Equations for formally real meadows

We consider the signatures $Σ_m=(0,1,-,+, \cdot, \ ^{-1})$ of meadows and $(Σ_m, {\mathbf s})$ of signed meadows. We give two complete axiomatizations of the equational theories of the real numbers with respect to these signatures. In the first case, we extend the axiomatization of zero-totalized fields by a single axiom scheme expressing formal realness; the second axiomatization presupposes an ordering. We apply these completeness results in order to obtain complete axiomatizations of the complex numbers.

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