A note on homomorphisms between products of algebras

Let 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 K to an HDI algebra from K is essentially unary. Hen...

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Bibliographic Details
Published in:Algebra universalis 2018-06, Vol.79 (2), p.1-7, Article 25
Main Authors: Chajda, Ivan, Goldstern, Martin, Länger, Helmut
Format: Article
Language:English
Subjects:
Algebra
Homomorphisms
Mathematics
Mathematics and Statistics
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Summary:Let 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 K to an HDI algebra from K is essentially unary. Hence, every homomorphism from a finite direct product of algebras A i ( i ∈ I ) from K to an arbitrary direct product of HDI algebras C j ( j ∈ J ) from K can be expressed as a product of homomorphisms from A σ ( j ) to  C j for a certain mapping σ from J to  I . A homomorphism from an infinite direct product of elements of K to an HDI algebra will in general not be essentially unary, but will always factor through a suitable ultraproduct.
ISSN:0002-5240
1420-8911
DOI:10.1007/s00012-018-0517-9