Free \((\ell r, rr)\)-dibands

Abstract

We prove that varieties of \((\ell r, rr)\)-dibands and \((\ell n, rn)\)-dibands coincide and describe the structure of free \((\ell r, rr)\)-dibands. We also show that operations of an idempotent dimonoid with left (right) regular bands coincide, construct a new class of dimonoids and for such dimonoids give an example of a semiretraction.