Inversive meadows and divisive meadows

Bergstra, Jan A.; Middelburg, Cornelis A.

doi:10.1016/j.jal.2011.03.001

Cited by 31 publications

(61 citation statements)

References 33 publications

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“…Some other specifications of Z in the "language of rings" are discussed in [1], but these have negative normal forms Below we define three more types of canonical terms and their associated canonical term algebras. The (involutive) meadow Q 0 is defined as the field Q of rational numbers with a zero-totalized inverse (so 0 −1 = 0 and ( ) −1 is an involution; see, e.g., [8,4,3]). With Q 0 we denote the canonical term algebra for the abstract datatype Q 0 with these canonical terms.…”

Section: Ddrses and Canonical Termsmentioning

confidence: 99%

“…Let R be a reduced commutative ring that satisfies property (4). The cancellation equivalence generated by the rational fracpair axiom RF defined in Table 6 and the common cancellation axiom CC for fracpairs (defined in Table 2) is called rf-equivalence, notation = rf .…”

Section: An Initial Algebra Of Fractions For Rational Numbersmentioning

confidence: 99%

“…4 Meadows are commutative von Neumann regular rings (vNrr's) equipped with a weak multiplicative inverse x −1 (thus 0 −1 = 0) that is an involution (thus (x −1 ) −1 = x). In particular, the class of meadows is a variety, so each substructure of a meadow is a meadow, which is not the case for commutative vNrr's (cf.…”

Section: Common Cancellation Fractions Constitute a Common Meadowmentioning

confidence: 99%

“…The meadow of rationals Q 0 admits an equational initial algebra specification (see [7] and a subsequent simplification in [4]). Now the conjecture is that no finite equational initial algebra specification of Q 0 is both confluent and strongly terminating (interpreting the equations as left-to-right rewrite rules).…”

Section: Conclusion and Digressionmentioning

confidence: 99%

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Fracpairs and fractions over a reduced commutative ring

Bergstra

Ponse

2016

Indagationes Mathematicae

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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 (that is, 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 preimage 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 (and therewith normal forms) for cc-fractions over the integers and some meadows that model the rational numbers expanded with a totalized inverse, and we 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 some negative conjectures concerning alternative specifications for these (concrete) datatypes.

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Section: Ddrses and Canonical Termsmentioning

confidence: 99%

Section: An Initial Algebra Of Fractions For Rational Numbersmentioning

confidence: 99%

Section: Common Cancellation Fractions Constitute a Common Meadowmentioning

confidence: 99%

Section: Conclusion and Digressionmentioning

confidence: 99%

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Fracpairs and fractions over a reduced commutative ring

Bergstra

Ponse

2016

Indagationes Mathematicae

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“…In [6], divisive meadows are proposed. A divisive meadow is a commutative ring with a multiplicative identity element and a total division operation satisfying the three equations 1 / (1 / x) = x, (x · x) / x = x, and x / y = x · (1 / y).…”

Section: Introductionmentioning

confidence: 99%

Transformation of fractions into simple fractions in divisive meadows

Bergstra¹,

Middelburg²

2016

Journal of Applied Logic

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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

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The Wheel of Rational Numbers as an Abstract Data Type

Bergstra

Tucker

2021

Lecture Notes in Computer Science

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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 infinity ∞ or error element ⊥. A wheel is an algebra in which division is totalised by setting 1/0 = ∞ but which also contains an error element ⊥ to help control its use. We construct the wheel of rational numbers as an abstract data type Qw and give it an equational specification without auxiliary operators under initial algebra semantics.

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Inversive meadows and divisive meadows

Cited by 31 publications

References 33 publications

Fracpairs and fractions over a reduced commutative ring

Fracpairs and fractions over a reduced commutative ring

Transformation of fractions into simple fractions in divisive meadows

The Wheel of Rational Numbers as an Abstract Data Type

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