Division by zero in common meadows

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that admit a certain cancellation law.Comment: 17 pages, 4 tables; differences with v3: axiom (14) of Mda (Table 2) has been replaced by the stronger axiom (12), this appears to be necessary for the proof of Theorem 3.2.

Fracpairs and fractions over a reduced commutative ring

In the well-known construction of the field of fractions of an integral domain, division by zero is excluded. We introduce "fracpairs" as pairs subject to laws consistent with the use of the pair as a fraction, but do not exclude denominators to be zero. We investigate fracpairs over a reduced commutative ring (a commutative ring that has no nonzero nilpotent elements) and provide these with natural definitions for addition, multiplication, and additive and multiplicative inverse. We find that modulo a simple congruence these fracpairs constitute a "common meadow", which is a commutative monoid both for addition and multiplication, extended with a weak additive inverse, a multiplicative inverse except for zero, and an additional element "a" that is the image of the multiplicative inverse on zero and that propagates through all operations. Considering "a" as an error-value supports the intuition. The equivalence classes of fracpairs thus obtained are called common cancellation fractions (cc-fractions), and cc-fractions over the integers constitute a homomorphic pre-image of the common meadow Qa, the field Q of rational numbers expanded with an a-totalized inverse. Moreover, the initial common meadow is isomorphic to the initial algebra of cc-fractions over the integer numbers. Next, we define canonical term algebras for cc-fractions over the integers and some meadows that model the rational numbers expanded with a totalized inverse, and provide some negative results concerning their associated term rewriting properties. Then we consider reduced commutative rings in which the sum of two squares plus one cannot be a zero divisor: by extending the equivalence relation on fracpairs we obtain an initial algebra that is isomorphic to Qa. Finally, we express negative conjectures concerning alternative specifications for these (concrete) datatypes.Comment: 25 pages, 8 table

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

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

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

Circular 114

Trials were begun in 1989 at the Georgeson Botanical Garden (64°51’N, 147° 52’W, elevation 475 feet; 136 meters) to evaluate the hardiness and ornamental potential of trees, shrubs, and herbaceous perennial ornamentals. Woody ornamentals are tested for 10 years, and herbaceous perennials for five years. This report is the first summary of perennials that have survived the trial period with a winter hardiness rating between zero and 2.5. Each plant in the trial is evaluated annually for winter injury and rated on a scale of zero through four. A zero rating denotes no visible injury, and four is death. A score of 2.5 and lower indicates the plant grew well in the Garden. It may have shown symptoms of winter injury but recovered in subsequent seasons. The species and cultivars listed in Table 1 are recommended for further trial throughout Interior Alaska. Plants are grown on a south-facing slope in Fairbanks silt loam soil. The plots have been cultivated since about 1910. All plants receive full sun except those located in the shade house. Plants receive supplemental irrigation, mostly hand weeding, and an annual application of 500 lb per acre (560.5 kg/ha) 10-20-20S fertilizer. Lilies receive 1500 lb (1,681.5 kg/ha) per acre of the same fertilizer. No plant receives winter protection such as mulches, wind barriers or snow fences. Weather data are compiled annually from U.S. Weather Service station (elevation 475 feet; 136 meters) located approximately 350 feet (105 meters) west of the Garden. A summary of pertinent weather statistics is shown in Table 2

Division by zero in non-involutive meadows

Meadows have been proposed as alternatives for fields with a purely equational axiomatization. At the basis of meadows lies the decision to make the multiplicative inverse operation total by imposing that the multiplicative inverse of zero is zero. Thus, the multiplicative inverse operation of a meadow is an involution. In this paper, we study `non-involutive meadows', i.e.\ variants of meadows in which the multiplicative inverse of zero is not zero, and pay special attention to non-involutive meadows in which the multiplicative inverse of zero is one.Comment: 14 page

Square root meadows

Let Q_0 denote the rational numbers expanded to a meadow by totalizing inversion such that 0^{-1}=0. Q_0 can be expanded by a total sign function s that extracts the sign of a rational number. In this paper we discuss an extension Q_0(s ,\sqrt) of the signed rationals in which every number has a unique square root.Comment: 9 page