Logics of Involutive Stone Algebras

An involutive Stone algebra (IS-algebra) is a structure that is simultaneously a De Morgan algebra and a Stone algebra (i.e. a pseudo-complemented distributive lattice satisfying the well-known Stone identity, ∼ x ∨ ∼ ∼ x ≈ 1). IS-algebras have been studied algebraically and topologically since the 1980’s, but a corresponding logic (here denoted IS ≤ ) has been introduced only very recently. The logic IS ≤ is the departing point for the present study, which we then extend to a wide family of previously unknown logics defined from IS-algebras. We show that IS ≤ is a conservative expansion of the Belnap-Dunn four-valued logic (i.e. the order-preserving logic of the variety of De Morgan algebras), and we give a finite Hilbert-style axiomatization for it. More generally, we introduce a method for expanding conservatively every super-Belnap logic so as to obtain an extension of IS ≤ . We show that every logic thus defined can be axiomatized by adding a fixed finite set of rule schemata to the corresponding super-Belnap base logic. We also consider a few sample extensions of IS ≤ that cannot be obtained in the above- described way, but can nevertheless be axiomatized finitely by other methods. Most of our axiomatization results are obtained in two steps: through a multiple-conclusion calculus first, which we then reduce to a traditional one. The multiple-conclusion axiomatizations introduced in this process, being analytic, are of independent interest from a proof-theoretic standpoint. Our results entail that the lattice of super-Belnap logics (which is known to be uncountable) embeds into the lattice of extensions of IS ≤ . Indeed, as in the super-Belnap case, we establish that the finitary extensions of IS ≤ are already uncountably many.

Gradation of Continuity for Fuzzy Soft Mappings

This paper is devoted to describe the notion of a parameterized degree of continuity for mappings between L -fuzzy soft topological spaces, where L is a complete De Morgan algebra. The degrees of openness, closedness, and being a homeomorphism for the fuzzy soft mappings are also presented. The properties and characterizations of the proposed notions are pictured. Besides, the degree of continuity for a fuzzy soft mapping is unified with the degree of compactness and connectedness in a natural way.

Some De Morgan algebras associated with a double fuzzy topological space

This paper identifies a De Morgan algebra associated with the families of (r, s)-regular fuzzy open sets and (r, s)-regular fuzzy closed sets in a double fuzzy topological space. The situation under which this De Morgan algebra becomes a Boolean algebra is characterized. Certain other properties of this algebra are also investigated.

Constructing some logical algebras from EQ-algebras

EQ-algebras were introduced by Nov?ak in [16] as an algebraic structure of truth values for fuzzy type theory (FTT). Nov?k and De Baets in [18] introduced various kinds of EQ-algebras such as good, residuated, and lattice ordered EQ-algebras. In any logical algebraic structures, by using various kinds of filters, one can construct various kinds of other logical algebraic structures. With this inspirations, by means of fantastic filters of EQ-algebras we construct MV-algebras. Also, we study prelinear EQ-algebras and introduce a new kind of filter and named it prelinear filter. Then, we show that the quotient structure which is introduced by a prelinear filter is a distributive lattice-ordered EQ-algebras and under suitable conditions, is a De Morgan algebra, Stone algebra and Boolean algebra.

De Morgan Functions and Free De Morgan Algebras

It is commonly known that the free Boolean algebra on n free generators is isomorphic to the Boolean algebra of Boolean functions of n variables. The free bounded distributive lattice on n free generators is isomorphic to the bounded lattice of monotone Boolean functions of n variables. In this paper, we introduce the concept of De Morgan function and prove that the free De Morgan algebra on n free generators is isomorphic to the De Morgan algebra of De Morgan functions of n variables. This is a solution of the problem suggested by B. I. Plotkin.

Strong N β-Compactness in L-Topological Spaces

In this paper,the new compactness which is strong-compactness is introduced for an arbitrary-subset and for a complete distributive De Morgan algebra. The strong-compactness implies strong-III-compactness,hence it also implies strong-II-compactness,strong-I-compactness,-compactness,-compactness and Lowen's fuzzy compactness. But it is different from-compactness.When ,strong-compactness is equivalent to-compactness.

Semicompactness inL-topological spaces

The concepts of semicompactness, countable semicompactness, and the semi-Lindelöf property are introduced inL-topological spaces, whereLis a complete de Morgan algebra. They are defined by means of semiopenL-sets and their inequalities. They do not rely on the structure of basis latticeLand no distributivity inLis required. They can also be characterized by semiclosedL-sets and their inequalities. WhenLis a completely distributive de Morgan algebra, their many characterizations are presented.

INVOLUTED SEMILATTICES AND UNCERTAINTY IN TERNARY ALGEBRAS

An involuted semilattice <S,∨,-> is a semilattice <S,∨> with an involution-: S→S, i.e., <S,∨,-> satisfies [Formula: see text], and [Formula: see text]. In this paper we study the properties of such semilattices. In particular, we characterize free involuted semilattices in terms of ordered pairs of subsets of a set. An involuted semilattice <S,∨,-,1> with greatest element 1 is said to be complemented if it satisfies a∨ā=1. We also characterize free complemented semilattices. We next show that complemented semilattices are related to ternary algebras. A ternary algebra <T,+,*,-,0,ϕ,1> is a de Morgan algebra with a third constant ϕ satisfying [Formula: see text], and (a+ā)+ϕ=a+ā. If we define a third binary operation ∨ on T as a∨b=a*b+(a+b)*ϕ, then <T,∨,-,ϕ> is a complemented semilattice.