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Fundamentalnaya i Prikladnaya Matematika, 2022, Volume 24, Issue 2, Pages 181–196
(Mi fpm1930)
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Interpolation pseudo-ordered algebras over partially ordered fields
A. V. Mikhaleva, E. E. Shirshovab a Lomonosov Moscow State University, Moscow, Russia
b Moscow Pedagogical State University, Moscow, Russia
Abstract:
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.
Citation:
A. V. Mikhalev, E. E. Shirshova, “Interpolation pseudo-ordered algebras over partially ordered fields”, Fundam. Prikl. Mat., 24:2 (2022), 181–196; J. Math. Sci., 275:4 (2023), 502–512
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https://www.mathnet.ru/eng/fpm1930 https://www.mathnet.ru/eng/fpm/v24/i2/p181
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Abstract page: | 81 | Full-text PDF : | 19 | References: | 16 |
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