The globals of pseudovarieties of ordered semigroups containing B2 and an application to a problem proposed by Pin

Given a basis of pseudoidentities for a pseudovariety of ordered semigroups containing the 5-element aperiodic Brandt semigroup B2, under the natural order, it is shown that the same basis, over the most general graph over which it can be read, defines the global. This is used to show that the global of the level 3/2 of the refinement of Straubing-Thérien’s concatenation hierarchy introduced by Pin and Weil has infinite vertex rank.