We show that the state reduction problem for fuzzy automata is related to the problem of finding a solution to a particular system of fuzzy relation equations in the set of all fuzzy equivalences on its set of states. This system may consist of infinitely many equations, and finding its non-trivial solutions may be a very difficult task. For that reason we aim our attention to some instances of this system which consist of finitely many equations and are easier to solve. First, we study right invariant fuzzy equivalences, and their duals, the left invariant ones. We prove that each fuzzy automaton possesses the greatest right (resp. left) invariant fuzzy equivalence, which provides the best reduction by means of fuzzy equivalences of this type, and we give an effective procedure for computing this fuzzy equivalence, which works if the underlying structure of truth values is a locally finite residuated lattice. Moreover, we show that even better reductions can be achieved alternating reductions by means of right and left invariant fuzzy equivalences. We also study strongly right and left invariant fuzzy equivalences, which give worse reductions than right and left invariant ones, but whose computing is much easier. We give an effective procedure for computing the greatest strongly right (resp. left) invariant fuzzy equivalence, which is applicable to fuzzy automata over an arbitrary complete residuated lattice.

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In our recent paper we have established close relationships between state reduction of a fuzzy recognizer and resolution of a particular system of fuzzy relation equations. In that paper we have also studied reductions by means of those solutions which are fuzzy equivalences. In this paper we will see that in some cases better reductions can be obtained using the solutions of this system that are fuzzy quasiorders. Generally, fuzzy quasi-orders and fuzzy equivalences are equally good in the state reduction, but we show that right and left invariant fuzzy quasi-orders give better reductions than right and left invariant fuzzy equivalences. We also show that alternate reductions by means of fuzzy quasi-orders give better results than alternate reductions by means of fuzzy equivalences. Furthermore we study a more general type of fuzzy quasi-orders, weakly right and left invariant ones, and we show that they are closely related to determinization of fuzzy recognizers. We also demonstrate some applications of weakly left invariant fuzzy quasi-orders in conflict analysis of fuzzy discrete event systems.

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