1. Pseudo-BCK Algebras

BCK algebras were introduced originally by K. Isèki with a binary operation ∗ modeling the set-theoretical difference and with a constant element 0, that is, a least element. Another motivation is from classical and non-classical prepositional calculi modeling logical implications. Such algebras contain as a special subfamily the family of MV-algebras where some important fuzzy structures can be studied. Pseudo-BCK algebras were introduced by G. Georgescu and A. lorgulescu as algebras with “two differences”, a left- and right-difference, instead of one ∗ and with a constant element 0 as the least element. Nowadays pseudo-BCK algebras are used in a dual form, with two implications, → and ⇝ and with one constant element 1, that is, the greatest element. Thus such pseudo-BCK algebras are in the “negative cone” and are also called “left-ones”. More properties of pseudo-BCK algebras and their connection with other fuzzy structures were established by A. lorgulescu. In this chapter we prove new properties of pseudo-BCK algebras with pseudo-product and pseudo-BCK algebras with pseudo-double negation and we show that every pseudo-BCK algebra can be extended to a good one. Examples of proper pseudo-BCK algebras, good pseudo-BCK algebras and pseudo-BCK lattices are given and the orthogonal elements in a pseudo-BCK algebra are characterized. Finally, we define the maximal and normal deductive systems of a pseudo-BCK algebra with pseudo-product and we study their properties.