An algebra of quasiordered logic
An algebraic and Kripke-style approach to a certain extension of intuitionistic logic [Book]
An ordered structure of pseudo-BCI-algebras
In Chajda's paper (2014), to an arbitrary BCI-algebra the author assigned an ordered structure with one binary operation which possesses certain antitone mappings. In the present paper, we show that a similar construction can be done also for pseudo-BCI-algebras, but the resulting structure should have two binary operations and a set of couples of antitone mappings which are in a certain sense mutually inverse. The motivation for this approach is the well-known fact that every commutative BCK-algebra...
Anneaux monadiques rationnels
Annihilators and deductive systems in commutative Hilbert algebras
The properties of deductive systems in Hilbert algebras are treated. If a Hilbert algebra considered as an ordered set is an upper semilattice then prime deductive systems coincide with meet-irreducible elements of the lattice of all deductive systems on and every maximal deductive system is prime. Complements and relative complements of are characterized as the so called annihilators in .
Annihilators in BCK-algebras
We introduce the concepts of an annihilator and a relative annihilator of a given subset of a BCK-algebra . We prove that annihilators of deductive systems of BCK-algebras are again deductive systems and moreover pseudocomplements in the lattice of all deductive systems on . Moreover, relative annihilators of with respect to are introduced and serve as relative pseudocomplements of w.r.t. in .
Archimedean atomic lattice effect algebras in which all sharp elements are central
We prove that every Archimedean atomic lattice effect algebra the center of which coincides with the set of all sharp elements is isomorphic to a subdirect product of horizontal sums of finite chains, and conversely. We show that every such effect algebra can be densely embedded into a complete effect algebra (its MacNeille completion) and that there exists an order continuous state on it.
Autonomous posets and quantales
Basic pseudorings
The concept of a basic pseudoring is introduced. It is shown that every orthomodular lattice can be converted into a basic pseudoring by using of the term operation called Sasaki projection. It is given a mutual relationship between basic algebras and basic pseudorings. There are characterized basic pseudorings which can be converted into othomodular lattices.
bi-BL-algebra
In this paper, we introduce the notion of a bi-BL-algebra, bi-filter, bi-deductive system and bi-Boolean elements of a bi-BL-algebra and deal with bi-filters in bi-BL-algebra. We study this structure and construct the quotient of bi-BL-algebra. Also present a classification for examples of proper bi-BL-algebras.
BL-algebra of fractions relative to an -closed system.
Boolean part of BL-algebras
Bounded dually residuated lattice ordered monoids as a generalization of fuzzy structures
Bounded lattices with antitone involutions and properties of MV-algebras
We introduce a bounded lattice L = (L;∧,∨,0,1), where for each p ∈ L there exists an antitone involution on the interval [p,1]. We show that there exists a binary operation · on L such that L is term equivalent to an algebra A(L) = (L;·,0) (the assigned algebra to L) and we characterize A(L) by simple axioms similar to that of Abbott's implication algebra. We define new operations ⊕ and ¬ on A(L) which satisfy some of the axioms of MV-algebra. Finally we show what properties must be satisfied by...
Cantor-Bernstein theorem for -algebras
Classes of filters in generalizations of commutative fuzzy structures
Bounded commutative residuated lattice ordered monoids (-monoids) are a common generalization of -algebras and Heyting algebras, i.e. algebras of basic fuzzy logic and intuitionistic logic, respectively. In the paper we develop the theory of filters of bounded commutative -monoids.
Classes of fuzzy filters of residuated lattice ordered monoids
The logical foundations of processes handling uncertainty in information use some classes of algebras as algebraic semantics. Bounded residuated lattice ordered monoids (monoids) are common generalizations of -algebras, i.e., algebras of the propositional basic fuzzy logic, and Heyting algebras, i.e., algebras of the propositional intuitionistic logic. From the point of view of uncertain information, sets of provable formulas in inference systems could be described by fuzzy filters of the corresponding...
Commutative directoids with sectional involutions
The concept of a commutative directoid was introduced by J. Ježek and R. Quackenbush in 1990. We complete this algebra with involutions in its sections and show that it can be converted into a certain implication algebra. Asking several additional conditions, we show whether this directoid is sectionally complemented or whether the section is an NMV-algebra.
Commutative directoids with sectionally antitone bijections
We study commutative directoids with a greatest element, which can be equipped with antitone bijections in every principal filter. These can be axiomatized as algebras with two binary operations satisfying four identities. A minimal subvariety of this variety is described.
Completion closed algebras and models of Peano arithmetic