Construction of some algebras of logics by using intuitionistic fuzzy filters on hoops

<abstract><p>In this paper, we define the notions of intuitionistic fuzzy filters and intuitionistic fuzzy implicative (positive implicative, fantastic) filters on hoops. Then we show that all intuitionistic fuzzy filters make a bounded distributive lattice. Also, by using intuitionistic fuzzy filters we introduce a relation on hoops and show that it is a congruence relation, then we prove that the algebraic structure made by it is a hoop. Finally, we investigate the conditions that quotient structure will be different algebras of logics such as Brouwerian semilattice, Heyting algebra and Wajesberg hoop.</p></abstract>

Frontal operators in distributive lattices with a generalized implication

We introduce a family of extensions of bounded distributive lattices. These extensions are obtained by adding two operations: an internal unary operation, and a function (called generalized implication) that maps pair of elements to ideals of the lattice. A bounded distributive lattice with a generalized implication is called gi-lattice in [J. E. Castro and S. A. Celani, Quasi-modal lattices, Order 21 (2004) 107–129]. The main goal of this paper is to introduce and study the category of frontal gi-lattices (and some subcategories of it). This category can be seen as a generalization of the category of frontal weak Heyting algebras (see [S. A. Celani and H. J. San Martín, Frontal operators in weak Heyting algebras, Studia Logica 100(1–2) (2012) 91–114]). In particular, we study the case of frontal gi-lattices where the generalized implication is defined as the annihilator (see [B. A. Davey, Some annihilator conditions on distributive lattices, Algebra Universalis 4(1) (1974) 316–322; M. Mandelker, Relative annihilators in lattices, Duke Math. J. 37 (1970) 377–386]). We give a Priestley’s style duality for each one of the new classes of structures considered.

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.

The Lattice of Intuitionistic Fuzzy Filters in Residuated Lattices

The notion of tip-extended pair of intuitionistic fuzzy filters is introduced by which it is proved that the set of all intuitionistic fuzzy filters in a residuated lattice forms a bounded distributive lattice.

Coherence classes of ideals of normal lattices with applications to C(X)

Given a topological space X, Jenkins and McKnight have shown how ideals of the ring C(X) are partitioned into equivalence classes — called coherence classes — defined by declaring ideals to be equivalent if their pure parts are identical. In this paper we consider a similar partitioning of the lattice of ideals of a normal bounded distributive lattice. We then apply results obtained herein to augment some of those of Jenkins and McKnight. In particular, for Tychonoff spaces, new results include the following:(a)all members of any coherence class have the same annihilator(b)every ideal is alone in its coherence class if and only if the space is a P-space.

Profinite Heyting Algebras and Profinite Completions of Heyting Algebras

This paper surveys recent developments in the theory of profinite Heyting algebras (resp. bounded distributive lattices, Boolean algebras) and profinite completions of Heyting algebras (resp. bounded distributive lattices, Boolean algebras). The new contributions include a necessary and sufficient condition for a profinite Heyting algebra (resp. bounded distributive lattice) to be isomorphic to the profinite completion of a Heyting algebra (resp. bounded distributive lattice). This results in simple examples of profinite bounded distributive lattices that are not isomorphic to the profinite completion of any bounded distributive lattice. We also show that each profinite Boolean algebra is isomorphic to the profinite completion of some Boolean algebra. It is still an open question whether each profinite Heyting algebra is isomorphic to the profinite completion of some Heyting algebra.

Bounded distributive lattice expansions

A new notion of a canonical extension $\mathbf{A}^{\sigma }$ is introduced that applies to arbitrary bounded distributive lattice expansions (DLEs) $\mathbf{A} $. The new definition agrees with the earlier ones whenever they apply. In particular, for a bounded distributive lattice $\mathbf{A}, \mathbf{A}^{\sigma }$ has the same meaning as before. A novel feature is the introduction of several topologies on the universe of the canonical extension of a DL. One of these topologies is used to define the canonical extension $f^{\sigma }:\mathbf{A}^{\sigma }\rightarrow \mathbf{B}^{\sigma }$ of an arbitrary map $f:\mathbf{A}\rightarrow \mathbf{B}$ between DLs, and hence to define the canonical extension $\mathbf{A}^{\sigma }$ of an arbitrary DLE $\mathbf{A}$. Together the topologies form a powerful tool for showing that many properties of DLEs are preserved by canonical extensions.

FREE TERNARY ALGEBRAS

A ternary algebra is a bounded distributive lattice with additonal operations e and ~ that satisfies (a+b)~=a~b~, a~~=a, e≤a+a~, e~= e and 0~=1. This article characterizes free ternary algebras by giving necessary and sufficient conditions on a set X of free generators of a ternary algebra L, so that X freely generates L. With this characterization, the free ternary algebra on one free generator is displayed. The poset of join irreducibles of finitely generated free ternary algebras is characterized. The uniqueness of the set of free generators and their pseudocomplements is also established.

Maximal subalgebras of Heyting algebras

AHeyting algebra is an algebra H;∨,∧ →, 0,1) of type (2,2,2,0,0) for which H;∨,∧,0,1) is a bounded distributive lattice and → is the binary operation of relative pseudocomplementation (i.e., for a,b,c∈H,ac ∧≦birr c≦a→b). Associated with every subalgebra of a Heyting algebra is a separating set. Those corresponding to maximal subalgebras are characterized in Proposition 8 and, subsequently, are used in an investigation of Heyting algebras.