Preservation theorems for MTL-chains

A general algebraic system M is considered with two binary operations. The family of all measurable functions with values in the unit interval is a motivating example. A state is a morphism from M to the unit interval, an observable is a morphism from the family of Borel sets to M. A joint distribution of two observables is constructed. It is applied for the construction of the sum of observables and for a representation of conditional probability.

Uninorms, as binary operations on the unit interval, have been widely applied in information aggregation.The class of almost equitable uninorms appears when the contradictory information is aggregated.It is proved that among various uninorms of which either underlying t-norm or t-conorm is continuous, only the representable uninorms belong to the class of almost equitable uninorms.As a byproduct, a characterization for the class of representable uninorms is obtained.

Uninorms, as binary operations on the unit interval, have been widely applied in fuzzy set theory. In this paper, the class of conjunctive left-continuous uninorms of which the underlying triangular norm is weak nilpotent minimum are discussed. This class of uninorms are characterized via the ordinal sum construction of Clifford for some cases.

On the role of logical connectives for primality and functional completeness of algebras of logics

An illustrative overview is laid out in this paper detailing the different classes of binary aggregation operations on the unit interval [0, 1] with an absorbing element. The authors focused on the characterization of specific well-known classes of these operations, including nullnorms, T-uninorms, S-uninorms and bi-uninorms, along with their interaction.

An MV-algebra is an algebraic structure with a binary operation ⊕, a unary operation ′ and the constant 0 satisfying certain axioms. MV-algebras are the algebraic semantics of Lukasiewicz logic. This work includes a type of derivation research on MV-algebras. Our aim is to introduce the concept of permuting tri-derivation on MV-algebras and to discuss some results.

We give the general form of an idempotent, associative, nondecreasing and continuous binary aggregation operation in a connected order topological space. The particular case of the unit interval is studied and the choice of weights is also analized. Possible generalizations for more than two arguments are also proposed.

We investigate algebraically the fragment of the intuitionistic propositional calculus consisting of equivalence together with conjunction on the intuitionistic regularizations. We find that this fragment is strongly algebraizable with the equivalent algebraic semantics being the variety of equivalential algebras with an additional binary operation that can be interpreted as the meet on regular elements. We give a finite equational base for this variety, and investigate its properties, in particular the commutator. As applications, we prove that the fragment is hereditarily structurally complete and we describe the top of the lattice of its axiomatic extensions.

We study some inequalities relating the behavior of certain binary operations of the unit interval on some numerical series.

We study a system, μŁΠ, obtained by an expansion of ŁΠ logic with fixed points connectives. The first main result of the paper is that μŁΠ is standard complete, i.e., complete with regard to the unit interval of real numbers endowed with a suitable structure. We also prove that the class of algebras which forms algebraic semantics for this logic is generated, as a variety, by its linearly ordered members and that they are precisely the interval algebras of real closed fields. This correspondence is extended to a categorical equivalence between the whole category of those algebras and another category naturally arising from real closed fields. Finally, we show that this logic enjoys implicative interpolation.