相关论文: Towards an Enumeration of Finite Common Meadows
Meadows are a sort of commutative rings with a multiplicative identity element and a total multiplicative inverse operation. In this paper we study algebraic properties of common meadows, which are meadows that introduce, as the inverse of…
We introduce the notion of Artinian meadow as an algebraic structure constructed from an Artinian ring which is also a common meadow, i.e.\ a commutative and associative structure with two operations (addition and multiplication) with…
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…
An inversive meadow is a commutative ring with identity equipped with a multiplicative inverse operation made total by choosing 0 as its value at 0. Previously, inversive meadows were shortly called meadows. A divisive meadow is an…
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…
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…
A common meadow is an enrichment of a field with a partial division operation that is made total by assuming that division by zero takes the a default value, a special element $\bot$ adjoined to the field. To a common meadow of real numbers…
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 value $\bot$ whose main purpose is to always return a value for division. To rings…
The rational, real and complex numbers with their standard operations, including division, are partial algebras specified by the axiomatic concept of a field. Since the class of fields cannot be defined by equations, the theory of…
A \emph{meadow} is a commutative ring with an inverse operator satisfying $0^{-1}=0$. We determine the initial algebra of the meadows of characteristic 0 and show that its word problem is decidable.
We examine the consequences of having a total division operation $\frac{x}{y}$ on commutative rings. We consider two forms of binary division, one derived from a unary inverse, the other defined directly as a general operation; each are…
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…
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…
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…
Reverse search is a convenient method for enumerating structured objects, that can be used both to address theoretical issues and to solve data mining problems. This method has already been successfully developed to handle unordered trees.…
An order-theoretic forest is a countable partial order such that the set of elements larger than any element is linearly ordered. It is an order-theoretic tree if any two elements have an upper-bound. The order type of a branch can be any…
$\mathbb{Q}_0$ - the involutive meadow of the rational numbers - is the field of the rational numbers where the multiplicative inverse operation is made total by imposing $0^{-1}=0$. In this note, we prove that $\mathbb{Q}_0$ cannot be…
Meadows - commutative rings equipped with a total inversion operation - can be axiomatized by purely equational means. We study subvarieties of the variety of meadows obtained by extending the equational theory and expanding the signature.
Let Q_0 denote the rational numbers expanded to a "meadow", that is, after taking its zero-totalized form (0^{-1}=0) as the preferred interpretation. In this paper we consider "cancellation meadows", i.e., meadows without proper zero…
We bridge sheaves of rings over a topological space with common meadows (algebraic structures where the inverse for multiplication is a total operation). More specifically, we show that the subclass of pre-meadows with $\mathbf{a}$, coming…