Minkowski difference and Sallee elements in an ordered semigroup

• Autorzy: Borowska D. , Grzybowski J.
• Języki publikacji: EN

Abstrakty

EN

In the manner of Pallaschke and Urbański ([5], chapter 3) we generalize the notions of the Minkowski difference and Sallee sets to a semigroup. Sallee set (see [7], definition of the family S on p. 2) is a compact convex subset A of a topological vector space X such that for all subsets B the Minkowski difference A-B of the sets A and B is a summand of A. The family of Sallee sets characterizes the Minkowski subtraction, which is important to the arithmetic of compact convex sets (see [5]). Sallee polytopes are related to monotypie poly topes (see [4]). We generalize properties of Minkowski difference and Sallee sets to semigroup and investigate the families of Sallee elements in several specific semigroups.

Słowa kluczowe

EN

Minkowski difference; Sallee elements; semigroup

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
• Rocznik: 2007
• Tom: Vol. 47, [Z] 1
• Strony: 77--83
• Opis fizyczny: bibliogr. 9 poz.

Twórcy

autor

Borowska D.

autor

Grzybowski J.

• Adam Mickiewicz University Umultowska 87, 61-614 Poznań,

Bibliografia

• [1] D. Borowska, H. Przybycien and R. Urbanski: On Summands Properties and Minkowski subtraction. Journal for Analysis and its Applications Volume 26(2) (2007), 247-260.
• [2] J. Grzybowski, H. Przybycien and R. Urbanski: On Summands of Closed Bounded Convex Sets. Journal for Analysis and its Applications Volume 21(4) (2002), 845-850.
• [3] H. Maehara: Convex bodies forming pairs of constant width. Journal of Geometry vol. 22 (1984), 101-107.
• [4] P. McMullen, R. Schneider, G. C. Shepard: Monotypic polytopes and their intersection properties.Geometriae Dedicata 3 (1974) 99-129.
• [5] D. Pallaschke, R. Urbanski: Pairs of Compact Convex Sets, Fractional Arithmetic with Convex Sets. Serie Mathematics and Its Applications, vol.548, Kluwer Academic Publisher, Dortrecht-Boston-London, 2002.
• [6] H. Przybycien, Doctoral Thesis: Certain Properties of closed bounded convex sets. Adam Mickiewicz University Poznan 2003 (in Polish).
• [7] G. T. Sallee: Pairs of sets of constant relative width. Journal of Geometry vol. 29 (1987) 1-11.
• [8] R. Schneider: Convex Bodies: The Brunn-Minkowski Theory (Encyklopedia of Mathematics and its Applications: Vol 44). Cambridge Univ. Press 1993.
• [9] R. Urbanski: A generalization of the Minkowski R°adstr¨om-H¨ormander theorem, Bull. Acad. Polon. Sci. S´er. Sci. Math. Astr. Phys. 24 (1976) 709-715.