Condition on Congruence Relations for Cancellable Monoid to be Group
Theorem
Let $\struct {S, \circ}$ be a cancellable monoid whose identity element is $e$.
Then:
- $\struct {S, \circ}$ is a group
- every non-trivial congruence relation on $\struct {S, \circ}$ is induced by a normal subgroup of $\struct {S, \circ}$.
Counterexample
Let $\struct {S, \circ}$ be a monoid which is not cancellable.
Let every non-trivial congruence relation on $\struct {S, \circ}$ be induced by a normal subgroup of $\struct {S, \circ}$.
Then it is not necessarily the case that $\struct {S, \circ}$ is a group.
Proof
Necessary Condition
Let $\struct {S, \circ}$ be such that every non-trivial congruence relation on $\struct {S, \circ}$ is induced by a normal subgroup of $\struct {S, \circ}$.
Hence, let $\RR$ be an arbitrary non-trivial congruence relation.
From Condition for Subgroup of Monoid to be Normal, there exists a normal subgroup $\struct {H, \circ}$ such that:
- the set of left cosets of $\struct {H, \circ}$ in $\struct {S, \circ}$ form a partition of $S$
and:
- the set of right cosets of $\struct {H, \circ}$ in $\struct {S, \circ}$ form a partition of $S$
such that the equivalence relations induced by those partitions is $\RR$.
By the definition of normal subgroup, the set of left cosets is the same as the set of right cosets.
It remains to be shown that $\struct {S, \circ}$ is a group.
We already have that
- Group Axiom $\text G 0$: Closure
- Group Axiom $\text G 1$: Associativity
- Group Axiom $\text G 2$: Existence of Identity Element
are a priori satisfied by the fact that $\struct {S, \circ}$ is a monoid.
Hence it remains to prove Group Axiom $\text G 3$: Existence of Inverse Element.
This theorem requires a proof. In particular: The suggestion is to use Condition for Partition between Invertible and Non-Invertible Elements to induce Congruence Relation on Monoid You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Sufficient Condition
Let $\struct {S, \circ}$ be a group.
This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Exercise $11.18 \ \text {(a)}$