A schizophrenic operation which aids the efficient transfer of strong dualities (Q2725477)

Let \(\mathbf D\) and \(\mathbf M\) be finite algebras of the same type generating the same quasivariety such that \(\mathbf D\) is a subalgebra of \(\mathbf M\). It is known [see \textit{M.~J.~Saramago}, ``Some remarks on dualisability and endodualisability'', Algebra Univers. 43, No.~2-3, 197-212 (2000)] that if we consider a full duality determined by \(\mathbf D\) as the duality determined by \(\mathbf M\) and we add a family \(\Omega\) of endomorphisms of \(\mathbf M\) with images contained in \(\mathbf D\) and separating vertices of \(\mathbf M\), then we obtain a full duality determined by \(\mathbf M\). The aim of this paper is to generalize this theorem for strong duality. It is proved that if an initial duality determined by \(\mathbf D\) is strong and we add \(\Omega\) and another special partial operation on \(\mathbf D\), then the obtained duality determined by \(\mathbf M\) is also strong. The obtained result is illustrated on distributive lattices, semilattices and Abelian groups.