Definition:Restricted Dipper Semigroup

From ProofWiki
Jump to navigation Jump to search

Definition

Let $m, n \in \N_{>0}$ be non-zero natural numbers.


Let $\RR^*_{m, n}$ be the restricted dipper relation on $\N$:

$\forall x, y \in \N_{>0}: x \mathrel {\RR^*_{m, n} } y \iff \begin {cases} x = y \\ m \le x < y \text { and } n \divides \paren {y - x} \\ m \le y < x \text { and } n \divides \paren {x - y} \end {cases}$

Let $\map {D^*} {m, n} := \N_{>0} / \RR^*_{m, n}$ be the quotient set of $\N_{>0}$ induced by $\RR^*_{m, n}$.

Let $\oplus^*_{m, n}$ be the operation induced on $\map {D^*} {m, n}$ by addition on $\N_{>0}$.


A restricted dipper (semigroup) is a semigroup which is isomorphic to the algebraic structure $\struct {\map {D^*} {m, n}, \oplus^*_{m, n} }$.


Also see

  • Results about restricted dipper semigroups can be found here.


Linguistic Note

The term restricted dipper semigroup was invented by $\mathsf{Pr} \infty \mathsf{fWiki}$ in order to be referred to compactly in conjunction with the dipper semigroup.

As such, it is not generally expected to be seen in this context outside $\mathsf{Pr} \infty \mathsf{fWiki}$.

The term dipper was coined by Seth Warner in the context of inductive semigroups.

Warner's approach is to bracket both the dipper semigroup and the restricted dipper semigroup into the same conceopt, referring to them both as dippers.


Sources