It is a classical result that the ring of unary polynomial functions on the
direct product of commutative rings with identity is isomorphic to the direct
product of the polynomial rings on the factors. When we generalize the concept
of polynomial functions to arbitrary algebraic structures, this simple correspondence
is no longer true. Clearly a polynomial function on an algebra A
preserves all congruences and induces polynomial functions on all quotients of
A. However, a congruence preserving function that induces polynomials on all
subdirectly irreducible quotients is not necessarily polynomial.
Still we can characterize the polynomial functions on certain direct and subdirect
products of algebras with Mal’cev term or with majority term by their
behaviour on the factors. This is joint work with Kalle Kaarli.