Abstract
Let \({\mathcal {K}}\) be a congruence distributive variety and call an algebra hereditarily directly irreducible (HDI) if every of its subalgebras is directly irreducible. It is shown that every homomorphism from a finite direct product of arbitrary algebras from \({\mathcal {K}}\) to an HDI algebra from \({\mathcal {K}}\) is essentially unary. Hence, every homomorphism from a finite direct product of algebras \({\mathbf {A}}_i\) (\(i\in I\)) from \({\mathcal {K}}\) to an arbitrary direct product of HDI algebras \({\mathbf {C}}_j\) (\(j\in J\)) from \({\mathcal {K}}\) can be expressed as a product of homomorphisms from \({\mathbf {A}}_{\sigma (j)}\) to \({\mathbf {C}}_j\) for a certain mapping \(\sigma \) from J to I. A homomorphism from an infinite direct product of elements of \({\mathcal {K}}\) to an HDI algebra will in general not be essentially unary, but will always factor through a suitable ultraproduct.
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Open access funding provided by TU Wien (TUW). We are grateful to the referee of a previous version of this paper for alerting us to [4], and to Gábor Czédli for suggesting the definition of HDI algebras.
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I. Chajda and H. Länger gratefully acknowledge support of this research by ÖAD, project CZ 04/2017, as well as by IGA, project PřF 2018 012. M. Goldstern gratefully acknowledges support by the Austrian Science Fund (FWF), project I 3081-N35. H. Länger gratefully acknowledges support of this research by the Austrian Science Fund (FWF), project I 1923-N25.
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Chajda, I., Goldstern, M. & Länger, H. A note on homomorphisms between products of algebras. Algebra Univers. 79, 25 (2018). https://doi.org/10.1007/s00012-018-0517-9
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DOI: https://doi.org/10.1007/s00012-018-0517-9