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A emph{meadow} is a commutative ring with an inverse operator satisfying $0^{-1}=0$. We determine the initial algebra of the meadows of characteristic 0 and show that its word problem is decidable.
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A meadow is a zero totalised field (0^{-1}=0), and a cancellation meadow is a meadow without proper zero divisors. In this paper we consider differential meadows, i.e., meadows equipped with differentiation operators. We give an equational axiomatization of these operators and thus obtain a finite basis for differential cancellation meadows. Using the Zariski topology we prove the existence of a differential cancellation meadow.
Meadows - commutative rings equipped with a total inversion operation - can be axiomatized by purely equational means. We study subvarieties of the variety of meadows obtained by extending the equational theory and expanding the signature.
The aim of this note is to describe the structure of finite meadows. We will show that the class of finite meadows is the closure of the class of finite fields under finite products. As a corollary, we obtain a unique representation of minimal meadows in terms of prime fields.
We consider the signatures $Sigma_m=(0,1,-,+, cdot, ^{-1})$ of meadows and $(Sigma_m, {mathbf s})$ 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.
Common meadows are fields expanded with a total inverse function. Division by zero produces an additional value denoted with a that propagates through all operations of the meadow signature (this additional value can be interpreted as an error element). We provide a basis theorem for so-called common cancellation meadows of characteristic zero, that is, common meadows of characteristic zero that admit a certain cancellation law.
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