Magma is Submagma of Itself
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Theorem
Let $\struct {S, \circ}$ be a magma.
Then $\struct {S, \circ}$ is a submagma of itself.
Proof
By definition, a magma is an algebraic structure $\struct {S, \circ}$ where $\circ$ is closed.
That is:
- $\forall x, y \in S: x \circ y \in S$
From Set is Subset of Itself, $S \subseteq S$.
By definition, $\struct {T, \circ}$ is a submagma of $S$ if:
- $\forall x, y \in T: x \circ y \in T$
It follows from the above that $\struct {S, \circ}$ is, by definition, a submagma of itself.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets: Example $8.3$