Existence of Subgroup of Dipper Semigroup

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Theorem

Let $m, n \in \Z$ be integers such that $m \ge 0, n > 0$.

Let $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$ denote the dipper semigroup.

Consider the subset $H \subseteq N_{< \paren {m \mathop + n} }$ defined as:

$H = \set {k \in \N: m \le k < m + n} = \set {m, m + 1, \ldots, m + n - 1}$


Then the substructure $\struct {H, +_{m, n} }$ is a subgroup of $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$.


Proof

Recall the definition of the dipper semigroup:

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 $+_{m, n}$ 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:

$m + k n \le a + b$

Taking the group axioms in turn:


Group Axiom $\text G 0$: Closure

Let $a, b \in H$.

Then:

$2 m \le a + b \le 2 \paren {m + n - 1}$

Thus in all cases $a + b > m$.


Thus by definition of $+_{m, n}$:

$a +_{m, n} b = a + b - k n$

where $k$ is the largest such that $m + k n < a + b$

Hence:

\(\ds m + k n\) \(\le\) \(\, \ds a + b \, \) \(\, \ds < \, \) \(\ds m + \paren {k + 1} n\) Definition of $k$
\(\ds \leadsto \ \ \) \(\ds m\) \(\le\) \(\, \ds a + b - k n \, \) \(\, \ds < \, \) \(\ds m + n\) subtracting $k n$ from all parts
\(\ds \leadsto \ \ \) \(\ds m\) \(\le\) \(\, \ds a +_{m, n} b \, \) \(\, \ds < \, \) \(\ds m + n\) Definition of $+_{m, n}$

That is:

$a +_{m, n} b \in H$

and so $\struct {H, +_{m, n} }$ is closed.

$\Box$


Group Axiom $\text G 1$: Associativity

From Restriction of Associative Operation is Associative, associativity is inherited by $\struct {H, +_{m, n} }$ from $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$.

$\Box$


Group Axiom $\text G 2$: Existence of Identity Element

We have:

$m \le a < m + n$
\(\ds \forall a \in H: \, \) \(\ds m\) \(\le\) \(\, \ds a \, \) \(\, \ds < \, \) \(\ds m + n\) Group Axiom $\text G 0$: Closure
\(\ds \leadstoandfrom \ \ \) \(\ds m + n\) \(\le\) \(\, \ds a + n \, \) \(\, \ds < \, \) \(\ds m + 2 n\)

Hence

$a +_{m, n} n = a$


From Dipper Semigroup is Commutative Semigroup we have that:

$n +_{m, n} a = a +_{m, n} n = n$

Thus $n$ is the identity element of $\struct {H, +_{m, n} }$.

$\Box$


Group Axiom $\text G 3$: Existence of Inverse Element

We have that $n$ is the identity element of $\struct {H, +_{m, n} }$.

Hence we need to find $x \in H$ such that:

$a +_{m, n} x = n$

That is:

$a + x - k n = n$

So:

$x = \paren {k + 1} n - a$

such that $m \le x < m + n$

where:

$m - n + a \le k n < m + a$

Thus every element $x$ of $\struct {H, +_{m, n} }$ has an inverse $x^{-1}$ where:

$\paren {k + 1} n - a$

such that:

$m - n + a \le k n < m + a$

$\Box$


All the group axioms are thus seen to be fulfilled, and so $\struct {H, +_{m, n} }$ is a group.

$\blacksquare$


Examples

Example: $\struct {H, +_{3, 4} }$

Consider the dipper semigroup $\struct {N_{<7}, +_{3, 4} }$.

Let $H = \set {x \in \N: 3 \le x < 7} = \set {3, 4, 5, 6}$.

Then:

$\struct {H, +_{3, 4} }$ is a subgroup of $\struct {N_{<7}, +_{3, 4} }$

where:

the identity of $\struct {H, +_{3, 4} }$ is $4$
the inverse $a^{-1}$ of $a \in H$ is given by:
$a^{-1} = 4 \paren {k - 1} - a$
such that:
$a - 1 \le 4 k < a + 3$
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