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
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[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.
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[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.
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