Straight-line Instruction Sequence Completeness for Total Calculation on Cancellation Meadows

Meadows are 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. Divisive meadows are meadows with the multiplicative inverse operation replaced by a division operation. Viewing a fraction as a term over the signature of divisive meadows that is of the form p / q, we investigate which divisive meadows admit transformation of fractions into simple fractions, i.e. fractions without proper subterms that are fractions.Comment: 23 pages; one theorem and two corollaries of it added at end of Sect. 5; some minor errors correcte

Section: Inversive Meadows and Divisive Meadowsmentioning

confidence: 99%

“…Let f (x) and g(x) be polynomials such that M |= 1+1/x = f (x)/g(x). Substitution of x by 0 yields 2 , and then applying the axiom…”

Section: ⊓ ⊔mentioning

confidence: 99%

Transformation of fractions into simple fractions in divisive meadows

Bergstra¹,

Middelburg²

2016

“…A meadow is a total algebra over the signature Σ Md that satisfies the equations given in Tables 1 and 2. 1 We write: E CR for the set of all equations in Table 1 , E mi 0 for the set of all equations in Table 2 ,…”

Section: Meadowsmentioning

confidence: 99%

“… 1. We write: E CR for the set of all equations in Table1, E mi 0 for the set of all equations in Table2, E Md for E CR ∪ E mi 0 .…”

mentioning

confidence: 99%

Division by zero in non-involutive meadows

Bergstra

Middelburg

2015

Abstract. 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 'noninvolutive 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.

“…This means that it is still a matter of design which logic of partial functions will be used when working with this partial algebra. 2 As soon as the logic is fixed, the above-mentioned questions are no longer open: it is anchored in the logic whether 0 −1 = 0 −1 is satisfied, 0 −1 = 0 −1 is satisfied, or neither of the two is satisfied. Similar remarks apply to the other two partial algebras introduced above.…”

Section: Partial Meadowsmentioning

confidence: 99%

Inversive meadows and divisive meadows

Bergstra

Middelburg

2011

Inversive meadows are commutative rings with a multiplicative identity element and a total multiplicative inverse operation satisfying 0 −1 = 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.