Polynomial interpolation in expanded groups

Abstract

We call an algebra strictly 1-affine complete iff every unary congruence preserving partial function with finite domain is a restriction of a polynomial. We characterize finite strictly 1-affine complete groups with operations, and, in particular, all finite strictly 1-affine complete groups and commutative rings with unit.

O polinomima u algebrama Maljceva

“…The proof of this proposition is identical with the proof of Proposition 7.16 in [5] except that k-ary polynomial functions have to be used instead of unary polynomial functions. Proposition 1.…”

“…Now we assume k ≥ 2. Defining The following proposition is the consequence of Propositions 2.2 and 6.1 in [5], but here, we give a direct proof. …”

“…In [5], homogeneous elements of a bounded lattice played an important role. It is not hard to show that this definition is equivalent to the definition of homogeneous elements of finite lattices given in [5…”

“…Furthermore, in this case we have ( In Section 7 of [5], one finds information on those unary polynomial functions whose range is contained in a homogeneous ideal. We will use straightforward generalizations of these results to k-ary functions.…”

“…Furthermore, we know by Theorem §1. 5.15 that a prime quotient of congruences α and β, α ≺ β, denoted by α, β is of type 2 if and only if [β, β] ≤ α. As we can see from Definition §1.…”