Convex ideals of partially pseudo-ordered algebras over partially ordered fields

E. E. Shirshova

Moscow State Pedagogical University

Abstract: Characteristics of partially pseudo-ordered ($K$-ordered) algebras over partially ordered fields are considered. Properties of the set of all convex directed ideals in partially pseudo-ordered algebras are described. It is shown that convex directed ideals play for the theory of partially pseudo-ordered algebras the same role as convex directed subgroups for the theory of partially ordered groups. Necessary and sufficient conditions for a convex directed ideal of an $AO$-pseudo-ordered algebra over a partially ordered field to be a rectifying ideal are obtained. We show that the set of all rectifying directed ideals of an $AO$-pseudo-ordered algebra over a partially ordered field forms a root system for the lattice of all convex directed ideals of that algebra. Properties of regular ideals for partially pseudo-ordered algebras over partially ordered fields are investigated. Some results are proved concerning convex directed ideals of pseudo-lattice pseudo-ordered algebras over directed fields.

UDC: 512.545