The law that applying the same binary idempotent operator to identical operands yields the same operand: for a connective ⊗ that is idempotent, A ⊗ A = A (examples: A ∧ A = A, A ∨ A = A; likewise set union A ∪ A = A).

Principle

Repetition of an operand under the same idempotent connective has no additional effect beyond a single occurrence.

Demonstration

In set theory, the union of a set with itself is the set: S ∪ S = S. In Boolean algebra, repeating A under ∧ or ∨ does not change truth value compared to A alone.

Misapplication

Treating non-idempotent operations as idempotent — for instance assuming numeric addition or multiplication satisfy A + A = A — produces false simplifications and data loss in algebraic manipulation.

Consequence

Permits elimination of duplicated literals in logical expressions, reduces redundancy in formulas and circuits, and supports canonical forms where duplicates are collapsed.

Reversal

Non-idempotency: operations where repeating an operand changes the result (e.g., numeric addition A + A = 2A, logical XOR A ⊕ A = 0 under Boolean arithmetic but is not equal to A).

Boundary

Applies only to operators proven idempotent in the given algebra; idempotency is operator- and context-specific and does not extend to composite or mixed operators without proof.

Semantic Tension

Tension with multiplicative intuitions—idempotency collapses repetition while many algebraic systems track multiplicity or frequency, so information-preserving systems often reject idempotency.

Synthesis

Idempotent law signals when duplication of the same operand is semantically redundant under a particular operator, enabling removal of duplicates and streamlined representations while distinguishing contexts that preserve multiplicity.