Papers by Sándor Radeleczki
Acta et commentationes Universitatis Tartuensis de mathematica, Nov 30, 2023
Given a lattice L, we denote by Res(L) the lattice of all residuated maps on L. The main objectiv... more Given a lattice L, we denote by Res(L) the lattice of all residuated maps on L. The main objective of the paper is to study the atoms of Res(L) where L is a complete lattice. Note that the description of dual atoms of Res(L) easily follows from earlier results of Shmuely (1974). We first consider lattices L for which all atoms of Res(L) are mappings with 2-element range and give a sufficient condition for this. Extending this result, we characterize these atoms of Res(L) which are weakly regular residuated maps in the sense of Blyth and Janowitz (Residuation Theory, 1972). In the rest of the paper we investigate the atoms of Res(M) where M is the lattice of a finite projective plane, in particular, we describe the atoms of Res(F), where F is the lattice of the Fano plane.
arXiv (Cornell University), Sep 7, 2023
By the means of lower and upper fuzzy approximations we define quasiorders. Their properties are ... more By the means of lower and upper fuzzy approximations we define quasiorders. Their properties are used to prove our main results. First, we characterize those pairs of fuzzy sets which form fuzzy rough sets w.r.t. a t-similarity relation θ on U, for certain t-norms and implicators. Then we establish conditions under which fuzzy rough sets form lattices. We show that for the min t-norm and any S-implicator defined by the max co-norm with an involutive negator, the fuzzy rough sets form a complete lattice, whenever U is finite or the range of θ and of the fuzzy sets is a fixed finite chain.
Algebra universalis, Mar 18, 2024
Equivalence relations or, more general, quasiorders (i.e., reflexive and transitive binary relati... more Equivalence relations or, more general, quasiorders (i.e., reflexive and transitive binary relations) have the property that an n-ary operation f preserves , i.e., f is a polymorphism of , if and only if each translation (i.e., unary polynomial function obtained from f by substituting constants) preserves , i.e., it is an endomorphism of. We introduce a wider class of relations-called generalized quasiorders-of arbitrary arities with the same property. With these generalized quasiorders we can characterize all algebras whose clone of term operations is determined by its translations by the above property, what generalizes affine complete algebras. The results are based on the characterization of so-called u-closed monoids (i.e., the unary parts of clones with the above property) as Galois closures of the Galois connection End-gQuord, i.e., as endomorphism monoids of generalized quasiorders. The minimal u-closed monoids are described explicitly.
International Journal of Approximate Reasoning
In this paper, we examined by using Formal Concept Analysis methods, the interrelation between th... more In this paper, we examined by using Formal Concept Analysis methods, the interrelation between the lattices of upper (lower) approximations induced by two tolerance relations \(R\subseteq \rho \subseteq U\times U \). These lattices are isomorphic (dually isomorphic) to the concept lattice \(\mathcal {L}(U,U,R^{c})\), \(\mathcal {L}(U,U,\rho ^{c})\) respectively, where \(R^{c}\) and \(\rho ^{c}\) stand for the complements of the corresponding relations. We proved sufficient conditions and we characterized the case when the concept lattice \(\mathcal {L}(U,U,\rho ^{c})\) is a complete sublattice of \(\mathcal {L}(U,U,R^{c})\). We used the so-called compatibility condition introduced recently and we showed that in the case when \(\rho \) is R-compatible and \(\mathcal {L}(U,U,\rho ^{c})\) is a complete sublattice of \(\mathcal {L} (U,U,R^{c})\), \(\rho \) must be an equivalence. Detailed examples for each case were presented.
International Journal of Approximate Reasoning, 2020
Since the theory of rough sets was introduced by Zdzislaw Pawlak, several approaches have been pr... more Since the theory of rough sets was introduced by Zdzislaw Pawlak, several approaches have been proposed to combine rough set theory with fuzzy set theory. In this paper, we examine one of these approaches, namely fuzzy rough sets, from a lattice theoretic point of view. We connect the lower and upper approximations of a fuzzy relation R to the approximations of the core and support of R. We also show that the lattice of fuzzy rough sets corresponding to a fuzzy equivalence relation R and the crisp subsets of its universe is isomorphic to the lattice of rough sets for the (crisp) equivalence relation E, where E is the core of R. We establish a connection between the exact (fuzzy) sets of R and the exact (crisp) sets of the support of R. Additionally, we examine some properties of a special case of a fuzzy relation.
Production Systems and Information Engineering
In this paper, within the framework of process mining we examine the Maximal Pattern Mining metho... more In this paper, within the framework of process mining we examine the Maximal Pattern Mining method introduced by Liesaputra et al. in [1]. This method constructs a transition graph, i.e. a labelled directed graph for traces with similar structure. The idea behind the algorithm is to analyze the traces in the event log, identify loops, parallel events and optionality between them, in order to determine the maximal patterns. In [1], the authors provide a pseudo code for the skeleton of their algorithm and discuss some parts, but other parts are not detailed. Here, we briefly discuss the steps of the algorithm and elaborate the steps that are not explained in [1]. We introduce some new subroutines to handle the loops, parallel and optional sequences.
International Journal of Approximate Reasoning, Oct 1, 2022
International Journal of Approximate Reasoning, 2021
In this paper, we examined by using Formal Concept Analysis methods, the interrelation between th... more In this paper, we examined by using Formal Concept Analysis methods, the interrelation between the lattices of upper (lower) approximations induced by two tolerance relations \(R\subseteq \rho \subseteq U\times U \). These lattices are isomorphic (dually isomorphic) to the concept lattice \(\mathcal {L}(U,U,R^{c})\), \(\mathcal {L}(U,U,\rho ^{c})\) respectively, where \(R^{c}\) and \(\rho ^{c}\) stand for the complements of the corresponding relations. We proved sufficient conditions and we characterized the case when the concept lattice \(\mathcal {L}(U,U,\rho ^{c})\) is a complete sublattice of \(\mathcal {L}(U,U,R^{c})\). We used the so-called compatibility condition introduced recently and we showed that in the case when \(\rho \) is R-compatible and \(\mathcal {L}(U,U,\rho ^{c})\) is a complete sublattice of \(\mathcal {L} (U,U,R^{c})\), \(\rho \) must be an equivalence. Detailed examples for each case were presented.
J. Multiple Valued Log. Soft Comput., 2021
Algebra Universalis, Mar 1, 2018
The central result of the paper claims that every integral quantale Q has a natural embedding int... more The central result of the paper claims that every integral quantale Q has a natural embedding into the quantale of complete tolerances on the underlying lattice of Q. As an application, we show that the underlying lattice of any finite integral quantale is distributive in 1 and dually pseudocomplemented. Besides, we exhibit relationships between several earlier results. In particular, we give an alternative approach to Valentini's ordered sets and show how the ordered sets are related to tolerances.
Trends in mathematics, 2018
This chapter deals with rough approximations defined by tolerance relations that represent simila... more This chapter deals with rough approximations defined by tolerance relations that represent similarities between the elements of a given universe of discourse. We consider especially tolerances induced by irredundant coverings of the universe U. This is natural in view of Pawlak’s original theory of rough sets defined by equivalence relations: any equivalence E on U is induced by the partition U∕E of U into equivalence classes, and U∕E is a special irredundant covering of U in which the blocks are disjoint. Here equivalence classes are replaced by tolerance blocks which are maximal sets in which all elements are similar to each other. The blocks of a tolerance R on U always form a covering of U which induces R, but this covering is not necessarily irredundant and its blocks may intersect. In this chapter we consider the semantics of tolerances in rough sets, and in particular the algebraic structures formed by the rough approximations and rough sets defined by different types of tolerances.
arXiv (Cornell University), Sep 13, 2019
stands for the reflexive weak ordered relation ≤ • T • ≤. Weak ordered relations constitute the g... more stands for the reflexive weak ordered relation ≤ • T • ≤. Weak ordered relations constitute the generalization of the ordered relations introduced by S. Valentini. Reflexive weak ordered relations can be characterized as compatible reflexive relations R ⊆ L 2 satisfying R = ≤ • R • ≤.
arXiv (Cornell University), Apr 25, 2020
In relational approach to general rough sets, ideas of directed relations are supplemented with a... more In relational approach to general rough sets, ideas of directed relations are supplemented with additional conditions for multiple algebraic approaches in this research paper. The relations are also specialized to representations of general parthood that are upper-directed, reflexive and antisymmetric for a better behaved groupoidal semantics over the set of roughly equivalent objects by the first author. Another distinct algebraic semantics over the set of approximations, and a new knowledge interpretation are also invented in this research by her. Because of minimal conditions imposed on the relations, neighborhood granulations are used in the construction of all approximations (granular and pointwise). Necessary and sufficient conditions for the lattice of local upper approximations to be completely distributive are proved by the second author. These results are related to formal concept analysis. Applications to student centered learning and decision making are also outlined.
J. Multiple Valued Log. Soft Comput., 2017
Order, Sep 26, 2015
In this paper a result of B. Leclerc and B. Monjardet concerning meet-projections in finite congr... more In this paper a result of B. Leclerc and B. Monjardet concerning meet-projections in finite congruence-simple atomistic lattices is generalized. We prove that the result remains valid for any finite tolerance-simple lattice; moreover, we extend it to a type of subdirect product of such lattices, introducing the notion of a generalized oligarchy.
arXiv (Cornell University), Jul 4, 2023
Equivalence relations or, more general, quasiorders (i.e., reflexive and transitive binary relati... more Equivalence relations or, more general, quasiorders (i.e., reflexive and transitive binary relations) ϱ have the property that an n-ary operation f preserves ϱ, i.e., f is a polymorphism of ϱ, if and only if each translation (i.e., unary polynomial function obtained from f by substituting constants) preserves ϱ, i.e., it is an endomorphism of ϱ. We introduce a wider class of relations-called generalized quasiorders-of arbitrary arities with the same property. With these generalized quasiorders we can characterize all algebras whose clone of term operations is determined by its translations by the above property, what generalizes affine complete algebras. The results are based on the characterization of so-called u-closed monoids (i.e., the unary parts of clones with the above property) as Galois closures of the Galois connection End − gQuord, i.e., as endomorphism monoids of generalized quasiorders. The minimal u-closed monoids are described explicitly.
arXiv (Cornell University), Feb 21, 2017
An alternative proof is given of the existence of greatest lower bounds in the imbalance order of... more An alternative proof is given of the existence of greatest lower bounds in the imbalance order of binary maximal instantaneous codes of a given size. These codes are viewed as maximal antichains of a given size in the infinite binary tree of 0-1 words. The proof proposed makes use of a single balancing operation instead of expansion and contraction as in the original proof of the existence of glb. 1 Terminology of codes and introduction The set {0, 1} * of all finite sequences (words) of the symbols 0 and 1 is partially ordered by the prefix order ≤ pref defined by v ≤ pref w ⇔ ∃z vz = w The prefix-ordered set of words is an infinite binary tree having the empty word as root. Instantaneous codes are defined as the finite antichains in this tree. (This finiteness shall be assumed throughout the paper, thus excluding infinite prefix-free sets.) By lexicographic (lex) order we mean the (only) linear extension of the prefix order in which words incomparable in the prefix order are compared by the "telephone book principle", i.e. v0x always precedes (is smaller than) v1y. We call an instantaneous code lex monotone if the sequence of lengths of the codewords taken in lex order is monotone (non-decreasing). With respect to a given lex monotone instantaneous code C whose codewords in lex order are c 1 , c 2 , ..., each codeword c i is then identified by its index i.
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Papers by Sándor Radeleczki