State reduction of fuzzy automata aims to efficiently construct a suitably small fuzzy automaton equivalent to a given one. It is a significant and well-studied problem in automata theory due to its practical applications in various fields. If we relax the requirement for exact equivalence, then we talk about the approximate state reduction problem, which has gained attention only recently. There are two approaches to approximate state reduction: one seeks approximate equivalence to a specified threshold, while the other aims for exact equivalence for length-bounded words. These two approaches have been considered separately. In this paper, we demonstrate that both approaches, and even their combination, can be achieved by merging indistinguishable states of a fuzzy automaton through the use of sequences of fuzzy relations that we introduce in this paper. We provide characterizations of these sequences, and show that they are closely related to certain approximate simulations for fuzzy automata that emerged in the recent literature. However, their subtle differences significantly affect the process of approximate state reduction. By formally proving this distinction, we generalize some well-known results and offer new insight into approximate state reduction. We discuss how all forms of approximate state reduction can be realized and provide algorithms for calculating the proposed sequences and performing the reductions, along with illustrative examples.

In this article, we define$\lambda$-approximate simulations and bisimulations for fuzzy automata over complete Heyting algebras. The value$\lambda$presents the degree of language similarity or equality between observed fuzzy automata. Algorithms for computing the greatest$\lambda$-approximate simulations and bisimulations are given. We show that$\lambda$-approximate simulations and bisimulations on a fuzzy automaton can be effectively used for factorization of fuzzy automata. We present the algorithm that splits the interval of the degrees of language similarity or equality into subintervals with the same minimal corresponding factor fuzzy automata.

Determinization of fuzzy finite automata is understood here as a procedure of their conversion into equivalent crisp-deterministic fuzzy automata, which can be viewed as being deterministic with possible infinitely many states, but with fuzzy sets of terminal states. Particularly, significant determinization methods are those that provide a minimal crisp-deterministic fuzzy automaton equivalent to the original fuzzy finite automaton called canonization methods. One canonization method for fuzzy finite automata, the Brzozowski type determinization, has been developed recently by Jancic and Ciric in [10]. Here we provide another canonization method for a fuzzy finite automaton A = (A, σ, δ, r) over a complete residuated lattice £, based on the degrees of inclusion of the right fuzzy languages associated with states of A into the left derivatives of the fuzzy language recognized by A. The proposed procedure terminates in a finite number of steps, whenever the membership values taken by δ, σ, and r generate a finite subsemiring of the semiring reduct of £. This procedure is generally faster than the Brzozowski type determinization, and if the basic operations in the residuated lattice £ can be performed in constant time, it has the same computational time as all other determinization procedures provided in [8], [11], and [12].

Subdirect decompositions of unary algebras are studied in connection with one-element subalgebras, cores, Rees extensions of congruences of subalgebras, dense extensions and disjunctive elements. In particular, subdirectly irreducible unary algebras