Interpolation pseudo-ordered algebras over partially ordered fields

A. V. Mikhalev ; E. E. Shirshova

Fundamentalʹnaâ i prikladnaâ matematika, Tome 24 (2022) no. 2, pp. 181-196

Résumé

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