Relational Representation of Groupoid Quantales

In Palmigiano and Re (J Pure Appl Algebra 215(8):1945-1957, 2011), spatial SGF-quantales are axiomatically introduced and proved to be representable as sub unital involutive quantales of quantales arising from set groupoids. In the present paper, spatial SGF-quantales of this class are shown to be optimally representable as unital involutive quantales of relations. The results of the present paper have several aspects in common with Jnsson and Tarski's representation theory for relation algebras (Jnsson and Tarski, Am J Math 74(2):127-162, 1952).