Monoid enumeration

This enumeration has 7 instances in total.

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Breakdown of solved instances

• Every group is cancellative.
• A finite monoid is cancellative if and only if it is a group.
• Otherwise, cancellativity is only decided for commutative monoids.
• Finite monoids, finite groups, and infinite Abelian groups are classified up to isomorphism.
• All other instances are opportunistically uniqued by comparing complete rewriting systems.

All solved instances

• ℤ is the free group with one generator, also known as the integers.
• ℕ is the free monoid with one generator, also known as the natural numbers.
• ℤ_n is the cyclic group with n elements, so ℤ₁ is the trivial group.
• ℕ_(m=n) with m > n is the non-cancellative monogenic monoid with m elements and a^m = aⁿ.
• ⟨x₁, ..., x_n⟩ ⊂ ℤ_n ⊕ ... ⊕ ℤ is the cancellative submonoid generated by the elements x₁, ..., x_n of an Abelian group.
• X ⊕ Y is the commutative direct sum of X and Y.
• X ∗ Y is the non-commutative free product of X and Y.