety theorems. We also introduce locally finite $\mathtt {DNA}$ -varieties and show that these varieties are axiomatised by the analogues of Jankov formulas. Finally, we prove that the lattice of extensions of $\mathtt {InqB}$ is dually isomorphic to the ordinal $\omega +1$ and give an axiomatisation of these logics via Jankov $\mathtt {DNA}$ -formulas. This shows that these extensions coincide with the so-called inquisitive hierarchy of 9. 1
Sahlqvist via Translation Willem Conradie; Alessandra Palmigiano; Zhiguang Zhao
Logical methods in computer science,
01/2019, Volume:
15, Issue 1
Journal Article
Peer reviewed
Open access
In recent years, unified correspondence has been developed as a generalized Sahlqvist theory which applies uniformly to all signatures of normal and regular (distributive) lattice expansions. This ...includes a general definition of the Sahlqvist and inductive formulas and inequalities in every such signature, based on order theory. This definition covers in particular all (bi-)intuitionistic modal logics. The theory of these logics has been intensively studied over the past seventy years in connection with classical polyadic modal logics, using suitable versions of Goedel-McKinsey-Tarski translations as main tools. It is therefore natural to ask (1) whether a general perspective on Goedel-McKinsey-Tarski translations can be attained, also based on order-theoretic principles like those underlying the general definition of Sahlqvist and inductive formulas and inequalities, which accounts for the known Goedel-McKinsey-Tarski translations and applies uniformly to all signatures of normal (distributive) lattice expansions; (2) whether this general perspective can be used to transfer correspondence and canonicity theorems for Sahlqvist and inductive formulas and inequalities in all signatures described above under Goedel-McKinsey-Tarski translations. In the present paper, we set out to answer these questions. We answer (1) in the affirmative; as to (2), we prove the transfer of the correspondence theorem for inductive inequalities of arbitrary signatures of normal distributive lattice expansions. We also prove the transfer of canonicity for inductive inequalities, but only restricted to arbitrary normal modal expansions of bi-intuitionistic logic. We also analyze the difficulties involved in obtaining the transfer of canonicity outside this setting, and indicate a route to extend the transfer of canonicity to all signatures of normal distributive lattice expansions.
The notion of
n
×
m
-valued Łukasiewicz algebras with negation (or
N
S
n
×
m
-algebras) was introduced by C. Sanza in Notes on
n
×
m
-valued Łukasiewicz algebras with negation, Logic J. of the IGPL ...12, 6 (2004), 499–507. These algebras constitute a non-trivial generalization of
n
-valued Łukasiewicz–Moisil algebras and they are a particular case of matrix Łukasiewicz algebras, which were introduced by W. Suchoń in 1975. In this note, we focus on
N
S
3
×
3
-algebras. We prove that they are Heyting algebras and in case that they are centered we describe the Heyting implication in terms of their centers. We also establish a relationship between centered
N
S
3
×
3
-algebras and a class of symmetrical Heyting algebras with operators. Finally, we define symmetrical Heyting algebras of order
3
×
3
(or
S
H
3
×
3
-algebras) and we present a discrete duality for them.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
