Weakly linear systems of fuzzy relation inequalities: The heterogeneous case

TL;DR

This paper extends methods for solving weakly linear fuzzy relation systems from homogeneous to heterogeneous cases, enabling applications in fuzzy automata and social network analysis.

Contribution

It adapts the existing solution method for homogeneous systems to heterogeneous systems involving relations between different sets, and explores related quotient systems.

Findings

Method for computing greatest solutions of heterogeneous systems

Relationships established between heterogeneous and homogeneous systems

Applications demonstrated in fuzzy automata and social networks

Abstract

New types of systems of fuzzy relation inequalities and equations, called weakly linear, have been recently introduced in [J. Ignjatovi\'c, M. \'Ciri\'c, S. Bogdanovi\'c, On the greatest solutions to weakly linear systems of fuzzy relation inequalities and equations, Fuzzy Sets and Systems 161 (2010) 3081--3113.]. The mentioned paper dealt with homogeneous weakly linear systems, composed of fuzzy relations on a single set, and a method for computing their greatest solutions has been provided. 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. Here we adapt this method for computing the greatest solutions of heterogeneous weakly linear systems, where the unknown fuzzy relation relates two possibly different sets. We also introduce and study quotient fuzzy relational systems and establish relationships between solutions to heterogeneous and homogeneous weakly linear systems. Besides, we point out to applications of the obtained results in the state reduction of fuzzy automata and computing the greatest simulations and bisimulations between fuzzy automata, as well as in the positional analysis of fuzzy social networks.