Quotient Structure on Subset Product

Theorem

Let $\struct {S, \circ}$ be an algebraic structure.

Let $\RR$ be a congruence for $\circ$ on $S$.

Then:

$\forall X, Y \in S / \RR: X \circ_\PP Y \subseteq X \circ_\RR Y$

where:

$S / \RR$ is the quotient of $S$ by $\RR$

$\circ_\PP$ is the operation induced on $\powerset S$ by $\circ$

$\circ_\RR$ is the operation induced on $S / \RR$ by $\circ$

Proof

By definition of subset product:

$X \circ_\PP Y = \set {x \circ y: x \in X, y \in Y}$

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Thus:

$X \circ_\RR Y = \set {x \circ y: x \in X, y \in Y} \cup \set {x \circ y: x \in \eqclass X \RR, y \in \eqclass Y \RR}$

The result follows from Subset of Union.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures