Restriction of Canonical Surjection to Restricted Dipper Semigroup is Isomorphism
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 $\N_{< \paren {m \mathop + n} } \setminus \set 0$:
- $\N^*_{< \paren {m \mathop + n} } := \set {1, 2, \ldots, m + n - 1}$
Let $+_{m, n}$ be the dipper operation on $\N_{< \paren {m \mathop + n} }$:
- $\forall a, b \in \N_{< \paren {m \mathop + n} }: a +_{m, n} b = \begin{cases}
a + b & : a + b < m \\ a + b - k n & : a + b \ge m \end{cases}$ where $k$ is the largest integer satisfying:
- $m + k n \le a + b$
The $+^*_{m, n}$ be the restricted dipper operation on $\N^*_{< \paren {m \mathop + n} }$ defined as:
- $\forall a, b \in \N^*_{< \paren {m \mathop + n} }: a +^*_{m, n} b = \begin{cases}
a + b & : a + b < m \\ a + b - k n & : a + b \ge m \end{cases}$ where $k$ is the largest integer satisfying:
- $m + k n \le a + b$
Consider the algebraic structure $\struct {\N^*_{< \paren {m \mathop + n} }, +^*_{m, n} }$.
Let $\phi^*_{m, n}$ be the restriction of the canonical surjection from $\N_{>0}$ to the restricted dipper semigroup $\struct {\map {D^*} {m, n}, \oplus^*_{m, n} }$.
Then $\phi^*_{m, n}$ is an isomorphism from $\struct {\N^*_{< \paren {m \mathop + n} }, +^*_{m, n} }$ to $\struct {\map {D^*} {m, n}, \oplus^*_{m, n} }$.
Proof
This theorem requires a proof. In particular: Formalise the notion that everything is a restriction of everything else You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers: Exercise $16.7 \ \text {(b)}$