Inversive meadows and divisive meadows

An inversive meadow is a commutative ring with identity and a total multiplicative inverse operation whose value at 0 is 0. Previously, inversive meadows were shortly called meadows. In this paper, we introduce divisive meadows, which are inversive meadows with the multiplicative inverse operation replaced by a division operation. We introduce a translation from the terms over the signature of divisive meadows into the terms over the signature of inversive meadows and a translation the other way round to show that it depends on the angle from which they are viewed whether inversive meadows or divisive meadows must be considered more basic. Divisive meadows are more basic if variants with a partial multiplicative inverse or division operation are considered as well. We also take a survey of first-order logics that are appropriate to handle those partial variants of inversive and divisive meadows.