Document Type : Research Paper

Authors

Department of Mathematics, Sahand University of Technology, Tabriz, Iran.

Abstract

In this paper, a Krein-Milman  type theorem in $T_0$ semitopological cone is proved,  in general. In fact, it is shown that in any locally convex $T_0$ semitopological cone, every convex compact saturated subset is the compact saturated convex hull of its extreme points, which improves the results of Larrecq.

Keywords

Main Subjects

References

[1] S. Abramsky and A. Jung, Domain theory. In S. Abramsky, D. M. Gabbay, and T. S. E. Maibaum, editors, handbook of logic in computer science, vol. 3, Clarendon Press, 1994.

[2] C.D. Aliprantis and K.C. Border, Infinite dimensional analysis: A hitchhiker's guide, Springer; 3rd editions, 2007.

[3] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, and D.S. Scott, Continuous lattices and domains, vol. 93, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2003.

[4] J. Goubault-Larrecq, A cone theoretic Krein-Milman theorem, Rapport derecherche LSV-08-18, ENS Cachan, France, 2008.

[5] J. Goubault-Larrecq, Non-Hausdorff topology and domain theory: Selected topics in point-set topology, Cambridge University Press, 2013.

[6] K. Keimel, Topological cones: Foundations for a domain theoretical semantics combining probability and nondeterminism, Electronic Notes in Theoretical Computer Science 155 (2006), 423-443.

[7] K. Keimel, Topological cones: Functional analysis in a T0-setting, Semigroup Forum 77 (2008), 109-142.

[8] K. Keimel and W. Roth, Ordered cones and approximation, Vol. 1517, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1992.

[9] J.D. Lawson, Domains, integration and positive analysis, Math. Structures Comput. Sci. 14 (2004), no. 6, 815-832.

[10] G. Plotkin, A domain-theoretic Banach-Alaoglu theorem, Math. Structures Comput. Sci. 16 (2006), no. 2, 299-311.

[11] W. Roth, Hahn-Banach type theorems for locally convex cones, J. Aust. Math. Soc. 68 (2000), no. 1, 104-125.

[12] W. Roth, Separation properties for locally convex cones, J. Convex Anal. 9 (2002), No. 1, 301-307.

[13] W. Roth, Operator-valued measures and integrals for cone-valued functions, Vol. 1964, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2009.

[14] D.S. Scott, Continuous lattices. in toposes, algebraic geometry and logic (conf., dalhousie univ., halifax, n. s., 1971), Lecture Notes in Math. 274 (1972), 97-136.

[15] R. Tix, Continuous $d$-cones: convexity and powerdomain constructions, PhD thesis, Technische Universität Darmstadt. Shaker, Aachen, 1999.

[16] R. Tix, Some results on Hahn-Banach type theorems for continuous $d$-cones, Theor. Comput. Sci. 264 (2001), 205-218.

[17] R. Tix, K. Keimel, and G. Plotkin, Semantic domains combining probability and nondeterminism, Electron. Notes Theor. Comput. Sci. 129 (2005), 1-104.