Prime radicals of lattice $\mathcal K$-ordered algebras

The Kopytov order for any algebra over a field is considered. Necessary and sufficient conditions for an algebra to be a linearly ordered algebra are presented. Some results concerning the properties of ideals of linearly ordered algebras are obtained. Some examples of algebras with the Kopytov order are described. The Kopytov order for these examples induces the order on other algebraic objects. The purpose of this paper is to investigate a generalization of the concept of prime radical to lattice ordered algebras over partially ordered fields. Prime radicals of $l$-algebras over partially ordered and directed fields are described. Some results concerning the properties of the lower weakly solvable $l$-radical of $l$-algebras are obtained. Necessary and sufficient conditions for the $l$-prime radical of an $l$-algebra to be equal to the lower weakly solvable $l$-radical of the $l$-algebra are presented.