Restricted Dipper Operation is Commutative
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Theorem
Let $m, n \in \N_{>0}$ be non-zero natural numbers.
Let $\N_{< \paren {m \mathop + n} }$ denote the initial segment of the natural numbers:
- $\N_{< \paren {m \mathop + n} } := \set {0, 1, \ldots, m + n - 1}$
Let $\N^*_{< \paren {m \mathop + n} }$ denote the set defined as $\N_{< \paren {m \mathop + n} } \setminus \set 0$:
- $\N^*_{< \paren {m \mathop + n} } := \set {1, 2, \ldots, m + n - 1}$
The restricted dipper operation $+^*_{m, n}$ on $\N^*_{< \paren {m \mathop + n} }$ is commutative.
Proof
By definition, $+^*_{m, n}$ is the restriction of the dipper relation $+_{m, n}$ to $\N_{>0}$.
We have from Dipper Operation is Associative that $+_{m, n}$ is a commutative operation.
The result follows from Restriction of Commutative Operation is Commutative.
$\blacksquare$