On Summands of Closed Bounded Convex Sets

Grzybowski, Jerzy; Przybycień, Hubert; Urbański, Ryszard

doi:10.4171/zaa/1111

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The Abstract Difference1

The Operators D N and ω N1

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2007

2008

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Cited by 2 publications

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“…Our idea is based on the definition of the Minkowski-Pontryagin difference for convex sets (see [3,4]). Note, that we consider only F-semigroups in this section.…”

Section: The Abstract Differencementioning

confidence: 99%

On Partially Ordered Semigroups and an Abstract Set-Difference

2008

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In this paper we prove the independence of a system of five axioms (S1)-(S5), which was proposed in the book of Pallaschke and Urbański (Pairs of Compact Convex Sets, vol. 548, Kluwer Academic Publishers, Dordrecht, 2002) for partially ordered commutative semigroups. These five axioms (S1)-(S5) are stated in the introduction below. A partially ordered commutative semigroup satisfying these axioms is called a F-semigroup. By the use of a further axiom (S6) we define an abstract difference for the elements of a F-semigroup and prove some basic properties. The most interesting example of a F-semigroup are the nonempty compact convex sets of a topological vector space endowed with the Minkowski sum as operation and the inclusion as partial order. In Section 4 we apply the abstract difference to the problem of minimality of convex fractions.

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“…Our idea is based on the definition of the Minkowski-Pontryagin difference for convex sets (see [3,4]). Note, that we consider only F-semigroups in this section.…”

Section: The Abstract Differencementioning

confidence: 99%

On Partially Ordered Semigroups and an Abstract Set-Difference

2008

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“…Replacing B by A− B in the above inclusion we get A− B ⊂ A− (A− (A− B)) which is equivalent to the following expression D 1 (A, B) ⊂ D 3 (A, B). On the other hand, putting M = B, N = A− (A− B) and applying the equalities (2) and (1)…”

Section: The Operators D N and ω Nmentioning

confidence: 99%