Mappings to Algebraic Structure form Similar Algebraic Structure

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Theorem

Let $X$ be a nonempty set.

Let $G$ be a magma with respect to the binary operations $\circ_1, \ldots, \circ_n$ on $G$.

Let $G^X$ be the set of all mappings from $X$ to $G$.

Denote also by $\circ_1, \ldots, \circ_n$ the binary operations defined on $G^X$ by pointwise addition.

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Also see

• Mappings to R-Algebraic Structure form Similar R-Algebraic Structure

Sources

• 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups: Exercise $2$