Existence of Subgroup of Big Dipper Semigroup/Examples/(m, n) = (3, 4)

Examples of Use of Existence of Subgroup of Big Dipper Semigroup
Consider the big 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:
 * $a - 1 \le 4 k < a + 3$