Topological closure of translation invariant preorders. (English) Zbl 1325.06002

Ok, Efe A.; Riella, Gil

Math. Oper. Res. 39, No. 3, 737-745 (2014).

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

Keywords:

topological closure; transitive closure; translation invariance; closures of preorders; vector preorders; topological groups; topological linear spaces

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