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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 |

### Keywords:

topological closure; transitive closure; translation invariance; closures of preorders; vector preorders; topological groups; topological linear spaces
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\textit{E. A. Ok} and \textit{G. Riella}, Math. Oper. Res. 39, No. 3, 737--745 (2014; Zbl 1325.06002)

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