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

Note on paraconsistency and reasoning about fractions

We apply a paraconsistent logic to reason about fractions.Comment: 6 page

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admit a certain cancellation law.Comment: 17 pages, 4 tables; differences with v3: axiom (14) of Mda (Table 2) has been replaced by the stronger axiom (12), this appears to be necessary for the proof of Theorem 3.2.

Architectural Adequacy and Evolutionary Adequacy as Characteristics of a Candidate Informational Money

For money-like informational commodities the notions of architectural adequacy and evolutionary adequacy are proposed as the first two stages of a moneyness maturity hierarchy. Then three classes of informational commodities are distinguished: exclusively informational commodities, strictly informational commodities, and ownable informational commodities. For each class money-like instances of that commodity class, as well as monies of that class may exist. With the help of these classifications and making use of previous assessments of Bitcoin, it is argued that at this stage Bitcoin is unlikely ever to evolve into a money. Assessing the evolutionary adequacy of Bitcoin is perceived in terms of a search through its design hull for superior design alternatives. An extensive comparison is made between the search for superior design alternatives to Bitcoin and the search for design alternatives to a specific and unconventional view on the definition of fractions.Comment: 25 page

A Complete Finite Equational Axiomatisation of the Fracterm Calculus for 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 flag $\bot$ 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 \textit{common meadows} wherein $x/0 = \bot$. The set of equations true in all common meadows is named the \textit{fracterm calculus of common meadows}. We give a finite equational axiomatisation of the fracterm calculus of common meadows and prove that it is complete and that the fracterm calculus is decidable

Datatype defining rewrite systems for naturals and integers

A datatype defining rewrite system (DDRS) is an algebraic (equational) specification intended to specify a datatype. When interpreting the equations from left-to-right, a DDRS defines a term rewriting system that must be ground-complete. First we define two DDRSs for the ring of integers, each comprising twelve rewrite rules, and prove their ground-completeness. Then we introduce natural number and integer arithmetic specified according to unary view, that is, arithmetic based on a postfix unary append constructor (a form of tallying). Next we specify arithmetic based on two other views: binary and decimal notation. The binary and decimal view have as their characteristic that each normal form resembles common number notation, that is, either a digit, or a string of digits without leading zero, or the negated versions of the latter. Integer arithmetic in binary and decimal notation is based on (postfix) digit append functions. For each view we define a DDRS, and in each case the resulting datatype is a canonical term algebra that extends a corresponding canonical term algebra for natural numbers. Then, for each view, we consider an alternative DDRS based on tree constructors that yields comparable normal forms, which for that view admits expressions that are algorithmically more involved. For all DDRSs considered, ground-completeness is proven