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We propose a new method for constructing nullnorms on bounded lattices based on the simultaneous use of a closure operator and an interior operator. This method requires as input a given underlying t-conorm on a join sub-semilattice (depending on the closure operator), a given underlying t-norm on a meet sub-semilattice (depending on the interior operator) and an auxiliary mapping. It is shown that a nullnorm constructed by Karacal et al. in 2015 and a nullnorm constructed by Ertugrul in 2018 can be retrieved as special cases. We also explain how to obtain the dual results.

Fuzzy signatures have been used for various applications, including medical diagnosis, communication of robots based on intention guessing, and residential building evaluation. However, there exist many questions about this research topic which have not been addressed in the literature. One of the most important aims is to formally define a family of fuzzy signatures from which an algebraic structure can be obtained allowing to make computations among fuzzy signatures. This paper studies this family and defines suitable meet and join operators satisfying the properties of a lattice as an algebraic structure. A partial ordering relation, the least and greatest elements are also defined on the family of fuzzy signatures. As a consequence, fuzzy signatures can be used as truth values of fuzzy sets, which provides a great level of representativity, completely different from the interval-valued and type-2 fuzzy sets, among others.

The main purpose of this paper is to describe the concrete form of the largest topological fuzzy closure operator on the power-set quantale of a Q-ordered semigroup. Meanwhile, we give a join- and meet-dense basis of the least Q-quantale completion of a Q-ordered semigroup. Moreover, we prove that the least Q-quantale completion is precisely an ϵ ≤ -injective hull of the Q-ordered semigroup in the category Q- Os g ≤ of Q-ordered semigroups and submultiplicative Q-order-preserving mappings. Here ϵ ≤ denotes the class of those morphisms h : S ⟶ T in Q- Os g ≤ for which e T ( h ( a 1 ) ⋯ h ( a n ) , h ( a ) ) ≤ e S ( a 1 ⋯ a n , a ) , where e S and e T are the Q-orders on the Q-ordered semigroups S and T, respectively.

The original Hutton interval I ( L ) can algebraically be identified with the tensor product I ⊗ L of the real unit interval I and a complete lattice L. Due to this, the tensor product M ⊗ L with M a completely distributive lattice is considered as a generalization of the lattice I ( L ) . When appropriately endowed with an L-topology, the tensor product M ⊗ L becomes also an L-topological extension of I ( L ) . If M is ⊲-separable (= it has a countable join base free of supercompact elements), many of the L-topological features of I ( L ) are retained. To wit, Urysohn lemma and Tietze–Urysohn extension theorem for ( M ⊗ L ) -valued functions are then proved. The relationship of M ⊗ L to the L-fuzzy topological modification of M in the sense of D. Zhang and Y.-M. Liu [27] is discussed.

This paper investigates the cardinality of a basis in semilinear spaces of n -dimensional vectors over join-semirings. First, it introduces the notion of an irredundant decomposition of an element in a join-semiring, then discusses the cardinality of a basis and proves that the cardinality of each basis is n if and only if the multiplicative identity element 1 is join-irreducible. If 1 is not a join-irreducible element then each basis need not have the same number of elements in semilinear spaces of n -dimensional vectors over join-semirings. This gives an answer to an open problem raised by Di Nola et al. in their work [Algebraic analysis of fuzzy systems, Fuzzy Sets and Systems 158 (2007) 1–22].

In some previous papers, the second author studied the possibility to associate a sequence of membership functions and of join spaces with a hypergroupoid H. Here we establish a sufficient condition in order to have two consecutive associated join spaces ^iH and ^i^+^1H non-isomorphic. We find also a property of the quotient hypergroup determined by the join space ^iH and the relation associated with the fuzzy set @[email protected]? and finally, we determine a necessary and sufficient condition for the join space ^iH in order to be a reduced hypergroup.

For finite L -fuzzy machines we develop behaviour, reduction and minimization theory. Here L stands for (L,∨,∧,0,1), where L is a totally ordered set with universal bounds 0 and 1 and the operations are join ∨ and meet ∧. The most essential results include investigating behaviour, equivalence, reduction and minimization problems and their algorithmical decidability.

This paper deals with the characterization of the isomorphisms of the structures related to De Morgan triplets. As is well known, the De Morgan triplets are used in order to define the basic connectives meet, join and negation in the setting of the Multiple-Valued Logic underlying Fuzzy Set Theory; therefore, such triplets will induce isomorphisms on the structures defined in [0,1] x by these connectives. A classification of the De Morgan triplets is given for which the De Morgan triplets of the type (Min, Max, n ) - and, therefore, the corresponding De Morgan algebras - constitute a class. The main resulting equivalence classes of De Morgan triplets are studied.