Some algebraic and topological properties of subspaces of (pre)orders

TL;DR

This paper investigates algebraic and topological characteristics of subsets within preorders on groups, extending existing theorems and analyzing properties of compositions and valuations in these structures.

Contribution

It generalizes a topological theorem from orders to preorders and explores properties of compositions and valuations in the context of preorders on groups.

Findings

Generalized a topological theorem to preorders

Analyzed properties of preorder compositions

Studied standard valuations in preorders

Abstract

We study algebraic and topological properties of subsets of preorders on a group. In particular we study properties of the composition of two preorders, generalize a topological theorem of \cite{S} in the case of standard orders and show the same theorem in the case of standard preorders. We also show a property of standard valuations.