Hereditary radical classes of linearly ordered groups
In this paper we deal with the of an -algebra , where and are nonzero cardinals. It is proved that if is singular and -distributive, then it is . We show that if is complete then it can be represented as a direct product of -algebras which are homogeneous with respect to higher degrees of distributivity.
We give two variations of the Holland representation theorem for -groups and of its generalization of Glass for directed interpolation po-groups as groups of automorphisms of a linearly ordered set or of an antilattice, respectively. We show that every pseudo-effect algebra with some kind of the Riesz decomposition property as well as any pseudo -algebra can be represented as a pseudo-effect algebra or as a pseudo -algebra of automorphisms of some antilattice or of some linearly ordered set.
We introduce the concept of a hyper BCI-algebra which is a generalization of a BCI-algebra, and investigate some related properties. Moreover we introduce a hyper BCI-ideal, weak hyper BCI-ideal, strong hyper BCI-ideal and reflexive hyper BCI-ideal in hyper BCI-algebras, and give some relations among these hyper BCI-ideals. Finally we discuss the relations between hyper BCI-algebras and hyper groups, and between hyper BCI-algebras and hyper -groups.