idempotentsus

Idempotentsus is a term used in some mathematical discussions to denote a generalization of the idea of idempotence to a set equipped with a family of binary operations. In this usage, an element is called idempotentsus if it is idempotent with respect to every operation in the specified family.

Definition. Let A be a set endowed with a collection F of binary operations, where each operation ⊕ is a map A × A → A. An element a ∈ A is idempotentsus with respect to F if a ⊕ a = a for every ⊕ ∈ F. In other words, a behaves as a fixed point under each operation in the family.

Relation to idempotence. Idempotentsus generalizes the standard notion of idempotence, which concerns a single operation. If F contains only one operation, the term reduces to the ordinary concept of an idempotent element for that operation.

Properties. If a is idempotentsus in A with respect to F and if a homomorphism preserves all operations in F, then the image of a under that homomorphism is idempotentsus in the target structure (assuming the target carries the corresponding family of operations). Idempotentsus elements often form a substructure that reflects the interaction of the operations in F.

Examples. In a set with two binary operations ⊕ and ⊗, an element a with a ⊕ a = a and a ⊗ a = a is idempotentsus. If F contains only one operation, the example reduces to ordinary idempotence.

Notes. Idempotentsus is not a widely adopted standard term in formal mathematics; it appears mainly in exploratory or pedagogical contexts to illustrate multi-operation idempotence. See also idempotent, multistructure, and universal algebra.