1,174 research outputs found
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
Social costs, limits to growth, right to growth: an approach to the global environment oriented to philosophy of law
The main Question of this paper is: how can we tackle the global warming in accordance with the economical growth especially in emerging countries?
K. W. Kapp, “The Social Costs of Private Enterprise” (1950), defines the social costs as direct or indirect damages which are not compensated by the producer, but added to the third parties. An example might be the disaster of the BP plant in April 2010, in which the polluter can hardly cover all the damages so as to make the seawater clean, to regenerate the harmed natural lives and to recover the jobs and the everyday life of the residents on site.
The Club of Rome, “The Limits to Growth” (1972), makes us aware of the five conditions which set the limits to growth: population, industrialization, pollution, consumption of food and natural resources, which tendentiously increase in a exponential progression. The GDP growth 10% a year means that it will be 2.59 times as large in ten years, whereas technology could resolve problematic concerning five elements at highest in arithmetical progression.
Remarkable would be that the modern industrial civilization has brought social damages in form of global warming. Developed nations have not payed for it yet. All the people in the world should have right to economical growth at any rate, which would however be limited by those five conditions. Conclusion: the developed nations should give up the consumption lifestyle for the sake of equal right of every citizen in the world to reasonable standard of living
Partial arithmetical data types of rational numbers and their equational specification
Upon adding division to the operations of a field we obtain a meadow. It is conventional toview division in a field as a partial function, which complicates considerably its algebra andlogic. But partiality is one out of a plurality of possible design decisions regarding division.Upon adding a partial division function ÷ to a field Q of rational numbers we obtain apartial meadow Q (÷) of rational numbers that qualifies as a data type. Partial data typesbring problems for specifying and programming that have led to complicated algebraicand logical theories – unlike total data types. We discuss four different ways of providingan algebraic specification of this important arithmetical partial data type Q (÷) via thealgebraic specification of a closely related total data type. We argue that the specificationmethod that uses a common meadow of rational numbers as the total algebra is themost attractive and useful among these four options. We then analyse the problem ofequality between expressions in partial data types by examining seven notions of equalitythat arise from our methods alone. Finally, based on the laws of common meadows, wepresent an equational calculus for working with fracterms that is of general interest outsideprogramming theory
A Complete Finite Equational Axiomatisation of the Fracterm Calculus for Common Meadows
We analyse abstract data types that model numerical structures with a concept
of error. Specifically, we focus on arithmetic data types that contain an error
flag whose main purpose is to always return a value for division. To
rings and fields we add a division operator and study a class of algebras
called \textit{common meadows} wherein . The set of equations true
in all common meadows is named the \textit{fracterm calculus of common
meadows}. We give a finite equational axiomatisation of the fracterm calculus
of common meadows and prove that it is complete and that the fracterm calculus
is decidable
The Transrational Numbers as an Abstract Data Type
In an arithmetical structure one can make division a total function by defining 1/0 to be an element of the structure, or by adding a new element, such as an error element also denoted with a new constant symbol, an unsigned infinity or one or both signed infinities, one positive and one negative. We define an enlargement of a field to a transfield, in which division is totalised by setting 1/0 equal to the positive infinite value and -1/0 equal to the negative infinite value , and which also contains an error element to help control their effects. We construct the transrational numbers as a transfield of the field of rational numbers and consider it as an abstract data type. We give it an equational specification under initial algebra semantics
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