Restricted Dipper Operation is Associative

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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 associative.


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 an associative operation.

The result follows from Restriction of Associative Operation is Associative.

$\blacksquare$


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