In this paper, using the notion of upper sets, we introduced the notions of complicated BE-Algebras and gave some related properties on complicated, self-distributive and commutative BE-algebras. In a self-distributive and complicated BE-algebra, characterizations of ideals are obtained.
In [6] da Costa has introduced a new hierarchy , of logics that are both paraconsistent and paracomplete. Such logics are now known as non-alethic logics. In this article we present an algebraic version of the logics and study some of their properties.
In this paper we shall give some results on irreducible deductive systems in BCK-algebras and we shall prove that the set of all deductive systems of a BCK-algebra is a Heyting algebra. As a consequence of this result we shall show that the annihilator of a deductive system is the the pseudocomplement of . These results are more general than that the similar results given by M. Kondo in [7].
The notion of bounded commutative residuated -monoid (-monoid, in short) generalizes both the notions of -algebra and of -algebra. Let be a -monoid; we denote by the underlying lattice of . In the present paper we show that each direct...
It is well-known that every MV-algebra is a distributive lattice with respect to the induced order. Replacing this lattice by the so-called directoid (introduced by J. Ježek and R. Quackenbush) we obtain a weaker structure, the so-called skew MV-algebra. The paper is devoted to the axiomatization of skew MV-algebras, their properties and a description of the induced implication algebras.
It is shown that every directoid equipped with sectionally switching mappings can be represented as a certain implication algebra. Moreover, if the directoid is also commutative, the corresponding implication algebra is defined by four simple identities.
We introduce a concept of implication groupoid which is an essential generalization of the implication reduct of intuitionistic logic, i.e. a Hilbert algebra. We prove several connections among ideals, deductive systems and congruence kernels which even coincide whenever our implication groupoid is distributive.
We modify slightly the definition of -partial functions given by Celani and Montangie (2012); these partial functions are the morphisms in the category of -space and this category is the dual category of the category with objects the Hilbert algebras with supremum and morphisms, the algebraic homomorphisms. As an application we show that finite pure Hilbert algebras with supremum are determined by the monoid of their endomorphisms.
In this note we describe the structure of dually residuated -monoids (-monoids) that have no non-trivial convex subalgebras.
We introduce Engel elements in a BCI-algebra by using left and right normed commutators, and some properties of these elements are studied. The notion of -Engel BCI-algebra as a natural generalization of commutative BCI-algebras is introduced, and we discuss Engel BCI-algebra, which is defined by left and right normed commutators. In particular, we prove that any nilpotent BCI-algebra of type is an Engel BCI-algebra, but solvable BCI-algebras are not Engel, generally. Also, it is proved that...
We prove that an equational class of Hilbert algebras cannot be defined by a single equation. In particular Hilbert algebras and implication algebras are not one-based. Also, we use a seminal theorem of Alfred Tarski in equational logic to characterize the set of cardinalities of all finite irredundant bases of the varieties of Hilbert algebras, implication algebras and commutative BCK algebras: all these varieties can be defined by independent bases of n elements, for each n > 1.