Parrelation

Parrelation is a concept in discrete mathematics describing a binary relation defined on a set that carries a parity labeling, typically separating elements into two classes such as even and odd. The term suggests a relation that respects or interacts with this parity partition.

Definition: Let S be a set equipped with a parity function par: S → {0,1}. A relation R ⊆ S × S is called a parrelation if, whenever (a,b) ∈ R, we have par(a) = par(b). In other words, a parrelation never relates elements from different parity classes. Equivalently, R is a subset of E × E ∪ O × O, where E = {x ∈ S | par(x) = 0} and O = {x ∈ S | par(x) = 1}. A parrelation need not be reflexive or symmetric unless specified; if it is symmetric, it can be viewed as a graph with two independent subgraphs on the parity classes.

Examples: On the set of integers with par(n) = n mod 2, the relation {(a,b) : par(a) = par(b)} is a parrelation. More restricted examples include the even–even subrelation {(a,b) : a and b are even} and the odd–odd subrelation {(a,b) : a and b are odd}.

Variants and properties: Parrelations naturally decompose into two subrelations on the parity classes, allowing separate analysis within each class. They are closed under union and intersection under appropriate constraints, and their composition preserves the parity-preserving property.

Applications: Parrelations arise in contexts where a built-in partition by parity is meaningful, such as modular arithmetic, parallel processing schemes that treat parity classes independently, and certain graph-theoretic constructions where edges are restricted to vertices of the same parity.

History: The term parrelation appears as a descriptive label in some mathematical discussions for parity-preserving relations and is not universally standardized. See also binary relation, equivalence relation, parity, and partitions.