Related Work

The work on meadows is concerned with different ways in which the multiplicative inverse operation can be totalized. The simplest and most straightforward way considered is the way in which the multiplicative inverse of zero is zero. This way has led to the equational axiomatization of involutive meadows (originally simply called meadows). Involutive meadows have been investigated in great detail and their usefulness have been demonstrated in various applications. Another way considered is the way in which the multiplicative inverse of zero is a value different from zero, but not an additional value. This way has led to the equational axiomatization of non-involutive meadows and the investigation of their connection with involutive meadows. Still another way considered is the way in which the multiplicative inverse of zero is an additional value that can be interpreted as an error value. This way has led to the equational axiomatization of common meadows.

Most closely related to the work on meadows is the work on transreal numbers and the work on wheels. Both consider a single way in which the multiplicative inverse operation is totalized. The way considered in the work on transreal numbers is the way in which the multiplicative inverse of zero is an additional value that can be interpreted as infinity. The transreal numbers include two other additional values. One of them can be interpreted as the additive inverse of infinity. The other one, confusingly called nullity, can be interpreted as the result of multiplying zero with infinity. The way considered in the work on wheels is the same as the one considered in the work on transreal numbers. However, because it is not required in wheels that each value has an additive inverse, the additional value that can be interpreted as the additive inverse of infinity is left out.

The work on transreal numbers started from a consistent set of equational and non-equational axioms that the intended algebraic structure of transreal arithmetic should satisfy. In Reis, Gomide, and Anderson (2016), the intended algebraic structure of transreal arithmetic is constructed from the field of real numbers. This paper is probably the best entry into the work on transreal numbers. It provides references to most earlier work on transreal numbers. Like the work on meadows, the work on wheels started from a purely equational axiomatization. So, like the class of meadows, the class of all wheels is a variety. In Carlström (2004), wheels are first introduced by means of their equational axiomatization and then investigated in some detail. To our knowledge, there are no more recent papers on wheels.