A note on homomorphisms between products of algebras. (English) Zbl 06904410

Summary: 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.


06B05 Structure theory of lattices
Full Text: DOI Link


[1] Birkhoff, G.: Lattice Theory. Corr. Repr. of the 1967 3rd Edn. American Mathematical Society Colloquium Publications, vol. 25. American Mathematical Society (AMS), Providence (1979) · Zbl 0505.06001
[2] Couceiro, M; Foldes, S; Meletiou, GC, On homomorphisms between products of Median algebras, Algebra Universalis, 78, 545-553, (2017) · Zbl 1420.06006
[3] Couceiro, M., Marichal, J.-L., Teheux, B.: Conservative median algebras and semilattices. Order 33, 121-132 (2016) · Zbl 1345.06004
[4] Fraser, GA; Horn, A, Congruence relations in direct products, Proc. Am. Math. Soc., 26, 390-394, (1970) · Zbl 0241.08004
[5] Grätzer, G.: Lattice Theory: Foundation. Birkhäuser, Basel (2011) · Zbl 1233.06001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.