  We introduce ($\ell$-)bimonoids as ordered algebras consisting of two
compatible monoidal structures on a partially ordered (lattice-ordered) set.
Bimonoids form an appropriate framework for the study of a general notion of
complementation, which subsumes both Boolean complements in bounded
distributive lattices and multiplicative inverses in monoids. The central
question of the paper is whether and how bimonoids can be embedded into
complemented bimonoids, generalizing the embedding of cancellative commutative
monoids into their groups of fractions and of bounded distributive lattices
into their free Boolean extensions. We prove that each commutative
($\ell$-)bimonoid indeed embeds into a complete complemented commutative
$\ell$-bimonoid in a doubly dense way reminiscent of the Dedekind--MacNeille
completion. Moreover, this complemented completion, which is term equivalent to
a commutative involutive residuated lattice, sometimes contains a tighter
complemented envelope analogous to the group of fractions. In the case of
cancellative commutative monoids this algebra of fractions is precisely the
familiar group of fractions, while in the case of Brouwerian (Heyting) algebras
it is a (bounded) idempotent involutive commutative residuated lattice. This
construction of the algebra of fractions in fact yields a categorical
equivalence between varieties of integral and of involutive residuated
structures which subsumes as special cases the known equivalences between
Abelian $\ell$-groups and their negative cones, and between Sugihara monoids
and their negative cones.
