# Poset properties of complex traces

## Abstract

This paper investigates PoSct properties of the monoid ℂ of infinite dependence graphs and of the monoid ℂ of complex traces. We show that a subset of G admits a least upper bound if and only if this set is coherent and countable. Hence, G is bounded complete. The compact and the prime graphs in G arc characterized and we prove that each graph is the least upper bound of its compact (resp. its prime) lower bounds. Therefore, up to the restriction to countable sets, G is a coherently complete Scott-Domain and is Prime Algebraic. We define very naturally two orders on ℂ: the product order and the prefix order. We show that ℂ with each order is a coherently complete CPO and we characterize the least upper bound (the greatest lower bound resp.) of a subset of ℂ when it exists. But contrary to the case of G, we prove that ℂ is not a Scott-Domain in general.

## Keywords

Partial Order Dependence Graph Prime Graph Product Order Real Trace## Preview

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