A note on homomorphisms between products of algebras
Ivan Chajda, Martin Goldstern, Helmut Länger · Algebra Universalis · 2018
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.