Chajda, I., Goldstern, M., & Länger, H. (2018). A note on homomorphisms between products of algebras. Algebra Universalis. https://doi.org/10.1007/s00012-018-0517-9
E104 - Institut für Diskrete Mathematik und Geometrie
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Journal:
Algebra Universalis
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ISSN:
0002-5240
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Date (published):
2018
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Publisher:
Birkhäuser
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Peer reviewed:
Yes
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Keywords:
Direct product of chains; Homomorphism; Essentially unary mapping; Ultrafilter
en
Abstract:
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 Ai (i∈I) from K to an arbitrary direct product of HDI algebras Cj (j∈J) from K can be expressed as a product of homomorphisms from Aσ(j) to Cj 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.