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Jouni Järvinen

Publications and source records attributed to Jouni Järvinen.

At least 19 recordsLinked to original sources
arXiv2026

Characterising regular double Stone and pseudocomplemented Kleene algebras in completions of rough sets induced by reflexive relations

We consider Kleene and Stone algebras defined on the completion DM(RS) of the ordered set of rough sets induced by a reflexive relation. We focus on the cases where the completion forms a spatial and completely distributive lattice. We derive the conditions under which DM(RS) is a regular pseudocomplemented Kleene algebra and a completely distributive double Stone algebra. Finally, we describe the reflexive relations for which DM(RS) forms a regular double Stone algebra, which is the same structure as in the case of equivalences. Our results generalise earlier findings on algebras of rough sets induced by equivalences, quasiorders, and tolerance relations.

math.RA↗
arXiv2025

The structure of rough sets defined by reflexive relations

For several types of information relations, the induced rough sets system RS does not form a lattice but only a partially ordered set. However, by studying its Dedekind-MacNeille completion DM(RS), one may reveal new important properties of rough set structures. Building upon D. Umadevi's work on describing joins and meets in DM(RS), we previously investigated pseudo-Kleene algebras defined on DM(RS) for reflexive relations. This paper delves deeper into the order-theoretic properties of DM(RS) in the context of reflexive relations. We describe the completely join-irreducible elements of DM(RS) and characterize when DM(RS) is a spatial completely distributive lattice. We show that even in the case of a non-transitive reflexive relation, DM(RS) can form a Nelson algebra, a property generally associated with quasiorders. We introduce a novel concept, the core of a relational neighborhood, and use it to provide a necessary and sufficient condition for DM(RS) to determine a Nelson algebra.

math.RA↗
arXiv2024

Nelson algebras, residuated lattices and rough sets: A survey

Over the past 50 years, Nelson algebras have been extensively studied by distinguished scholars as the algebraic counterpart of Nelson's constructive logic with strong negation. Despite these studies, a comprehensive survey of the topic is currently lacking, and the theory of Nelson algebras remains largely unknown to most logicians. This paper aims to fill this gap by focussing on the essential developments in the field over the past two decades. Additionally, we explore generalisations of Nelson algebras, such as N4-lattices which correspond to the paraconsistent version of Nelson's logic, as well as their applications to other areas of interest to logicians, such as duality and rough set theory. A general representation theorem states that each Nelson algebra is isomorphic to a subalgebra of a rough set-based Nelson algebra induced by a quasiorder. Furthermore, a formula is a theorem of Nelson logic if and only if it is valid in every finite Nelson algebra induced by a quasiorder.

math.LO↗
arXiv2023

Relational correspondences for L-fuzzy rough approximations defined on De Morgan Heyting algebras

We consider fuzzy rough sets defined on De Morgan Heyting algebras. We present a theorem that can be used to obtain several correspondence results between fuzzy rough sets and fuzzy relations defining them. We characterize fuzzy rough approximation operators corresponding to compositions of reflexive, transitive, mediate, Euclidean and adjoint fuzzy relations defined on De Morgan Heyting algebras by using only a single axiom.

math.RA↗
arXiv2023

Pseudo-Kleene algebras determined by rough sets

We study the pseudo-Kleene algebras of the Dedekind-MacNeille completion of the ordered set of rough set determined by a reflexive relation. We characterize the cases when PBZ and PBZ*-lattices can be defined on these pseudo-Kleene algebras.

math.RA↗
arXiv2021

Defining rough sets as core-support pairs of three-valued functions

We answer the question what properties a collection $\mathcal{F}$ of three-valued functions on a set $U$ must fulfill so that there exists a quasiorder $\leq$ on $U$ such that the rough sets determined by $\leq$ coincide with the core--support pairs of the functions in $\mathcal{F}$. Applying this characterization, we give a new representation of rough sets determined by equivalences in terms of three-valued Łukasiewicz algebras of three-valued functions.

math.RA↗
arXiv2020

The structure of multigranular rough sets

We study multigranulation spaces of two equivalences. The lattice-theoretical properties of so-called "optimistic" and "pessimistic" multigranular approximation systems are given. We also consider the ordered sets of rough sets determined by these approximation pairs.

math.RA↗
arXiv2019

Defining rough sets using tolerances compatible with an equivalence

We consider tolerances $T$ compatible with an equivalence $E$ on $U$, meaning that the relational product $E \circ T$ is included in $T$. We present the essential properties of $E$-compatible tolerances and study rough approximations defined by such $E$ and $T$. We consider rough set pairs $(X_E,X^T)$, where the lower approximation $X_E$ is defined as is customary in rough set theory, but $X^T$ allows more elements to be possibly in $X$ than $X^E$. Motivating examples of $E$-compatible tolerances are given, and the essential lattice-theoretical properties of the ordered set of rough sets $\{ (X_E,X^T) \mid X \subseteq U\}$ are established.

math.CO↗
arXiv2019

Prime filter structures of pseudocomplemented Kleene algebras and representation by rough sets

We introduce Kleene-Varlet spaces as partially ordered sets equipped with a polarity satisfying certain additional conditions. By applying Kleene-Varlet spaces, we prove that each regular pseudocomplemented Kleene algebra is isomorphic to a subalgebra of the rough set regular pseudocomplemented Kleene algebra defined by a tolerance induced by an irredundant covering. We also characterize the Kleene-Varlet spaces corresponding to the regular pseudocomplemented Kleene algebras satisfying the Stone identity.

math.LO↗
arXiv2015

Tolerances induced by irredundant coverings

In this paper, we consider tolerances induced by irredundant coverings. Each tolerance $R$ on $U$ determines a quasiorder $\lesssim_R$ by setting $x \lesssim_R y$ if and only if $R(x) \subseteq R(y)$. We prove that for a tolerance $R$ induced by a covering $\mathcal{H}$ of $U$, the covering $\mathcal{H}$ is irredundant if and only if the quasiordered set $(U, \lesssim_R)$ is bounded by minimal elements and the tolerance $R$ coincides with the product ${\gtrsim_R} \circ {\lesssim_R}$. We also show that in such a case $\mathcal{H} = \{ {\uparrow}m \mid \text{$m$ is minimal in $(U,\lesssim_R)$} \}$, and for each minimal $m$, we have $R(m) = {\uparrow} m$. Additionally, this irredundant covering $\mathcal{H}$ inducing $R$ consists of some blocks of the tolerance $R$. We give necessary and sufficient conditions under which $\mathcal{H}$ and the set of $R$-blocks coincide. These results are established by applying the notion of Helly numbers of quasiordered sets.

math.RA↗
arXiv2014

Characterizing intermediate tense logics in terms of Galois connections

We propose a uniform way of defining for every logic ${\sf L}$ intermediate between intuitionistic and classical logics, the corresponding intermediate minimal tense logic ${\sf LK_t}$. This is done by building the fusion of two copies of intermediate logic with a Galois connection ${\sf LGC}$, and then interlinking their operators by two Fischer Servi axioms. The resulting system is called here ${\sf L2GC{+}FS}$. In the cases of intuitionistic logic ${\sf Int}$ and classical logic ${\sf Cl}$, it is noted that ${\sf Int2GC{+}FS}$ is syntactically equivalent to intuitionistic minimal tense logic ${\sf IK_t}$ by W. B. Ewald and ${\sf Cl2GC{+}FS}$ equals classical minimal tense logic ${\sf K_t}$. This justifies to consider ${\sf L2GC{+}FS}$ as minimal ${\sf L}$-tense logic ${\sf LK_t}$ for any intermediate logic ${\sf L}$. We define H2GC+FS-algebras as expansions of HK1-algebras, introduced by E. Orlowska and I. Rewitzky. For each intermediate logic ${\sf L}$, we show algebraic completeness of ${\sf L2GC{+}FS}$ and its conservativeness over ${\sf L}$. We prove relational completeness of ${\sf Int2GC{+}FS}$ with respect to the models defined on ${\sf IK}$-frames introduced by G. Fischer Servi. We also prove a representation theorem stating that every H2GC+FS-algebra can be embedded into the complex algebra of its canonical ${\sf IK}$-frame.

math.LO↗
arXiv2014

Monteiro spaces and rough sets determined by quasiorder relations: Models for Nelson algebras

Rough sets induced by quasiorders appear in several constructions using binary relations in computer science. In this paper, a structural characterisation of rough sets induced by quasiorders is given. These rough sets form Nelson algebras defined on algebraic lattices. We prove that any Nelson algebra can be represented as a subalgebra of an algebra defined on rough sets induced by a suitable quasiorder. We also show that Monteiro spaces, rough sets induced by quasiorders and Nelson algebras defined on $\rm T_0$-spaces that are Alexandrov topologies can be considered as equivalent structures, because they determine each other up to isomorphism.

math.LO↗
arXiv2013

Representing distributive lattices with Galois connections in terms of rough sets

This paper studies expansions of bounded distributive lattices equipped with a Galois connection. We introduce GC-frames and canonical frames for these algebras. The complex algebras of GC-frames are defined in terms of rough set approximation operators. We prove that each bounded distributive lattice with a Galois connection can be embedded into the complex algebra of its canonical frame. We show that for every spatial Heyting algebra $L$ equipped with a Galois connection, there exists a GC-frame such that $L$ is isomorphic to the complex algebra of this frame, and an analogous result holds for weakly atomic Heyting-Brouwer algebras with a Galois connection. In each case of representation, given Galois connections are represented by rough set upper and lower approximations.

math.RA↗
arXiv2013

Rough sets determined by tolerances

We show that for any tolerance $R$ on $U$, the ordered sets of lower and upper rough approximations determined by $R$ form ortholattices. These ortholattices are completely distributive, thus forming atomistic Boolean lattices, if and only if $R$ is induced by an irredundant covering of $U$, and in such a case, the atoms of these Boolean lattices are described. We prove that the ordered set $\mathit{RS}$ of rough sets determined by a tolerance $R$ on $U$ is a complete lattice if and only if it is a complete subdirect product of the complete lattices of lower and upper rough approximations. We show that $R$ is a tolerance induced by an irredundant covering of $U$ if and only if $\mathit{RS}$ is an algebraic completely distributive lattice, and in such a situation a quasi-Nelson algebra can be defined on $\mathit{RS}$. We present necessary and sufficient conditions which guarantee that for a tolerance $R$ on $U$, the ordered set $\mathit{RS}_X$ is a lattice for all $X \subseteq U$, where $R_X$ denotes the restriction of $R$ to the set $X$ and $\mathit{RS}_X$ is the corresponding set of rough sets. We introduce the disjoint representation and the formal concept representation of rough sets, and show that they are Dedekind--MacNeille completions of $\mathit{RS}$.

math.RA↗
arXiv2012

Intuitionistic logic with two Galois connections combined with Fischer Servi axioms

Earlier, the authors introduced the logic IntGC, which is an extension of intuitionistic propositional logic by two rules of inference mimicking the performance of Galois connections (Logic J. of the IGPL, 18:837-858, 2010). In this paper, the extensions Int2GC and Int2GC+FS of IntGC are studied. Int2GC can be seen as a fusion of two IntGC logics, and Int2GC+FS is obtained from Int2GC by adding instances of duality-like connections $\Diamond(A \to\ B) \to (\Box A \to \Diamond B)$ and $(\Diamond A \to \Box B) \to \Box(A \to B)$, introduced by G. Fischer Servi (Rend. Sem. Mat. Univers. Politecn. Torino, 42:179-194, 1984), for interlinking the two Galois connections of Int2GC. Both Kripke-style and algebraic semantics are presented for Int2GC and Int2GC+FS, and the logics are proved to be complete with respect to both of these semantics. We show that rough lattice-valued fuzzy sets defined on complete Heyting algebras are proper algebraic models for Int2GC+FS. We also prove that Int2GC+FS is equivalent to the intuitionistic tense logic IKt, and an axiomatisation of IKt with the number of axioms reduced to the half of the number of axioms given by W. B. Ewald (J. Symb. Log, 51:166-179, 1986) is presented.

math.LO↗
arXiv2012

Information completeness in Nelson algebras of rough sets induced by quasiorders

In this paper, we give an algebraic completeness theorem for constructive logic with strong negation in terms of finite rough set-based Nelson algebras determined by quasiorders. We show how for a quasiorder $R$, its rough set-based Nelson algebra can be obtained by applying the well-known construction by Sendlewski. We prove that if the set of all $R$-closed elements, which may be viewed as the set of completely defined objects, is cofinal, then the rough set-based Nelson algebra determined by a quasiorder forms an effective lattice, that is, an algebraic model of the logic $E_0$, which is characterised by a modal operator grasping the notion of "to be classically valid". We present a necessary and sufficient condition under which a Nelson algebra is isomorphic to a rough set-based effective lattice determined by a quasiorder.

math.RA↗
arXiv2011

Representation of Nelson Algebras by Rough Sets Determined by Quasiorders

In this paper, we show that every quasiorder $R$ induces a Nelson algebra $\mathbb{RS}$ such that the underlying rough set lattice $RS$ is algebraic. We note that $\mathbb{RS}$ is a three-valued Łukasiewicz algebra if and only if $R$ is an equivalence. Our main result says that if $\mathbb{A}$ is a Nelson algebra defined on an algebraic lattice, then there exists a set $U$ and a quasiorder $R$ on $U$ such that $\mathbb{A} \cong \mathbb{RS}$.

math.RT↗