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Weakly linear systems of fuzzy relation inequalities: the heterogeneous case. (English) Zbl 1268.03073

In the paper heterogeneous weakly linear systems of fuzzy relation inequalities and equations are introduced and examined. These systems of fuzzy relation inequalities and equations are composed of fuzzy relations on two possible different sets and an unknown is a fuzzy relation between these two sets. The domain of investigations is a complete residuated lattice.
The iterative method for computing the greatest solutions presented here is adapted from the paper [J. Ignjatović et al., Fuzzy Sets Syst. 161, No. 24, 3081–3113 (2010; Zbl 1231.03044)], where homogeneous weakly linear systems of fuzzy relation inequalities and equations were studied. This method is based on the computing of the greatest post-fixed point, contained in a given fuzzy relation, of an isotone function on the lattice of fuzzy relations.
It is proven that every heterogeneous weakly linear system has a greatest solution (which may be empty). Next, it is shown how the applied procedure can be modified to compute the greatest crisp solution of the system. Moreover, the concept of a quotient fuzzy relational system with respect to a fuzzy equivalence is introduced. Some theorems similar to the well-known theorems from classical algebra are proven (these theorems deal with homomorphism, isomorphism etc.). Thanks to this concept, relationships between solutions of heterogeneous and homogeneous weakly linear systems are established.
Since weakly linear systems originate from the theory of automata, it is noticed that solutions of the heterogeneous systems were used in the study of simulations and bisimulations between fuzzy automata. Moreover, applications of the results obtained in the paper are shown in state reduction of fuzzy automata and computing the greatest simulations and bisimulations between fuzzy automata and in the positional analysis of fuzzy social networks.

MSC:

03E72 Theory of fuzzy sets, etc.
68Q45 Formal languages and automata
91D30 Social networks; opinion dynamics

Citations:

Zbl 1231.03044

Software:

Pajek
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Full Text: DOI arXiv

References:

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