Abstract
In this paper we introduce the concept of free ordered semigroups as
follows: If (X,≤X) is an ordered set, an ordered semigroup (F, .,≤F ) is said to be a free
ordered semigroup over (X,≤X), if there is an isotone mapping ε : (X,≤X) → (F,≤F )
satisfying the following ”universal” condition: for any ordered semigroup (S, ∗,≤S)
and any isotone mapping f : (X,≤X) → (S,≤S), there exists a unique homomorphism
ϕ : (F, .,≤F ) → (S, ∗,≤S) such that ϕ◦ε = f. Basing on the fact that the mapping ε is
reverse isotone, we find relationships between the mappings ε1 and ε2 which correspond
to free ordered semigroups ((F1, .,≤1), ε1) and ((F2, .,≤2), ε2).
