arXiv subjects

Jan A Bergstra

Publications and source records attributed to Jan A Bergstra.

4 recordsLinked to original sources

arXiv2026

On Ambiguity: The case of fraction, its meanings and roles

We contemplate the notion of ambiguity in mathematical discourse. We consider a general method of resolving ambiguity and semantic options for sustaining a resolution. The general discussion is applied to the case of `fraction' which is ill-defined and ambiguous in the literature of elementary arithmetic. In order to clarify the use of `fraction' we introduce several new terms to designate some of its possible meanings. For example, to distinguish structural aspects we use `fracterm', to distinguish purely numerical aspects `fracvalue' and, to distinguish purely textual aspects `fracsign' and `fracsign occurence'. These interpretations can resolve ambiguity, and we discuss the resolution by using such precise notions in fragments of arithmetical discourse. We propose that fraction does not qualify as a mathematical concept but that the term functions as a collective for several concepts, which we simply call a `category'. This analysis of fraction leads us to consider the notion of number in relation to fracvalue. We introduce a way of specifying number systems, and compare the analytical concepts with those of structuralism.

cs.LO↗

arXiv2025

Expressing entropy and cross-entropy in expansions of common meadows

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 add a binary logarithm $\log_2(-)$, which we also assume to be total with $\log_2(p) = \bot$ for $p \leq 0$. With these and other auxiliary operations, such as a sign function, we form algebras over which entropy and cross entropy can be defined for probability mass functions on a finite sample space by algebraic formulae that are simple terms built from the operations of the algebras and without case distinctions or conventions to avoid partiality. The discuss the advantages of algebras based on common meadows, whose theory is established, and alternate methods to define entropy and other information measures completely for all arguments using single terms.

cs.IT↗

arXiv2024

A Complete Finite Axiomatisation of the Equational Theory of 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 value $\bot$ whose main purpose is to always return a value for division. To rings and fields, we add a division operator $x/y$ and study a class of algebras called common meadows wherein $x/0 = \bot$. The set of equations true in all common meadows is named the equational theory of common meadows. We give a finite equational axiomatisation of the equational theory of common meadows and prove that it is complete and that the equational theory is decidable.

cs.LO↗

arXiv2024

Rings with common division, common meadows and their conditional equational theories

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 made total by setting $1/0$ equal to an error value $\bot$, which is added to the ring. Such totalised divisions we call common divisions. In a field the two forms are equivalent and we have a finite equational axiomatisation $E$ that is complete for the equational theory of fields equipped with common division, called common meadows. These equational axioms $E$ turn out to be true of commutative rings with common division but only when defined via inverses. We explore these axioms $E$ and their role in seeking a completeness theorem for the conditional equational theory of common meadows. We prove they are complete for the conditional equational theory of commutative rings with inverse based common division. By adding a new proof rule, we can prove a completeness theorem for the conditional equational theory of common meadows. Although, the equational axioms $E$ fail with common division defined directly, we observe that the direct division does satisfies the equations in $E$ under a new congruence for partial terms called eager equality.

cs.LO↗