#13091 ⟨a, b | bab=aaa, abba=b⟩

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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 21 elements
Not cancellative, because multiplication by a³ is not injective:
- a³ ⋅ a⁶ = a³ and a³ ⋅ 1 = a³, 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.

5 element center:
- 1, a², a⁴, a⁶, a⁸
1 non-trivial idempotent:
- a⁶
Order of generators:
- a: index 3, period 6
- b: index 1, period 3

Histogram:

index 1, period 1	1 element	a⁶
index 1, period 2	3 elements	a³, aba⁴, ba⁵
index 1, period 3	8 elements	b, aba, ba², a⁴, aba³, ...
index 1, period 6	6 elements	ab, ba, aba², ba³, a⁵, ...
index 2, period 3	1 element	a²
index 3, period 6	1 element	a

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	a⁷	aba⁵	a⁸
1	1	a	b	a²	ab	ba	a³	aba	ba²	a⁴	aba²	ba³	a⁵	aba³	ba⁴	a⁶	aba⁴	ba⁵	a⁷	aba⁵	a⁸
a	a	a²	ab	a³	ba²	aba	a⁴	ba³	aba²	a⁵	ba⁴	aba³	a⁶	ba⁵	aba⁴	a⁷	b	aba⁵	a⁸	ba	a³
b	b	ba	aba³	ba²	a³	aba⁴	ba³	a⁴	aba⁵	ba⁴	a⁵	ab	ba⁵	a⁶	aba	b	a⁷	aba²	ba	a⁸	ba²
a²	a²	a³	ba²	a⁴	aba²	ba³	a⁵	aba³	ba⁴	a⁶	aba⁴	ba⁵	a⁷	aba⁵	b	a⁸	ab	ba	a³	aba	a⁴
ab	ab	aba	ba⁵	aba²	a⁴	b	aba³	a⁵	ba	aba⁴	a⁶	ba²	aba⁵	a⁷	ba³	ab	a⁸	ba⁴	aba	a³	aba²
ba	ba	ba²	a³	ba³	aba⁵	a⁴	ba⁴	ab	a⁵	ba⁵	aba	a⁶	b	aba²	a⁷	ba	aba³	a⁸	ba²	aba⁴	ba³
a³	a³	a⁴	aba²	a⁵	ba⁴	aba³	a⁶	ba⁵	aba⁴	a⁷	b	aba⁵	a⁸	ba	ab	a³	ba²	aba	a⁴	ba³	a⁵
aba	aba	aba²	a⁴	aba³	ba	a⁵	aba⁴	ba²	a⁶	aba⁵	ba³	a⁷	ab	ba⁴	a⁸	aba	ba⁵	a³	aba²	b	aba³
ba²	ba²	ba³	aba⁵	ba⁴	a⁵	ab	ba⁵	a⁶	aba	b	a⁷	aba²	ba	a⁸	aba³	ba²	a³	aba⁴	ba³	a⁴	ba⁴
a⁴	a⁴	a⁵	ba⁴	a⁶	aba⁴	ba⁵	a⁷	aba⁵	b	a⁸	ab	ba	a³	aba	ba²	a⁴	aba²	ba³	a⁵	aba³	a⁶
aba²	aba²	aba³	ba	aba⁴	a⁶	ba²	aba⁵	a⁷	ba³	ab	a⁸	ba⁴	aba	a³	ba⁵	aba²	a⁴	b	aba³	a⁵	aba⁴
ba³	ba³	ba⁴	a⁵	ba⁵	aba	a⁶	b	aba²	a⁷	ba	aba³	a⁸	ba²	aba⁴	a³	ba³	aba⁵	a⁴	ba⁴	ab	ba⁵
a⁵	a⁵	a⁶	aba⁴	a⁷	b	aba⁵	a⁸	ba	ab	a³	ba²	aba	a⁴	ba³	aba²	a⁵	ba⁴	aba³	a⁶	ba⁵	a⁷
aba³	aba³	aba⁴	a⁶	aba⁵	ba³	a⁷	ab	ba⁴	a⁸	aba	ba⁵	a³	aba²	b	a⁴	aba³	ba	a⁵	aba⁴	ba²	aba⁵
ba⁴	ba⁴	ba⁵	aba	b	a⁷	aba²	ba	a⁸	aba³	ba²	a³	aba⁴	ba³	a⁴	aba⁵	ba⁴	a⁵	ab	ba⁵	a⁶	b
a⁶	a⁶	a⁷	b	a⁸	ab	ba	a³	aba	ba²	a⁴	aba²	ba³	a⁵	aba³	ba⁴	a⁶	aba⁴	ba⁵	a⁷	aba⁵	a⁸
aba⁴	aba⁴	aba⁵	ba³	ab	a⁸	ba⁴	aba	a³	ba⁵	aba²	a⁴	b	aba³	a⁵	ba	aba⁴	a⁶	ba²	aba⁵	a⁷	ab
ba⁵	ba⁵	b	a⁷	ba	aba³	a⁸	ba²	aba⁴	a³	ba³	aba⁵	a⁴	ba⁴	ab	a⁵	ba⁵	aba	a⁶	b	aba²	ba
a⁷	a⁷	a⁸	ab	a³	ba²	aba	a⁴	ba³	aba²	a⁵	ba⁴	aba³	a⁶	ba⁵	aba⁴	a⁷	b	aba⁵	a⁸	ba	a³
aba⁵	aba⁵	ab	a⁸	aba	ba⁵	a³	aba²	b	a⁴	aba³	ba	a⁵	aba⁴	ba²	a⁶	aba⁵	ba³	a⁷	ab	ba⁴	aba
a⁸	a⁸	a³	ba²	a⁴	aba²	ba³	a⁵	aba³	ba⁴	a⁶	aba⁴	ba⁵	a⁷	aba⁵	b	a⁸	ab	ba	a³	aba	a⁴

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.	a⁹ ⇒ a³	[16]
2.	ba⁶ ⇒ b	[15]
3.	a²b ⇒ ba²	[17]
4.	b² ⇒ aba³	[6]
5.	bab ⇒ a³	[1]

# ab:bab=aaa,abba=b a/b
aaaaaaaaa=aaa
baaaaaa=b
aab=baa
bb=abaaa
bab=aaa

Same cardinality

8 unique, 64 total

Σ	#	Presentation	Description	Related
8	402	⟨a, b \| aaa=ab, bbb=1⟩	Finite non-commutative monoid with 21 elements	12 iso, 27 anti-iso
9	1599	⟨a, b \| aaa=ab, bbb=b⟩	Finite non-commutative monoid with 21 elements	1 anti-iso
10	5309	⟨a, b \| aaa=aa, bbb=ab⟩	Finite non-commutative monoid with 21 elements
10	6265	⟨a, b \| aaa=a, abbbb=b⟩	Finite non-commutative monoid with 21 elements	2 iso
11	11127	⟨a, b \| aaaaa=b, abbbb=1⟩	Isomorphic to ℤ₂₁	13 iso
11	13248	⟨a, b \| bab=aaa, bbb=aa⟩	Finite non-commutative monoid with 21 elements
11	15158	⟨a, b \| aab=ba, ababab=1⟩	Finite non-Abelian group with 21 elements	1 iso
11	19212	⟨a, b \| aba=b, baaaab=a⟩	Finite non-commutative monoid with 21 elements

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