By constructions in monoid and group theory we exhibit an adjunction between the category of partially ordered monoids and lazy monoid homomorphisms, and the category of partially ordered groups and group homomorphisms, such that the unit of the adjunction is injective. We also prove a similar result for sets acted on by monoids and groups. We introduce the new notion of lazy homomorphism for a function f between partially-ordered monoids such that f (m ◦m′) ≤ f (m) ◦ f (m′). Every monoid can be endowed with the discrete partial ordering (m ≤ m ′ if and only if m = m′) so our constructions provide a way of embedding monoids into groups. A simple counterexample (the two-element monoid with a non-trivial idempotent) and some calculations sh...