Embedding of a uniquely divisible Abelian semigroup in a convex cone

Orlov, I. V.

doi:10.1134/s0001434617090061

Cited by 2 publications

(4 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The corresponding category denote by (RS). (4) Convex cone is an Abelian semigroup with respect to vector addition that forms a module over R + with respect to multiplication by scalars (see [2,3]). The corresponding category denote by (Con).…”

Section: Basic Objects: Terminology Notation Auxiliary Resultsmentioning

confidence: 99%

“…(5) Regular (cancellative) convex cone is a convex cone that satisfies the cancellation law (see [2,3]…”

Section: Basic Objects: Terminology Notation Auxiliary Resultsmentioning

confidence: 99%

“…Modulation. (see[3]) M : (US) → (Con), (URS) → (RCon), (UG) → (Lin).Let's introduce in uniquely divisible Abelian semigroup X an "additive multiplication" by non-negative scalars, first for rational case. (a) For x ∈ X, r = m n ∈ Q + set (y = r * x) ⇔ ( m x = n y).…”

mentioning

confidence: 99%

See 2 more Smart Citations

Sublinear extension of algebraic grothendieck theory

Orlov

2017

2017 Constructive Nonsmooth Analysis and Related Topics (Dedicated to the Memory of V.F. Demyanov) (CNSA)

View full text Add to dashboard Cite

A commutative diagram that connects the basic objects of commutative algebra with the main objects of commutative analysis is constructed. Namely, with the help of five types of canonical embeddings we constructed a diagram between two sets of objects: Abelian semigroups -Abelian regular (cancellative) semigroups -Abelian groups, on the first hand, and convex cones -regular convex cones -linear spaces, on the other hand. Thus, some extension of the Grothendieck algebraic K-theory arises, that includes the basic objects not only of linear (smooth) analysis but of sublinear (nonsmooth) analysis also.

show abstract

Section: Basic Objects: Terminology Notation Auxiliary Resultsmentioning

confidence: 99%

“…(5) Regular (cancellative) convex cone is a convex cone that satisfies the cancellation law (see [2,3]…”

Section: Basic Objects: Terminology Notation Auxiliary Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Sublinear extension of algebraic grothendieck theory

Orlov

2017

2017 Constructive Nonsmooth Analysis and Related Topics (Dedicated to the Memory of V.F. Demyanov) (CNSA)

View full text Add to dashboard Cite

show abstract

“…This lemma turned out to be a basic tool in various fields and hundreds of papers have used it by now. For instance, in nonsmooth analysis [7][8][9]14,[18][19][20]34], optimization theory [15,36,38], theory of convex sets and functions [10,12,16,17,[23][24][25][26][27][28][29][30][31][32][33]35,40,59,60], set-valued analysis [2,13,37,39,41,44,45,47,48], set-valued differential equations [3,4,11,22,43,49], set-valued functional equations [6,42,[51][52]…”

Section: Introductionmentioning

confidence: 99%

An extension of the Rådström Cancellation Theorem to Cornets

Molnár¹

2020

Preprint

View full text Add to dashboard Cite

The aim of this paper is to introduce the notion of cornets, which form a particular subclass of ordered semigroups also equipped with a multiplication by natural numbers. The most important standard examples for cornets are the families of the nonempty subsets and the nonempty fuzzy subsets of a vector space. In a cornet, the convexity, nonnegativity, Archimedean property, boundedness, closedness of an element can be defined naturally. The basic properties related to these notions are established. The main result extends the Cancellation Principle discovered by Rådström in 1952.Definition 2.1. An ordered triplet (X, +, ) is called an ordered commutative semigroup if (i) (X, +) is a commutative unital semigroup with a unit element 0;

show abstract

Embedding of a uniquely divisible Abelian semigroup in a convex cone

Cited by 2 publications

References 7 publications

Sublinear extension of algebraic grothendieck theory

Sublinear extension of algebraic grothendieck theory

An extension of the Rådström Cancellation Theorem to Cornets

Contact Info

Product

Resources

About