11 research outputs found
The structure of finite meadows
A meadow is a commutative ring with a total inverse operator satisfying
0^{-1}=0. We show that the class of finite meadows is the closure of the class
of Galois fields under finite products. As a corollary, we obtain a unique
representation of minimal finite meadows in terms of finite prime fields.Comment: 12 page
Some properties of finite meadows
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.Comment: 8 pages, 1 tabl
Equations for formally real meadows
We consider the signatures of meadows
and 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.Comment: 24 pages, 14 tables, revised, new Theorem 3.
Straight-line instruction sequence completeness for total calculation on cancellation meadows
A combination of program algebra with the theory of meadows is designed
leading to a theory of computation in algebraic structures which use in
addition to a zero test and copying instructions the instruction set . It is proven that total functions on cancellation
meadows can be computed by straight-line programs using at most 5 auxiliary
variables. A similar result is obtained for signed meadows.Comment: 24 page
Inversive Meadows and Divisive Meadows
Inversive meadows are commutative rings with a multiplicative identity
element and a total multiplicative inverse operation whose value at 0 is 0.
Divisive meadows are inversive meadows with the multiplicative inverse
operation replaced by a division operation. We give finite equational
specifications of the class of all inversive meadows and the class of all
divisive meadows. It depends on the angle from which they are viewed whether
inversive meadows or divisive meadows must be considered more basic. We show
that inversive and divisive meadows of rational numbers can be obtained as
initial algebras of finite equational specifications. In the spirit of
Peacock's arithmetical algebra, we study variants of inversive and divisive
meadows without an additive identity element and/or an additive inverse
operation. We propose simple constructions of variants of inversive and
divisive meadows with a partial multiplicative inverse or division operation
from inversive and divisive meadows. Divisive meadows are more basic if these
variants are considered as well. We give a simple account of how mathematicians
deal with 1 / 0, in which meadows and a customary convention among
mathematicians play prominent parts, and we make plausible that a convincing
account, starting from the popular computer science viewpoint that 1 / 0 is
undefined, by means of some logic of partial functions is not attainable.Comment: 18 pages; error corrected; 29 pages, combined with arXiv:0909.2088
[math.RA] and arXiv:0909.5271 [math.RA
The structure of finite meadows
A meadow is a commutative ring with a total inverse operator satisfying 0−1=0. We show that the class of finite meadows is the closure of the class of Galois fields under finite products. As a corollary, we obtain a unique representation of minimal finite meadows in terms of finite prime fields