Solution of Three Open Problems on Pseudo-Filters (Pseudo-Ideals) of Pseudo-BCK Algebras

Three open problems on pseudo-filters (ideals) of pseudo-BCK algebras, posed by Y.B. Jun, M. Kondo and K.H. Kim [Pseudo-ideals of pseudo-BCK algebras, Scientiae Mathematicae Japonicae, 2003, 1: 93-97], are solved. Moreover, some properties of pseudo-BCK algebras and some examples are presented, and the following important result is proved: a pseudo-BCK algebra becomes a positive implicative BCK-algebra if and only if its every pseudo-filter (ideals) is a normal positive implicative pseudo-filter (ideal).