Operation
What this is
An Operation on a carrier A is a map
for some arity n ≥ 0. It takes n elements of A and produces an element of A. It is internal: it maps A within itself, not to a different object.
This is the mathematical sense shared by:
- Universal algebra: an n-ary operation on a set A is a function A^n → A; the operations of an algebra are what give the carrier its algebraic structure.
- Machine operations: an instruction (ADD, LOAD, STORE) takes values from registers and writes a result back to a register — the carrier (register state) maps to itself.
- Surgical operations: a procedure acts on a patient (the carrier) and produces a changed state of that same patient — the carrier is the same before and after.
An Operation is atomic: it either applies or it does not. There is no partial application that leaves the carrier in an inconsistent state.
Distinction from Function
A Function f : A → B maps between objects. An Operation ω : A^n → A maps within an object.
The distinction matters when reasoning about whether something transforms a structure internally (Operation) or bridges two structures (Function). Both are present in any non-trivial algebraic structure.
Arity
A 0-ary operation is a constant: it picks a distinguished element of A. A 1-ary (unary) operation maps A → A: a closure operator or endomorphism. A 2-ary (binary) operation maps A × A → A: a product, a meet, a concatenation.
The nuclei σ, Δ are unary operations (closure operators) on the fiber H_t. The monoidal product ⊛ is a binary operation on R. The universe closure U_G is a unary operation on classes of objects of R.