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<p>Characteristics of partially pseudo-ordered (<i>K</i>-ordered) algebras over partially ordered fields are considered. Properties of the set of all convex directed ideals in partially pseudo-ordered algebras are described. It is shown that convex directed ideals play for the theory of partially pseudo-ordered algebras the same role as convex directed subgroups for the theory of partially ordered groups. Necessary and sufficient conditions for a&#xa0;convex directed ideal of an <i>AO</i>-pseudo-ordered algebra over a&#xa0;partially ordered field to be a&#xa0;rectifying ideal are obtained. We show that the set of all rectifying directed ideals of an <i>AO</i>-pseudo-ordered algebra over a&#xa0;partially ordered field forms a&#xa0;root system for the lattice of all convex directed ideals of that algebra. Properties of regular ideals for partially pseudo-ordered algebras over partially ordered fields are investigated. Some results are proved concerning convex directed ideals of pseudo-lattice pseudo-ordered algebras over directed fields.</p>

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CONVEX IDEALS OF PARTIALLY PSEUDO-ORDERED ALGEBRAS OVER PARTIALLY ORDERED FIELDS

  • 期刊论文

  • 发表日期：

    2026年3月20日

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  • E. E. Shirshova

摘要

Characteristics of partially pseudo-ordered (K-ordered) algebras over partially ordered fields are considered. Properties of the set of all convex directed ideals in partially pseudo-ordered algebras are described. It is shown that convex directed ideals play for the theory of partially pseudo-ordered algebras the same role as convex directed subgroups for the theory of partially ordered groups. Necessary and sufficient conditions for a convex directed ideal of an AO-pseudo-ordered algebra over a partially ordered field to be a rectifying ideal are obtained. We show that the set of all rectifying directed ideals of an AO-pseudo-ordered algebra over a partially ordered field forms a root system for the lattice of all convex directed ideals of that algebra. Properties of regular ideals for partially pseudo-ordered algebras over partially ordered fields are investigated. Some results are proved concerning convex directed ideals of pseudo-lattice pseudo-ordered algebras over directed fields.