Quotients and Homomorphisms of Relational Systems

Chajda, Ivan; Länger, Helmut

doi:10.1142/9789814366311_0007

Cited by 5 publications

(10 citation statements)

References 2 publications

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“…Some of our results were already published in [5] but, for the reader's convenience, these results are repeated.…”

mentioning

confidence: 80%

“…An algebraic approach to relational systems was developed by A. I. Mal'cev [6]. Some particular cases of relational systems and certain homomorphisms of them were treated by the first author and his co-authors in [1]- [5]. The method assigning a groupoid to a given relational system was initiated in [5] where certain concepts of this paper were introduced.…”

mentioning

confidence: 99%

“…Some particular cases of relational systems and certain homomorphisms of them were treated by the first author and his co-authors in [1]- [5]. The method assigning a groupoid to a given relational system was initiated in [5] where certain concepts of this paper were introduced. We are motivated by the fact that to certain relational systems, in particular to directed posets, a certain directoid (see e.g.…”

mentioning

confidence: 99%

“…We generalize these approaches in a way similar to that of [5] and produce several results which enable us to consider relational systems from a purely algebraic point of view.…”

mentioning

confidence: 99%

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Groupoids assigned to relational systems

Chajda¹,

Länger²

2013

Math. Bohem.

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By a relational system we mean a couple (A, R) where A is a set and R is a binary relation on A, i.e. R ⊆ A × A. To every directed relational system A = (A, R) we assign a groupoid G(A) = (A, •) on the same base set where xy = y if and only if (x, y) ∈ R. We characterize basic properties of R by means of identities satisfied by G(A) and show how homomorphisms between those groupoids are related to certain homomorphisms of relational systems.

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“…Some of our results were already published in [5] but, for the reader's convenience, these results are repeated.…”

mentioning

confidence: 80%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…We generalize these approaches in a way similar to that of [5] and produce several results which enable us to consider relational systems from a purely algebraic point of view.…”

mentioning

confidence: 99%

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Groupoids assigned to relational systems

Chajda¹,

Länger²

2013

Math. Bohem.

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“…We will see in Section 7 that quotient (fuzzy) relational systems can be also used to reduce the number of nodes of a (fuzzy) network, while keeping the basic structure of the network. It is also worth noting that quotient crisp relational systems have been recently defined in the same way in [27].…”

Section: Quotient Fuzzy Relational Systemsmentioning

confidence: 99%

Weakly linear systems of fuzzy relation inequalities and their applications: A brief survey

Ignjatović

Ćirić

2012

Filomat

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Weakly linear systems of fuzzy relation inequalities and equations have recently emerged from research in the theory of fuzzy automata. From the general aspect of the theory of fuzzy relation equations and inequalities homogeneous and heterogeneous weakly linear systems have been discussed in two recent papers. Here we give a brief overview of the main results from these two papers, as well as from a series of papers on applications of weakly linear systems in the state reduction of fuzzy automata, the study of simulation, bisimulation and equivalence of fuzzy automata, and in the social network analysis. Especially, we present algorithms for computing the greatest solutions to weakly linear systems.

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Orthogonal relational systems

2015

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In this paper, we discuss the concept of relational system with involution. This system is called orthogonal if, for every pair of non-zero orthogonal elements, there exists a supremal element in their upper cone and the upper cone of orthogonal elements is a singleton (i.e. x; are complements of each other). To every orthogonal relational system can be assigned a groupoid with involution. The conditions under which a groupoid is assigned to an orthogonal relational systems are investigated. We will see that many properties of the relational system can be captured by the associated groupoid. Moreover, these structures enjoy several desirable algebraic features such as, e.g. a direct decomposition representation and the strong amalgamation property

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Quotients and Homomorphisms of Relational Systems

Cited by 5 publications

References 2 publications

Groupoids assigned to relational systems

Groupoids assigned to relational systems

Weakly linear systems of fuzzy relation inequalities and their applications: A brief survey

Orthogonal relational systems

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