Prime radicals of directed pseudo-ordered algebras over directed fields

Characteristics of partially pseudo-ordered ($\mathcal K$-ordered) algebras are considered. Properties of ideals and order homomorphisms in partially pseudo-ordered algebras are described. A variation of the concept of the prime radical for an algebra in a subclass of directed pseudo-ordered algebras ($\mathcal{AO}$-ordered algebras) over directed fields is investigated. A description of elements of $\mathcal{AO}$-prime radicals for $\mathcal{AO}$-ordered algebras over directed fields is obtained.