proprials

Proprials are a fictional class of algebraic structures introduced to explore how proportional reasoning can be modeled within abstract systems. They are defined as sets equipped with two operations: a binary, commutative, associative product and a binary proportionality operator that assigns to each pair of elements another element of the same set.

Formally, a proprial consists of a nonempty set A with a product ·: A × A → A and a proportionality operator P: A × A → A. The product (A, ·) is required to be a commutative monoid, possessing an identity element e. The proportionality operator is governed by a proportionality law: for all a, b, c in A, a · P(b, c) = P(a · b, a · c). Additionally, P is normalized by P(a, a) = a for all a in A, and P(a, e) = a to ensure compatibility with the identity.

Examples are typically constructed in small finite sets to illustrate the axioms. The trivial proprial on a singleton set satisfies all conditions. More interesting examples are built by choosing a commutative monoid structure on A and then defining P to respect the proportionality law, often by deriving P from a valuation or grading on A.

Properties and variations of proprials include the study of subproprials (subsets closed under both operations), quotients by congruences, and automorphisms that preserve both · and P. They are occasionally combined with categorical perspectives, viewing proprials as objects in a category with morphisms preserving both operations.

Note: This article describes a fictional mathematical concept created for illustrative purposes and does not reflect established real-world terminology. See also semigroups, monoids, and abstract algebra concepts.