Dipper Operation/Examples/(m, n) = (3, 4)/Examples/x +(3,4) x = x
Examples of Equations on Dipper Operation $+_{3, 4}$
Let $\N_{<7}$ denote the initial segment of the natural numbers:
- $\N_{<7} := \set {0, 1, \ldots, 6}$
Let $+_{3, 4}$ be the dipper operation on $\N_{<7}$ defined as:
- $\forall a, b \in \N_{<7}: a +_{3, 4} b = \begin{cases}
a + b & : a + b < 3 \\ a + b - 4 k & : a + b \ge 3 \end{cases}$
where $k$ is the largest integer satisfying:
- $3 + 4 k \le a + b$
Consider the equation:
- $x +_{3, 4} x = x$
This has the solutions:
- $x = 0$
- $x = 4$
Proof
Apparent from direct inspection of the Cayley table:
- $\begin{array}{r|rrrrrrr}
+_m & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 0 & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ 1 & 1 & 2 & 3 & 4 & 5 & 6 & 3 \\ 2 & 2 & 3 & 4 & 5 & 6 & 3 & 4 \\ 3 & 3 & 4 & 5 & 6 & 3 & 4 & 5 \\ 4 & 4 & 5 & 6 & 3 & 4 & 5 & 6 \\ 5 & 5 & 6 & 3 & 4 & 5 & 6 & 3 \\ 6 & 6 & 3 & 4 & 5 & 6 & 3 & 4 \\ \end{array}$
The main diagonal contains the elements $x$ of $\N_{<7}$ such that $x +_{3, 4} x$.
As can be seen, $x +_{3, 4} x = x$ when $x = 0$ and $x = 4$.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Exercise $2.8 \ \text{(c)}$