In this paper we consider twelve classical laws of negation and study their relations in the context of BCKalgebras. A classification of the laws of negation is established and some characterizations are obtained. For example, using the concept of translation we obtain some characterizations of Hilbert algebras and commutative BCK-algebras with minimum. As a consequence we obtain a theorem rela...

Defects in the gene encoding Bruton's tyrosine kinase (Btk) result in a disease called X-linked agammaglobulinemia, in which there is a profound decrease of mature B cells due to a block in B cell development. Recent studies have shown that Btk is tyrosine phosphorylated and activated upon B cell antigen receptor (BCR) stimulation. To elucidate the functions of this kinase, we examined BCR sign...

Evidence is given that implication (and its special case, negation) carry the logical strength of a system of formal logic. This is done by proving normalization and cut elimination for a system based on combinatory logic or λ-calculus with logical constants for and, or, all, and exists, but with none for either implication or negation. The proof is strictly finitary, showing that this system i...

D. Buşneag 1 defined pseudo valuation on a Hilbert algebra and proved that every pseudo valuation induces a pseudometric on a Hilbert algebra. Also, D. Buşneag 2 provided several theorems on extensions of pseudo valuations. C. Buşneag 3 introduced the notions of pseudo valuations valuations on residuated lattices, and proved some theorems of extension for these using the model of Hilbert algebr...

We consider the relationship between ideals of a BCI-algebra and order ideals of its adjoint semigroup. We show that (1) if I is an ideal, then I =M−1(M(I)), (2) M(M−1(J)) is the order ideal generated by J∩R(X), (3) if X is a BCK-algebra, then J =M(M−1(J)) for any order ideal J of X, thus, for each BCK-algebra X there is a one-to-one correspondence between the set (X) of all ideals of X and the...