Characteristics of partially pseudo-ordered (K-ordered) rings are considered. Properties of the set of all convex directed ideals in pseudo-ordered rings are described. It is shown that convex directed ideals play for the theory of partially pseudo-ordered rings 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 ring to be a rectifying ideal are obtained. We show that the set of all rectifying directed ideals of an AO-pseudo-ordered ring form the root system for the lattice of all convex directed ideals of that ring. Properties of regular ideals for partially pseudo-ordered rings are investigated. Some results are proved concerning convex directed ideals of pseudo-lattice 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 Ia 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.

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 ∈ 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.

Characteristics of partially pseudo-ordered (𝒦-ordered) algebras are considered. Properties of ideals and order homomorphisms in partially pseudo-ordered algebras are described. A variation of the concept of the prime radical for an algebra in a subclass of directed pseudo-ordered algebras (𝒜𝒪-ordered algebras) over directed fields is investigated. A description of elements of 𝒜𝒪-prime radicals for 𝒜𝒪-ordered algebras over directed fields is obtained.

Victor Timofeevich was a man of very broad views, he was interested in almost everything in life, and everywhere he showed his outstanding talents and deep intelligence.In his young years, he supported many university traditions of leisure: sports, music, tourism

Derivative lattices associated with partially ordered linear spaces over partially ordered skew fields are considered. Properties of the convex projective geometry $$ \mathcal{L} $$ for a partially ordered linear space FV over a partially ordered skew field F are investigated. The convexity of linear subspaces for the linear space FV means the Abelian convexity (ab-convexity), which is based on the definition of a convex subgroup for a partially ordered group. It is shown that ab-convex directed linear subspaces plays for the theory of partially ordered linear spaces the same role as convex directed subgroups for the theory of partially ordered groups. We obtain the element-wise description of the smallest ab-convex directed linear subspace that contains a given positive element, for a linear space over a directed skew field. It is proved that if FV is an interpolation linear space over a partially ordered skew field F, then, in the lattice $$ \mathcal{L} $$ , the union operation is completely distributive with respect to intersection. Properties of the projective geometry for pseudo lattice-ordered linear spaces over partially ordered skew fields are investigated.

We show that all convex directed subgroups of a pl-group form a distributive lattice under inclusions that is a Brouwer lattice. We succeeded in extending some l-group results concerning rectifying and regular subgroups to the class of $$ \mathcal{AO} $$ -groups. Necessary and sufficient conditions are given for an element of a pl-group to be an element with a unique value. In order to prove this, some properties of lexicographic extensions of $$ \mathcal{AO} $$ -groups and pl-groups are investigated.

АннотацияВ статье, посвященной 75-летию Александра Юрьевича Ольшанского, коллеги, друзья и ученики отразили биографические данные о юбиляре

Characteristics of partially pseudo-ordered ( K -ordered) rings are considered. Properties of ideals in pseudo-ordered rings are described. A variation of the concept of the prime radical for a ring in the subclass of directed pseudo-ordered rings ( AO -ordered rings) is investigated. The description of elements of AO -prime radicals for AO -ordered rings is obtained.

A new notion of a partial ordering for rings is considered. Properties of arbitrary partially right \( \mathcal{K} \)-ordered rings are investigated. A series of results for linearly right \( \mathcal{K} \)-ordered rings is obtained. Some theorems are proved for ideals of those rings.

We consider properties of convex directed subgroups for the interpolation groups in which each element is a quotient of two almost orthogonal elements. A series of results on values for almost orthogonal elements in those groups is obtained. We investigate characteristics of lexicographic extensions for partially ordered groups in which each element is a quotient of two almost orthogonal elements.

Characteristics of groups with the interpolation relation (not necessarily directed) are considered. A necessary and sufficient condition for a partially ordered group to be an interpolation group is obtained. An almost orthogonality criterion for positive elements of an interpolation group is proved. Characteristics of minimal convex directed subgroups containing almost orthogonal elements are described. Properties of convex directed subgroups in the subclass of interpolation groups in which each element is a quotient of two almost orthogonal elements are investigated.

The Kopytov order for any algebra over a field is considered. Necessary and sufficient conditions for an algebra to be a linearly ordered algebra are presented. Some results concerning the properties of ideals of linearly ordered algebras are obtained. Some examples of algebras with the Kopytov order are described. The Kopytov order for these examples induces the order on other algebraic objects. The purpose of this paper is to investigate a generalization of the concept of prime radical to lattice-ordered algebras over partially ordered fields. Prime radicals of l -algebras over partially ordered and directed fields are described. Some results concerning the properties of the lower weakly solvable l -radical of l -algebras are obtained. Necessary and sufficient conditions for the l -prime radical of an l -algebra to be equal to the lower weakly solvable l -radical of the l -algebra are presented.

This paper deals with an approach to the ordering of rings, which was introduced by V. M. Kopytov for Lie algebras. The conditions of existence of a linear order in rings are found. The properties of ideals of linear ordered rings are investigated.

This paper deals with an approach to the ordering of rings, which was introduced by V. M. Kopytov for Lie algebras. The conditions of existence of a linear order in rings are found. The properties of ideals of linear ordered rings are investigated.

Partially ordered groups satisfying the interpolation condition (and not necessarily directed) are considered. It is proved that an isomorphism theorem holds for these groups (this theorem fails to hold for partially ordered groups in the general case). A criterion for almost orthogonality of positive elements of interpolation groups is found. The location of a subgroup associated with a pair of almost orthogonal elements in the lattice of subgroups of an interpolation group is described.

The Kopytov order for any algebras over a field is considered. Necessary and sufficient conditions for an algebra to be a linearly ordered algebra are presented. Some results concerning the properties of ideals of linearly ordered algebras are obtained. Some examples of algebras with the Kopytov order are described. The Kopytov order for these examples induces the order on other algebraic objects.

The notions of Cartesian and semidirect products for partially ordered groups are considered. A series of results on those products of \( \mathcal{A}\mathcal{O} \)-groups and interpolation groups is obtained. Some results concerning wreath products of directed groups are obtained.

The concept of prime radicals is important in the study of rings and groups. The purpose of this paper is to investigate a generalization of this concept to directed groups. Some results are obtained concerning convex directed subgroups of AO-groups. Prime radicals of pl-groups are described.