Definition:Extension

Definition

Extension of a Relation

Let:

• $\mathcal R_1 \subseteq X \times Y$ be a relation on $X \times Y$
• $\mathcal R_2 \subseteq S \times T$ be a relation on $S \times T$
• $X \subseteq S$
• $Y \subseteq T$
• $\mathcal R_2 \restriction_{X \times Y}$ be the restriction of $\mathcal R_2$ to $X \times Y$.

Let $\mathcal R_2 \restriction_{X \times Y} = \mathcal R_1$.

Then $\mathcal R_2$ extends or is an extension of $\mathcal R_1$.

Extension of a Mapping

As a mapping is, by definition, also a relation, the definition of an extension of a mapping is the same as that for an extension of a relation:

Let:

• $f_1 \subseteq X \times Y$ be a mapping on $X \times Y$
• $f_2 \subseteq S \times T$ be a mapping on $S \times T$
• $X \subseteq S$
• $Y \subseteq T$
• $f_2 \restriction_{X \times Y}$ be the restriction of $f_2$ to $X \times Y$.

Let $f_2 \restriction_{X \times Y} = f_1$.

Then $f_2$ extends or is an extension of $f_1$.

Extension of an Operation

Let $\left({S, \circ}\right)$ be a magma.

Let $\left({T, \circ \restriction_T}\right)$ be a submagma of $\left({S, \circ}\right)$, where $\circ \restriction_T$ denotes the restriction of $\circ$ to $T$.

Then:

$\left({S, \circ}\right)$ is an extension of $\left({T, \circ \restriction_T}\right)$

or

$\left({S, \circ}\right)$ extends $\left({T, \circ \restriction_T}\right)$

We can use the term directly to the operation itself and say:

$\circ$ is an extension of $\circ \restriction_T$

or:

$\circ$ extends $\circ \restriction_T$

Warning

Do not confuse this with expansion, which is a term used for a topology which is a superset of another topology.

Also see

• Restriction of a Relation
• Restriction of a Mapping
• Restriction of an Operation