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

Chajda, Ivan; Goldstern, Martin; Länger, Helmut

Algebra Univers. 79, No. 2, Paper No. 25, 7 p. (2018).

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

Keywords:

direct product of chains; homomorphism; essentially unary mapping; ultrafilter

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