Regalars

Regalars is a term used in some niche mathematical contexts to denote a class of algebraic structures that are intended to generalize the concept of scalars. The expression is not widely standardized, and in published work it can refer to different ideas depending on the author. As a result, there is no single accepted definition, but most treatments frame regulairs axiomatically as a set equipped with a conventional addition operation, a second binary operation described as reglar multiplication, and a regulator map that interacts with these components.

In informal terms, a reglar aims to extend the idea of scalar fields by incorporating a regulator that controls or records scaling within the algebraic framework. This regulator is typically required to be compatible with the two operations in a way that generalizes familiar laws such as distributivity and scalar action. Variants of regulairs differ in properties of the reglar operation (for example, whether it is associative or commutative) and whether the regulator satisfies additional conditions (such as idempotence) or an order structure is present.

Examples of regulairs are largely theoretical and used to illustrate how introducing a regulator affects standard algebraic results. A common, trivial example is when the regulator acts as the identity, in which case the reglar reduces to a conventional field or ring. In practice, regulairs appear mainly in exploratory discussions within module theory, enriched category theory, or models of scaling in theoretical computer science, where tracking the regulator provides a way to study generalized scalar behavior.

See also: scalars, regulators, regulated algebra, and enriched category theory.