Definition:Relation Compatible with Operation
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Definition
Let $\struct {S, \circ}$ be a closed algebraic structure.
Let $\RR$ be a relation on $S$.
Then $\RR$ is compatible with $\circ$ if and only if:
- $\forall x, y, z \in S: x \mathrel \RR y \implies \paren {x \circ z} \mathrel \RR \paren {y \circ z}$
- $\forall x, y, z \in S: x \mathrel \RR y \implies \paren {z \circ x} \mathrel \RR \paren {z \circ y}$
Also see
- Equivalence Relation is Congruence iff Compatible with Operation
- Preordering of Products under Operation Compatible with Preordering
- Results about compatible relations can be found here.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 15$: Ordered Semigroups