Associativity of the <Emphasis Type="Italic">∇</Emphasis>-operation on bands in rings

Given a multiplicative band of idempotents S in a ring R, for all e,f∈S the ∇-product e ∇ f=e+f+fe−efe−fef is an idempotent that lies roughly above e and f in R just as ef and fe lie roughly below e and f. In this paper we study ∇-bands in rings, that is, bands in rings that are closed under ∇, giving various criteria for ∇ to be associative, thus making the band a skew lattice. We also consider when a given band S in R generates a ∇-band.