Nuclear images and simple axioms for ordered quasivarieties

A nucleus on an ordered algebra is a closure operator compatible with its basic operations. We prove that, for every quasivariety K of isotone algebras in an arbitrary finitary signature, its nuclear images are exactly the models of the quasi-inequalities valid in K whose premise inequalities have variables on the right. Consequently, K is closed under nuclear images if and only if it admits an axiomatization of this form. For nuclei compatible with every basic operation, this answers the general-signature question raised by Adam Přenosil in 2022, extending his results for ordered monoids and semilattice-ordered monoids. Every algebra satisfying the simple theory is an actual nuclear image of a member of K. The representation also takes the form of a surjective homomorphism with an order-embedding right adjoint. Constants are allowed and need not be fixed by the nucleus. A version for operations without an isotonicity assumption holds under an explicit compatibility identity. Code and verification data: https://github.com/eulogica/nuclear-ordered-quasivarieties