#5033 ⟨a, b | aaa=aa, abba=b⟩

Properties

Presentation has sum-of-sides 10
Finite commutative monoid with 4 elements
Commutative Gröbner basis: ⟨a, b | b²=b, ab=b, a³=a²⟩
Not cancellative, because multiplication by b is not injective:
- b ≠ 1
- b ⋅ b = b
- b ⋅ 1 = b
- Cancellative quotient is isomorphic to ℤ₁
Grothendieck group is isomorphic to ℤ₁
Group of units is isomorphic to ℤ₁
3 Archimedian components:
- 1, a, b

Format:	Pretty Plain
Word to reduce:
	Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Reduction strategy:	Leftmost Rightmost
Path to normal form:	1 1

# ab:aaa=aa,abba=b ab
ab=b
ba=b
bb=b
aaa=aa

Idempotents are shown in bold.

Idempotents are shown in bold.

8 unique, 1605 total

Σ	#	Presentation	Description	Related
5	7	⟨a, b \| aa=b, bb=1⟩	Isomorphic to ℤ₄	1419 iso
6	57	⟨a, b \| aa=a, bb=a⟩	Isomorphic to ℕ_{(4 = 2)}	37 iso
6	61	⟨a, b \| aa=b, bb=a⟩	Isomorphic to ℕ_{(4 = 1)}	72 iso
7	158	⟨a, b \| ab=aa, ba=b⟩	Finite non-commutative monoid with 4 elements	8 iso, 6 anti-iso
7	159	⟨a, b \| ab=aa, bb=a⟩	Isomorphic to ℕ_{(4 = 3)}	16 iso
7	242	⟨a, b \| aa=a, abb=b⟩	Finite non-commutative monoid with 4 elements	14 iso
7	280	⟨a, b \| ab=a, bb=aa⟩	Finite commutative monoid with 4 elements	17 iso
9	2881	⟨a, b \| aa=a, abbba=b⟩	Finite commutative monoid with 4 elements	8 iso

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

2 total

Σ	#	Presentation	Mapping
11	15414	⟨a, b \| aaa=aa, aabba=b⟩	φ(a) = a, φ(b) = b
11	15420	⟨a, b \| aaa=aa, ababa=b⟩	φ(a) = a, φ(b) = b

Up:	Monoids with two generators and two relations
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Next:	#5034 ⟨a, b \| aaa=aa, abbb=a⟩