The aim of this paper is the study of a covering of a max-min general fuzzy automaton by another, ADMISSIBLE RELATIONs, ADMISSIBLE partitions of a max-min general fuzzy automaton, d~-orthogonality of ADMISSIBLE partitions, irreducible max-min general fuzzy automata. Then we obtain the RELATIONships between them.

The Lie-Santilli ADMISSIBLE theory gives in physics applications mainly in producing energy. The mathematical models need to be multivalued and the class of Hv-structures is the most appropriate, since this is the largest class of hyper-structures. The Hv-numbers, weak hyper-numbers, used satisfy all needed axioms. We focus on small and very-thin Hv-fields with P-hyper-operations on the product.

In this article, after the definitions of the reduced hesitant L-fuzzy automaton (RHLFA) and the minimal hesitant L-fuzzy automaton; we convert a hesitant L-fuzzy automaton (HLFA) to an RHLFA by reducing the number of its states such that its language is equal to the original HLFA language. Then, by defining an equivalence RELATION on the monoid X*,we construct an HLFA whose language is equal to the language of the transformed RHLFA, and we show that this HLFA is minimal. In conclusion, we delineate the criteria under which, the number of states in the minimal HLFA is equal to the number of states in the RHLFA

In this work, we examine several widely-used interval-valued fuzzy logical connectives with respect to ADMISSIBLE orders. We are concerned with interval-valued fuzzy negations, automorphisms, fuzzy implications and aggregation functions with respect to $K_{\alpha,\beta}$ orders and arbitrary intervals on $L([0, 1])$. We also make a discussion of width-preserving interval-valued fuzzy equivalence functions and fuzzy dissimilarity functions with respect to arbitrary ADMISSIBLE orders and the intervals with the same width on $L([0, 1])$. Then we bring some approaches to constructing the proposed interval-valued fuzzy logical connectives with respect to ADMISSIBLE orders.