16,879 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
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
Probability functions in the context of signed involutive meadows
The Kolmogorov axioms for probability functions are placed in the context of
signed meadows. A completeness theorem is stated and proven for the resulting
equational theory of probability calculus. Elementary definitions of
probability theory are restated in this framework.Comment: 20 pages, 6 tables, some minor errors are correcte
Meadow enriched ACP process algebras
We introduce the notion of an ACP process algebra. The models of the axiom
system ACP are the origin of this notion. ACP process algebras have to do with
processes in which no data are involved. We also introduce the notion of a
meadow enriched ACP process algebra, which is a simple generalization of the
notion of an ACP process algebra to processes in which data are involved. In
meadow enriched ACP process algebras, the mathematical structure for data is a
meadow.Comment: 8 pages; correction in Table
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.
Universality of Univariate Mixed Fractions in Divisive Meadows
Univariate fractions can be transformed to mixed fractions in the equational
theory of meadows of characteristic zero.Comment: 12 page
Probabilistic thread algebra
We add probabilistic features to basic thread algebra and its extensions with
thread-service interaction and strategic interleaving. Here, threads represent
the behaviours produced by instruction sequences under execution and services
represent the behaviours exhibited by the components of execution environments
of instruction sequences. In a paper concerned with probabilistic instruction
sequences, we proposed several kinds of probabilistic instructions and gave an
informal explanation for each of them. The probabilistic features added to the
extension of basic thread algebra with thread-service interaction make it
possible to give a formal explanation in terms of non-probabilistic
instructions and probabilistic services. The probabilistic features added to
the extensions of basic thread algebra with strategic interleaving make it
possible to cover strategies corresponding to probabilistic scheduling
algorithms.Comment: 25 pages (arXiv admin note: text overlap with arXiv:1408.2955,
arXiv:1402.4950); some simplifications made; substantially revise
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