The finite embeddability property for residuated groupoids

A very simple proof of the finite embeddability property for residuated distributive-lattice-ordered groupoids and some related classes of structures is presented. In particular, this gives an answer to the question, posed by Blok and van Alten, whether the class of residuated ordered groupoids has the property. The presented construction improves the computational-complexity upper bound of the universal theory of residuated distributive-lattice-ordered groupoids given by Buszkowski and Farulewski; for chains in the class, a tight bound is obtained.