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Tomi Kärki - One of the best experts on this subject based on the ideXlab platform.
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WORDS - Similarity Relations and Repetition-Freeness
Lecture Notes in Computer Science, 2013Co-Authors: Tomi KärkiAbstract:A similarity Relation is a Relation on words of equal length induced by a symmetric and Reflexive Relation on letters. The aim of this article is to give an overview of the results concerning repetition-freeness in connection with similarity Relations. We consider so called chain Relations, cyclic Relations and partial words, which can be seen as a special case of similarity Relations. As a new result, we prove that local 3+-repetitions can be avoided in binary partial words and the local avoidability index of $\mathring{R}$ -cubes is five, where $\mathring{R}$ is a Relation such that the graph of the Relation is a cycle.
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Repetition-freeness with Cyclic Relations and Chain Relations
Fundamenta Informaticae, 2012Co-Authors: Tomi KärkiAbstract:A similarity Relation R is a Relation on words of equal length induced by a symmetric and Reflexive Relation on letters. Such a Relation is called cyclic if the graph of the Relation on letters is a cycle. A chain Relation is obtained from a cyclic Relation by removing one symmetric Relation from the cycle. A word uv is an R-square if u and v are in Relation R. The avoidability index of R-squares is the size of the minimal alphabet such that there exists an R-square-free infinite word having infinitely many occurrences of each letter of the alphabet. We prove that the avoidability index of R-squares is 7 in the case of cyclic Relations and 6 in the case of chain Relations. We also consider R-overlaps and show that they are 5-avoidable with cyclic Relations and 4-avoidable with chain Relations.
Wan-rong Zhan - One of the best experts on this subject based on the ideXlab platform.
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On the topological properties of generalized rough sets
Information Sciences, 2014Co-Authors: Wan-rong ZhanAbstract:In this paper, we consider some topological properties of generalized rough sets induced by binary Relations and show that1.Any serial binary Relation can induce a topology. 2.Let R be a binary Relation on a universe U. t(R) and e(R) denote the transitive closure and the equivalence closure of R, respectively. If R is a Reflexive Relation on U, then R and t(R) induce the same topology, i.e. T(R)=T(t(R)). The interior and closure operators of the topology T(R) induced by R are the lower and upper approximation operators t(R) and t(R)@?, respectively. Moreover, R(T(R))=t(R), where R(T(R)) is the Relation induced by the topology T(R). 3.When R is a Reflexive and symmetric Relation, R and e(R) induce the same topology, i.e. T(R)=T(e(R)). The interior and closure operators of the topology T(R) induced by R are the lower and upper approximation operators e(R) and e(R)@?, respectively. Moreover, R(T(R))=e(R). 4.Based on the above conclusions, the notion of topological reduction of incomplete information systems is proposed, and characterizations of reduction of consistent incomplete decision tables are obtained.
Jari Kortelainen - One of the best experts on this subject based on the ideXlab platform.
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On Relationship between modified sets, topological spaces and rough sets
Fuzzy Sets and Systems, 1994Co-Authors: Jari KortelainenAbstract:Abstract In this paper we define modifiers by Relations. Especially, weakening and substantiating modifiers are defined by a so called accessibility Relation which is a Reflexive Relation on a non-empty set X. After presenting some main results we prove that this type of modifiers will satisfy the Kuratowski Closure Axioms. This means that modifiers in fact induce topological spaces. Also rough sets are considered to be a special case of modified sets.
Li Zheng - One of the best experts on this subject based on the ideXlab platform.
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Topology vs generalized rough sets
International Journal of Approximate Reasoning, 2011Co-Authors: Zhi Pei, Daowu Pei, Li ZhengAbstract:This paper investigates the Relationship between topology and generalized rough sets induced by binary Relations. Some known results regarding the Relation based rough sets are reviewed, and some new results are given. Particularly, the Relationship between different topologies corresponding to the same rough set model is examined. These generalized rough sets are induced by inverse serial Relations, Reflexive Relations and pre-order Relations, respectively. We point that inverse serial Relations are weakest Relations which can induce topological spaces, and that different Relation based generalized rough set models will induce different topological spaces. We proved that two known topologies corresponding to Reflexive Relation based rough set model given recently are different, and gave a condition under which the both are the same topology.
Idriss Tchoffo Nguefeu - One of the best experts on this subject based on the ideXlab platform.
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Variations of the Shifting Lemma and Goursat categories
Algebra universalis, 2019Co-Authors: Marino Gran, Diana Rodelo, Idriss Tchoffo NguefeuAbstract:We prove that Mal’tsev and Goursat categories may be characterized through variations of the Shifting Lemma, that is classically expressed in terms of three congruences R , S and T , and characterizes congruence modular varieties. We first show that a regular category $${\mathbb {C}}$$ C is a Mal’tsev category if and only if the Shifting Lemma holds for Reflexive Relations on the same object in $${\mathbb {C}}$$ C . Moreover, we prove that a regular category $${\mathbb {C}}$$ C is a Goursat category if and only if the Shifting Lemma holds for a Reflexive Relation S and Reflexive and positive Relations R and T in $${\mathbb {C}}$$ C . In particular this provides a new characterization of 2-permutable and 3-permutable varieties and quasi-varieties of universal algebras.
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Variations of the Shifting Lemma and Goursat categories
Algebra universalis, 2019Co-Authors: Marino Gran, Diana Rodelo, Idriss Tchoffo NguefeuAbstract:We prove that Mal'tsev and Goursat categories may be characterised through stronger variations of the Shifting Lemma, that is classically expressed in terms of three congruences $R$, $S$ and $T$, and characterises congruence modular varieties. We first show that a regular category $\mathcal C$ is a Mal'tsev category if and only if the Shifting Lemma holds for Reflexive Relations on the same object in $\mathcal C$. Moreover, we prove that a regular category $\mathcal C$ is a Goursat category if and only if the Shifting Lemma holds for a Reflexive Relation $S$ and Reflexive and positive Relations $R$ and $T$ in $\mathcal C$. In particular this provides a new characterisation of $2$-permutable and $3$-permutable varieties and quasi-varieties of universal algebras.