  The authors developed in a recent paper natural dualities for finitely
generated quasivarieties of Sugihara algebras. They thereby identified the
admissibility algebras for these quasivarieties which, via the Test Spaces
Method devised by Cabrer et al., give access to a viable method for studying
admissible rules within relevance logic, specifically for extensions of the
deductive system $R$-mingle.
  This paper builds on the work already done on the theory of natural dualities
for Sugihara algebras. Its purpose is to provide an integrated suite of
multisorted duality theorems of a uniform type, encompassing finitely generated
quasivarieties and varieties of both Sugihara algebras and Sugihara monoids,
and embracing both the odd and the even cases. The overarching theoretical
framework of multisorted duality theory developed here leads on to amenable
representations of free algebras. More widely, it provides a springboard to
further applications.
