Partial Actions of Ordered Groupoids on Rings

In this paper, we introduce the notion of a partial action of an ordered groupoid on a ring and we construct the corresponding partial skew groupoid ring. We present sufficient conditions under which the partial skew groupoid ring is either associative or unital. Also, we show that there is a one-to-one correspondence between partial actions of an ordered groupoid G on a ring R, in which the domain of each partial bijection is an ideal, and meet-preserving global actions of the Birget–Rhodes expansion G BR of G on R. Using this correspondence, we prove that the partial skew groupoid ring is a homomorphic image of the skew groupoid ring constructed through the Birget–Rhodes expansion.