Residuated, commutative $\ell$-monoid; orbit; translation

We prove that each residuated commutative l-monoid is a monoid of all residuated endomorphisms of some universal algebra. This result demonstrates that similarly to groups and semigroups, a monoidal operation of a residuated commutative l-monoid is represented by a composition of endomorphisms that are moreover, residuated.