Discrete dualities for groupoids

We present discrete dualities for groupoid-based algebras and their associated frames starting from a plain groupoid and then expanding step by step its signature and/or axioms assuming existence of a compatible partial ordering, commutativity, associativity, idempotency, existence of the identity element, and existence of left and right residuals of the groupoid operator. In the final section we present a brief overview of lattice-ordered groupoids and provide hints how to obtain discrete dualities for these from the dualities established in the preceding sections and relevant results in the literature.