Strong ideals and horizontal ideals in pseudo-BCH-algebras

Abstract

In this paper we define strong ideals and horizontal ideals in pseudo-BCH-algebras and investigate the properties and characterizations of them. 1. Introduction In 1966, Y. Imai and K. Iséki ([13, 14]) introduced BCKand BCI-algebras. In 1983, Q.P. Hu and X. Li ([11]) introduced BCH-algebras. It is known that BCKand BCI-algebras are contained in the class of BCH-algebras. In 2001, G. Georgescu and A. Iorgulescu ([10]) introduced pseudo-BCK-algebras as an extension of BCK-algebras. In 2008, W.A. Dudek and Y.B. Jun ([2]) introduced pseudo-BCI-algebras as a natural generalization of BCI-algebras and of pseudo-BCK-algebras. These algebras have also connections with other algebras of logic such as pseudo-MV-algebras and pseudo-BL-algebras defined by G. Georgescu and A. Iorgulescu in [8] and [9], respectively. Those algebras were investigated by several authors in a number of papers (see for example [3, 5, 6, 7, 15, 17, 18, 19]). Recently, A. Walendziak ([20]) introduced pseudo-BCH-algebras as an extension of BCH-algebras and studied the set Cen X of all minimal elements of a pseudo-BCH-algebra X, the so-called centre of X. He also considered ideals in pseudo-BCH-algebras and established a relationship between the ideals of a pseudo-BCH-algebra and the ideals of its centre. In this paper we define strong ideals and horizontal ideals in pseudo-BCHalgebras and investigate the properties and characterizations of them.