Multisorted dualisability: change of base
The basic theory of natural dualities applies to quasivarieties generated by a single finite algebra. \textit{B. A. Davey} and \textit{H. A. Priestley} [Houston J. Math. 13, 151--198 (1987; Zbl 0692.08012)] extended the theory to quasivarieties of the form \(\mathrm{ISP}(\mathcal{M})\), where \(\mathcal{M}\) is a finite indexed family of finite algebras. In such a case, one works with a multisorted alter ego with a sort for each algebra in \(\mathcal{M}\) and under additional conditions obtains a multisorted duality based on \(\mathcal{M}\). It is known (see [\textit{B. A. Davey} and \textit{R. Willard}, Algebra Univers. 45, No. 1, 103--106 (2001; Zbl 1039.08006)]) that if \(A\) and \(B\) are finite algebras generating the same quasivariety, i.e. \(\mathrm{ISP}(A)=\mathrm{ISP}(B)\), then if \(A\) is dualisable, so is \(B\). In this paper, the authors obtain a similar result for multisorted dualities, namely if \(\mathcal{A}=\mathrm{ISP}(\mathcal{M})=\mathrm{ISP}(\mathcal{N})\) for finite indexed families of finite algebras \(\mathcal{M}\) and \(\mathcal{N}\), then, if there is a multisorted duality for \(\mathcal{A}\) based on \(\mathcal{M}\), then there is a multisorted duality for \(\mathcal{A}\) based on \(\mathcal{N}\). The theorem provides then a change of base result for multisorted natural dualities. As a corollary one obtains that if \(\mathcal{A}=\mathrm{ISP}(M)=\mathrm{ISP}(\mathcal{N})\), where \(M\) is a finite algebra and \(\mathcal{N}\) a family defined as above, then, if \(\mathcal{A}\) has no natural duality based on \(M\), then it has no multisorted natural duality based on \(\mathcal{N}\). The proof of the theorem is constructive and based on two lemmas, presented in the paper, which show how to remove/add an algebra from/to the finite family generating the quasivariety. Also, an example for the described procedure is provided.
scientific article; zbMATH DE number 5988463