Interpolation pseudo-ordered rings

Characteristics of partially pseudo-ordered ($K$-ordered) rings are considered. Properties of the set $L(R)$ of all convex directed ideals in pseudo-ordered rings are described. The convexity of ideals has the meaning of the Abelian convexity, which is based on the definition of a convex subgroup for a partially ordered group. It is proved that if $R$ is an interpolation pseudo-ordered ring, then, in the lattice $L(R)$, the union operation is completely distributive with respect to the intersection. Properties of the lattice $L(R)$ for pseudo-lattice pseudo-ordered rings are investigated. The second and third theorems of ring order isomorphisms for interpolation pseudo-ordered rings are proved. Some theorems are proved for principal convex directed ideals of interpolation pseudo-ordered rings. The principal convex directed ideal $I_a$ of a partially pseudo-ordered ring $R$ is the smallest convex directed ideal of the ring $R$ that contains the element $a\in R$. The analog for the third theorem of ring order isomorphisms for principal convex directed ideals is demonstrated for interpolation pseudo-ordered rings.