#15933 ⟨a, b | aba=bb, baaab=a⟩

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Properties
Elements
Cayley table
Right Cayley graph
Left Cayley graph
Rewriting system
Same cardinality

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 20 elements
Not cancellative, because multiplication by a is not injective:
- a ⋅ a⁶ = a and a ⋅ 1 = a, however a⁶ ≠ 1

Elements

Elements in the center commute with all other elements.
An idempotent element x satisfies x² = x.
The index and period of x is the least m (index) and n (period) such that x^(m+n) = x^m.

4 element center:
- 1, a², a⁴, a⁶
1 non-trivial idempotent:
- a⁶
Order of generators:
- a: index 1, period 6
- b: index 2, period 3

Histogram:

index 1, period 1	1 element	a⁶
index 1, period 2	3 elements	ab, ba, a³
index 1, period 3	8 elements	a², aba, ba², a⁴, aba³, ...
index 1, period 6	6 elements	a, aba², ba³, a⁵, aba⁴, ...
index 2, period 3	1 element	b

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	ba	a³	aba	ba²	a⁴	aba²	ba³	a⁵	aba³	ba⁴	a⁶	aba⁴	ba⁵	aba⁵	ba⁶
1	1	a	b	a²	ab	ba	a³	aba	ba²	a⁴	aba²	ba³	a⁵	aba³	ba⁴	a⁶	aba⁴	ba⁵	aba⁵	ba⁶
a	a	a²	ab	a³	ba²	aba	a⁴	ba³	aba²	a⁵	ba⁴	aba³	a⁶	ba⁵	aba⁴	a	ba⁶	aba⁵	ba	ab
b	b	ba	aba	ba²	a⁵	aba²	ba³	a⁶	aba³	ba⁴	a	aba⁴	ba⁵	a²	aba⁵	ba⁶	a³	ab	a⁴	aba
a²	a²	a³	ba²	a⁴	aba²	ba³	a⁵	aba³	ba⁴	a⁶	aba⁴	ba⁵	a	aba⁵	ba⁶	a²	ab	ba	aba	ba²
ab	ab	aba	ba³	aba²	a⁶	ba⁴	aba³	a	ba⁵	aba⁴	a²	ba⁶	aba⁵	a³	ba	ab	a⁴	ba²	a⁵	ba³
ba	ba	ba²	a⁵	ba³	aba³	a⁶	ba⁴	aba⁴	a	ba⁵	aba⁵	a²	ba⁶	ab	a³	ba	aba	a⁴	aba²	a⁵
a³	a³	a⁴	aba²	a⁵	ba⁴	aba³	a⁶	ba⁵	aba⁴	a	ba⁶	aba⁵	a²	ba	ab	a³	ba²	aba	ba³	aba²
aba	aba	aba²	a⁶	aba³	ba⁵	a	aba⁴	ba⁶	a²	aba⁵	ba	a³	ab	ba²	a⁴	aba	ba³	a⁵	ba⁴	a⁶
ba²	ba²	ba³	aba³	ba⁴	a	aba⁴	ba⁵	a²	aba⁵	ba⁶	a³	ab	ba	a⁴	aba	ba²	a⁵	aba²	a⁶	aba³
a⁴	a⁴	a⁵	ba⁴	a⁶	aba⁴	ba⁵	a	aba⁵	ba⁶	a²	ab	ba	a³	aba	ba²	a⁴	aba²	ba³	aba³	ba⁴
aba²	aba²	aba³	ba⁵	aba⁴	a²	ba⁶	aba⁵	a³	ba	ab	a⁴	ba²	aba	a⁵	ba³	aba²	a⁶	ba⁴	a	ba⁵
ba³	ba³	ba⁴	a	ba⁵	aba⁵	a²	ba⁶	ab	a³	ba	aba	a⁴	ba²	aba²	a⁵	ba³	aba³	a⁶	aba⁴	a
a⁵	a⁵	a⁶	aba⁴	a	ba⁶	aba⁵	a²	ba	ab	a³	ba²	aba	a⁴	ba³	aba²	a⁵	ba⁴	aba³	ba⁵	aba⁴
aba³	aba³	aba⁴	a²	aba⁵	ba	a³	ab	ba²	a⁴	aba	ba³	a⁵	aba²	ba⁴	a⁶	aba³	ba⁵	a	ba⁶	a²
ba⁴	ba⁴	ba⁵	aba⁵	ba⁶	a³	ab	ba	a⁴	aba	ba²	a⁵	aba²	ba³	a⁶	aba³	ba⁴	a	aba⁴	a²	aba⁵
a⁶	a⁶	a	ba⁶	a²	ab	ba	a³	aba	ba²	a⁴	aba²	ba³	a⁵	aba³	ba⁴	a⁶	aba⁴	ba⁵	aba⁵	ba⁶
aba⁴	aba⁴	aba⁵	ba	ab	a⁴	ba²	aba	a⁵	ba³	aba²	a⁶	ba⁴	aba³	a	ba⁵	aba⁴	a²	ba⁶	a³	ba
ba⁵	ba⁵	ba⁶	a³	ba	aba	a⁴	ba²	aba²	a⁵	ba³	aba³	a⁶	ba⁴	aba⁴	a	ba⁵	aba⁵	a²	ab	a³
aba⁵	aba⁵	ab	a⁴	aba	ba³	a⁵	aba²	ba⁴	a⁶	aba³	ba⁵	a	aba⁴	ba⁶	a²	aba⁵	ba	a³	ba²	a⁴
ba⁶	ba⁶	ba	aba	ba²	a⁵	aba²	ba³	a⁶	aba³	ba⁴	a	aba⁴	ba⁵	a²	aba⁵	ba⁶	a³	ab	a⁴	aba

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Rewriting system

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

Reduction order:
- Left-to-right recursive path with deg(a) = 0; deg(b) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	a⁷ ⇒ a	[7]
2.	aba⁶ ⇒ ab	[6]
3.	a²b ⇒ ba²	[10]
4.	b² ⇒ aba	[1]
5.	bab ⇒ a⁵	[11]

# ab:aba=bb,baaab=a a/b
aaaaaaa=a
abaaaaaa=ab
aab=baa
bb=aba
bab=aaaaa

Same cardinality

20 unique, 65 total

Σ	#	Presentation	Description	Related
8	526	⟨a, b \| aab=a, bbbb=1⟩	Finite non-commutative monoid with 20 elements	10 iso, 9 anti-iso
9	2207	⟨a, b \| ab=aa, bbbb=b⟩	Finite non-commutative monoid with 20 elements	1 anti-iso
10	4095	⟨a, b \| baa=abb, abab=1⟩	Finite non-Abelian group with 20 elements	5 iso
10	4212	⟨a, b \| aaaaa=1, abbbb=1⟩	Isomorphic to ℤ₂₀	16 iso
10	5349	⟨a, b \| aaa=bb, bab=aa⟩	Finite non-commutative monoid with 20 elements	1 iso
10	6728	⟨a, b \| aba=a, aaaa=bb⟩	Finite non-commutative monoid with 20 elements
10	6764	⟨a, b \| aba=b, aaaa=bb⟩	Finite non-commutative monoid with 20 elements
10	7117	⟨a, b \| ab=aa, bbbb=bb⟩	Finite non-commutative monoid with 20 elements
11	12159	⟨a, b \| aaaa=aa, abbb=b⟩	Finite non-commutative monoid with 20 elements
11	12322	⟨a, b \| aaab=bb, bbba=a⟩	Finite non-commutative monoid with 20 elements
11	14367	⟨a, b \| aaaa=a, bbbbb=a⟩	Isomorphic to ℕ_{(20 = 5)}
11	14407	⟨a, b \| aaaa=b, bbbbb=a⟩	Isomorphic to ℕ_{(20 = 1)}
11	14408	⟨a, b \| aaaa=b, bbbbb=b⟩	Isomorphic to ℕ_{(20 = 4)}
11	16124	⟨a, b \| aab=aa, baba=bb⟩	Finite non-commutative monoid with 20 elements
11	16339	⟨a, b \| aba=aa, aaaa=bb⟩	Finite non-commutative monoid with 20 elements
11	16343	⟨a, b \| aba=aa, aaab=bb⟩	Finite non-commutative monoid with 20 elements
11	16545	⟨a, b \| aba=bb, abb=aaa⟩	Finite non-commutative monoid with 20 elements
11	18619	⟨a, b \| aba=b, aaaaabb=1⟩	Finite non-Abelian group with 20 elements	3 iso
11	19503	⟨a, b \| aab=a, bbbbb=bb⟩	Finite non-commutative monoid with 20 elements
11	21047	⟨a, b \| ab=aa, bbbb=bbb⟩	Finite non-commutative monoid with 20 elements

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