-structures applied to closed ideals in BCH-algebras.
-subalgebras in BCK/BCI-algebras based on point -structures.
A characterization of commutative basic algebras
A basic algebra is an algebra of the same type as an MV-algebra and it is in a one-to-one correspondence to a bounded lattice having antitone involutions on its principal filters. We present a simple criterion for checking whether a basic algebra is commutative or even an MV-algebra.
A construction of mirror -algebras.
A glimpse of deductive systems in algebra
The concept of a deductive system has been intensively studied in algebraic logic, per se and in connection with various types of filters. In this paper we introduce an axiomatization which shows how several resembling theorems that had been separately proved for various algebras of logic can be given unique proofs within this axiomatic framework. We thus recapture theorems already known in the literature, as well as new ones. As a by-product we introduce the class of pre-BCK algebras.
A groupoid characterization of Boolean algebras
We present a groupoid which can be converted into a Boolean algebra with respect to term operations. Also conversely, every Boolean algebra can be reached in this way.
A note on homomorphisms of Hilbert algebras.
A representation of bounded commutative BCK-algebras.
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...
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 .
Annihilators on weakly standard BCC-algebras.
Bipolar fuzzy subalgebras and bipolar fuzzy ideals of BCK/BCI-algebras.
Commutative pseudo valuations on BCK-algebras.
Companion -algebras
Complicated BE-algebras and characterizations of ideals
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.
Congruences, ideals and annihilators in standard QBCC-algebras
We characterize congruence lattices of standard QBCC-algebras and their connection with the congruence lattices of congruence kernels.
Deductive systems of BCK-algebras
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].
Doubt fuzzy BCI-algebras.
Dual commutative hyper K-ideals of type 1 in hyper K-algebras of order 3.
In this note we classify the bounded hyper K-algebras of order 3, which have D1 = {1}, D2 = {1,2} and D3 = {0,1} as a dual commutative hyper K-ideal of type 1. In this regard we show that there are such non-isomorphic bounded hyper K-algebras.
Engel BCI-algebras: an application of left and right commutators
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...