Commutative semigroups with cancellation law: a representation theorem

Grzybowski, Jerzy; Pallaschke, Diethard; Przybycień, Hubert; Urbański, Ryszard

doi:10.1007/s00233-011-9327-5

Cited by 5 publications

(6 citation statements)

References 11 publications

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“…The following theorem is a corollary from Theorem 2.7 in [10], which was proved for commutative semigroups. The proof is easy and we omit it.…”

Section: Definition 27mentioning

confidence: 88%

“…In this section first we recall (Proposition 2.4 and Theorem 2.6) main results from [10] concerning commutative semigroup S with 0 satisfying the law of cancellation and a semigroup symmetry T on S. Definition 2.1 We say that S is 2-torsion-free if s + s = t + t implies s = t. We say that a semigroup symmetry T is 2-divisible if for any s ∈ S there exists t ∈ S such that s + Ts = t + t.…”

Section: Decomposition Of Quotient Groupmentioning

confidence: 99%

“…For Definitions 2.7 and 2.8 and the notations were used in [10] in the case of commutative, cancellative semigroups with 0. Definition 2.…”

Section: Definition 27mentioning

confidence: 99%

See 2 more Smart Citations

Decomposition of Minkowski–Rådström–Hörmander Space to the Direct Sum of Symmetric and Asymmetric Subspaces

Grzybowski

Przybycień

Urbański

2013

Set-Valued Var. Anal

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Any abstract convex cone S with a uniformity satisfying the law of cancellation can be embedded in a topological vector space S (Urbański, Bull Acad Pol Sci, Sér Sci Math Astron Phys 24: [709][710][711][712][713][714][715] 1976). We introduce a notion of a cone symmetry and decompose in Theorem 2.12 a quotient vector space S into a topological direct sum of its symmetric subspace S s and asymmetric subspace S a . In Theorem 2.19 we prove a similar decomposition for a normed space S. In section 3 we apply decomposition to Minkowski-Rådström-Hörmander (MRH) space with three best known norms and four symmetries. In section 4 we obtain a continuous selection from a MRH space over R 2 to the family of pairs of nonempty compact convex subsets of R 2 .Keywords Quasidifferential · Abstract convex cone · Cone of nonempty bounded closed convex sets · Minkowski-Rådström-Hörmander space · Hausdorff metric · Demyanov metric · Bartels-Pallaschke norm Mathematics Subject Classifications (2010) 52A07 · 22A20 · 46N10 · 26A27

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“…The following theorem is a corollary from Theorem 2.7 in [10], which was proved for commutative semigroups. The proof is easy and we omit it.…”

Section: Definition 27mentioning

confidence: 88%

Section: Decomposition Of Quotient Groupmentioning

confidence: 99%

See 1 more Smart Citation

Decomposition of Minkowski–Rådström–Hörmander Space to the Direct Sum of Symmetric and Asymmetric Subspaces

Grzybowski

Przybycień

Urbański

2013

Set-Valued Var. Anal

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“…Moreover, let the following conditions be satisfied: Proof. For k ∈ N we have 2(i) [27] and Proposition 3.5 [15]).…”

Section: Cancellation Law In An Ordered Semigroup With An Operator Of...mentioning

confidence: 99%

Order cancellation law in the family of bounded convex sets

Grzybowski

Urbański

2019

J Glob Optim

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In this paper generalize Robinson's version of an order cancellation law for subsets of vector spaces in which we cancel by unbounded sets. We introduce the notion of weakly narrow sets in normed spaces, study their properties and prove the order cancellation law where the canceled set is weakly narrow. Also we prove the order cancellation law for closed convex subsets of topological vector space where the canceled set has bounded Hausdorff-like distance from its recession cone. We topologically embed the semigroup of closed convex sets sharing a recession cone having bounded Hausdorff-like distance from it into a topological vector space. This result extends Bielawski and Tabor's generalization of Rådström theorem.

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“…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][46]48], set-valued differential equations [3,4,11,22,43,49], set-valued functional equations [6,42,[51][52][53]…”

Section: Introductionmentioning

confidence: 99%

An extension of the Rådström cancellation theorem to cornets

Molnár

2021

Semigroup Forum

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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.

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Commutative semigroups with cancellation law: a representation theorem

Cited by 5 publications

References 11 publications

Decomposition of Minkowski–Rådström–Hörmander Space to the Direct Sum of Symmetric and Asymmetric Subspaces

Decomposition of Minkowski–Rådström–Hörmander Space to the Direct Sum of Symmetric and Asymmetric Subspaces

Order cancellation law in the family of bounded convex sets

An extension of the Rådström cancellation theorem to cornets

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