Identity of Subgroup of Dipper Semigroup is not Identity of Dipper/Examples/(m, n) = (3, 4)

Examples of Use of Identity of Subgroup of Dipper Semigroup is not Identity of Dipper

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

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

From Existence of Subgroup of Dipper Semigroup Example: $\struct {H, +_{3, 4} }$

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

whose identity is $4$.

We have that:

$0 +_{3, 4} 4 = 4$

and so $4$ is not the identity of $\struct {N_{<7}, +_{3, 4} }$.

Sources

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