Strolling through common meadows

Abstract

Meadows are a sort of commutative rings with a multiplicative identity element and a total multiplicative inverse operation. In this paper we study algebraic properties of common meadows, which are meadows that introduce, as the inverse of zero, an error term $\abf$ which is absorbent for addition. We show that common meadows are unions of rings which are ordered by a partial order that defines a lattice. These results allow us to generalize some classical algebraic constructions to the setting of common meadows. We also briefly consider common meadows from a categorical perspective.