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

Examples of Use of Existence of Subgroup of Dipper Semigroup

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:

$a - 1 \le 4 k < a + 3$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets: Exercise $8.4$