  The rational, real and complex numbers with their standard operations,
including division, are partial algebras specified by the axiomatic concept of
a field. Since the class of fields cannot be defined by equations, the theory
of equational specifications of data types cannot use field theory in
applications to number systems based upon rational, real and complex numbers.
We study a new axiomatic concept for number systems with division that uses
only equations: a meadow is a commutative ring with a total inverse operator
satisfying two equations which imply that the inverse of zero is zero. All
fields and products of fields can be viewed as meadows. After reviewing
alternate axioms for inverse, we start the development of a theory of meadows.
We give a general representation theorem for meadows and find, as a corollary,
that the conditional equational theory of meadows coincides with the
conditional equational theory of zero totalized fields. We also prove
representation results for meadows of finite characteristic.
