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**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.

### MSC:

06B05 | Structure theory of lattices |

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\textit{I. Chajda} et al., Algebra Univers. 79, No. 2, Paper No. 25, 7 p. (2018; Zbl 06904410)

### References:

[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 |

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