Towards an enumeration of finite common meadows

Abstract

Common meadows are commutative and associative algebraic structures with two operations (addition and multiplication) with additive and multiplicative identities and for which inverses are total. The inverse of zero is an error term [Formula: see text] which is absorbent for addition. We study the problem of enumerating all finite common meadows of it order [Formula: see text] (that is, common meadows with [Formula: see text] elements). This problem turns out to be deeply connected with both the number of finite rings of order [Formula: see text] and with the number of a certain kind of partition of positive integers.