idempodentsuses

Idempodentsuses is a fictional algebraic concept introduced in speculative discussions to denote a class of idempotent-like operators that are idempotent with respect to two compatible operations. Formally, given an object X in a category with two binary operations, composition ∘ and a secondary operation ⋆, an endomorphism f: X -> X is called an idempodentsus if f∘f = f and f⋆f = f, and if ∘ and ⋆ satisfy a coherence condition such as f∘(g⋆h) = (f∘g)⋆(f∘h) for all endomorphisms g,h. In other formulations, the property is stated as f being a projection with respect to both operations.

Properties: Idempodentsuses act as projections in both operational contexts; they form a subfamily closed under the two-operand composition and are stable under conjugation by endomorphisms that preserve both operations. They often arise in settings that combine linear structure with a parallel composition, such as product categories or bimonoids.

Examples and interpretations: A toy example can be framed in a setting with two compatible projection-like operations on a vector space, where an endomorphism acts as a projection with respect to both operations. In computational contexts, analogous ideas appear in retry-safe designs and idempotent endpoints that remain stable under duplicate requests, reflecting a dual idempotence across two interaction modes.

History and usage: The term idempodentsuses is not standardized and appears mainly in niche or speculative writings. It is typically discussed as a hypothetical generalization of idempotence, intended to explore how two independent idempotent mechanisms might interact within a single framework.

See also: idempotent, projection, endomorphism, bimonoid.