Interpolation Pseudo-Ordered Rings

• Published in: Journal of mathematical sciences (New York, N.Y.) Vol. 269; no. 5; pp. 734 - 743
• Main Authors: Mikhalev, A. V., Shirshova, E. E.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-02-2023; Springer; Springer Nature B.V
• Subjects: Convexity; Interpolation; Isomorphism; Lattices (mathematics); Mathematics; Mathematics and Statistics; Subgroups; Theorems
• ISSN: 1072-3374; 1573-8795
• DOI: 10.1007/s10958-023-06310-7

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
∈
R
. The analog for the third theorem of ring order isomorphisms for principal convex directed ideals is demonstrated for interpolation pseudo-ordered rings.