Definition:Dipper Semigroup
Definition
Let $m \in \N$ be a natural number.
Let $n \in \N_{>0}$ be a non-zero natural number.
Let $\N_{< \paren {m \mathop + n} }$ denote the initial segment of the natural numbers:
- $\N_{< \paren {m \mathop + n} } := \set {0, 1, \ldots, m + n - 1}$
Definition 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:
- $m + k n \le a + b$
A dipper (semigroup) is a semigroup which is isomorphic to the algebraic structure $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$.
Definition 2
Let $\RR_{m, n}$ be the dipper relation on $\N$:
- $\forall x, y \in \N: x \mathrel {\RR_{m, n} } y \iff \begin {cases} x = y \\ m \le x < y \text { and } n \divides \paren {y - x} \\ m \le y < x \text { and } n \divides \paren {x - y} \end {cases}$
Let $\map D {m, n} := \N / \RR_{m, n}$ be the quotient set of $\N$ induced by $\RR_{m, n}$.
Let $\oplus_{m, n}$ be the operation induced on $\map D {m, n}$ by addition on $\N$.
A dipper (semigroup) is a semigroup which is isomorphic to the algebraic structure $\struct {\map D {m, n}, \oplus_{m, n} }$.
Also see
- Results about dipper semigroups can be found here.
Linguistic Note
The term dipper semigroup appears to have been coined by Seth Warner in an article exploring commutative inductive semigroups.
The name arises as a result of a suggestion by Seth Warner in his Modern Algebra, where he exploits the analogy by means of the shape of the Big Dipper constellation.
Sources
- 1960: Seth Warner: Mathematical Induction in Commutative Semigroups (Amer. Math. Monthly Vol. 67, no. 6: pp. 533 – 537) www.jstor.org/stable/2309170