Imaginary groups: lazy monoids and reversible computation

We use constructions in monoid and group theory to 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 a lazy homomorphism for a function f between partially ordered monoids such that f(m circle m')

Every monoid can be endowed with the discrete partial ordering (m

Informally, but perhaps informatively, we can describe this work as follows: we give an abstract analysis of how we can sensibly add 'undo' (in the sense of 'control-Z').