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Topological closure of translation invariant preorders. (English) Zbl 1325.06002
Summary: Our primary query is to find conditions under which the closure of a preorder on a topological space remains transitive. We study this problem for translation invariant preorders on topological groups. The results are fairly positive; we find that the closure of preorders and normal orders remain as such in this context. The same is true for factor orders as well under quite general conditions. In turn, in the context of topological linear spaces, these results allow us to obtain a simple condition under which the order-duals with respect to a vector order and its closure coincide. Various order-theoretic applications of these results are also provided in the paper.
##### MSC:
 06A06 Partial orders, general 06F15 Ordered groups 54H11 Topological groups (topological aspects) 46A40 Ordered topological linear spaces, vector lattices
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##### References:
 [1] Aliprantis CD, Tourky R (2007) Cones and Duality (AMS, Providence, RI). [2] Bosi G, Herden G (2006) On a possible continuous analogue of the Szpilrajn theorem and its strengthening by Dushnik and Miller. Order 23(4):271-296. CrossRef · Zbl 1117.06002 [3] Bridges DS, Mehta G (1995) Representations of Preference Orderings (Springer-Verlag, Berlin). CrossRef [4] Candeal-Haro JC, Indurain-Eraso E (1992) Utility functions on partially ordered topological groups. Proc. Amer. Math. Soc. 115(3):765-767. CrossRef · Zbl 0754.90003 [5] Carruth JH (1968) A note on partially ordered compacta. Pacific J. Math. 24(2):229-231. CrossRef · Zbl 0157.53604 [6] Dubra J, Maccheroni F, Ok EA (2004) Expected utility theory without the completeness axiom. J. Econom. Theory 115(1):118-133. CrossRef · Zbl 1062.91025 [7] Evren O, Ok EA (2011) On the multi-utility representation of preference relations. J. Math. Econom. 47(4-5):554-563. CrossRef [8] Graham RL, Knuth DE, Motzkin TS (1972) Complements and transitive closures. Discrete Math. 2(1):17-29. CrossRef · Zbl 0309.04002 [9] Herden G, Pallack A (2002) On the continuous analogue of the Szpilrajn theorem I. Math. Soc. Sci. 43(2):115-134. CrossRef · Zbl 1023.91012 [10] Kuratowski K (1922) Sur l’operation a de l’analysis situs. Fund. Math. 3(1):182-199. · JFM 48.0210.04 [11] Lee GM, Shin JY, Sim HS (2001) A counterexample on the closedness of the convex hull of a closed cone in $$\mathbb{R}$$^n. Acta Math. Vietnamica 26(3):297-299. · Zbl 1039.52002 [12] Levin VL (1983) Measurable utility theorems for closed and lexicographic preference relations. Soviet Math. Dokl. 27(3):639-643. · Zbl 0543.90008 [13] Li J, Ok EA (2012) On the topological and transitive closures of binary relations. Mimeo, New York University. [14] Mabrouk MBR (2011) Translation invariance when utility streams are infinite and unbounded. Internat. J. Econom. Theory 7(4):317-329. CrossRef [15] Nachbin L (1965) Topology and Order (Van Nostrand, Princeton, NJ). [16] Ok EA, Riella G (2013) Fully preorderable groups. Mimeo, New York University. [17] Ok EA, Ortoleva P, Riella G (2012) Incomplete preferences under uncertainty: Indecisiveness in beliefs versus tastes. Econometrica 80(4):1791-1808. CrossRef · Zbl 1274.91149 [18] Peressini AL (1967) Ordered Topological Vector Spaces (Harper & Row, New York). · Zbl 0169.14801 [19] Royden HL (1988) Real Analysis (Prentice-Hall, Upper Saddle River, NJ). [20] Sagi JS (2006) Anchored preference relations. J. Econom. Theory 130(1):283-295. CrossRef · Zbl 1141.91362 [21] Wallace AD (1945) A fixed-point theorem. Bull. Amer. Math. Soc. 51(6):413-416. CrossRef · Zbl 0060.40104 [22] Ward LE (1954) Partially ordered topological spaces. Proc. Amer. Math. Soc. 5(1):144-161. CrossRef · Zbl 0055.16101
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