#16053 ⟨a, b | aaa=bb, aaba=ab⟩

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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 19 elements
Not cancellative, because multiplication by a⁴ is not injective:
- a⁴ ⋅ a⁶ = a⁴ and a⁴ ⋅ 1 = a⁴, 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, a³, a⁶, a⁹
1 non-trivial idempotent:
- a⁶
Order of generators:
- a: index 4, period 6
- b: index 3, period 4

Histogram:

index 1, period 1	1 element	a⁶
index 1, period 2	1 element	a⁹
index 1, period 3	2 elements	a⁴, a⁸
index 1, period 4	6 elements	ba³, ba⁴, ba⁵, ba⁶, ba⁷, ...
index 1, period 6	2 elements	a⁵, a⁷
index 2, period 2	1 element	a³
index 2, period 3	1 element	a²
index 2, period 4	2 elements	ba, ba²
index 3, period 4	1 element	b
index 4, period 6	1 element	a

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸
1	1	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸
a	a	a²	ba⁵	a³	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a⁴	ba⁷
b	b	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a⁴	ba³	a⁵
a²	a²	a³	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba³	a⁹	ba⁴	a⁴	ba⁵	a⁵	ba⁶
ba	ba	ba²	a⁸	ba³	a⁹	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba³	a⁹	ba⁴	a⁴
a³	a³	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵
ba²	ba²	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹
a⁴	a⁴	a⁵	ba⁸	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba³	a⁷	ba⁴
ba³	ba³	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸
a⁵	a⁵	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba³	a⁹	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba³
ba⁴	ba⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba³	a⁹	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷
a⁶	a⁶	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸
ba⁵	ba⁵	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶
a⁷	a⁷	a⁸	ba⁵	a⁹	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a⁴	ba⁷
ba⁶	ba⁶	ba⁷	a⁹	ba⁸	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a⁴	ba³	a⁵
a⁸	a⁸	a⁹	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba³	a⁹	ba⁴	a⁴	ba⁵	a⁵	ba⁶
ba⁷	ba⁷	ba⁸	a⁸	ba³	a⁹	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba³	a⁹	ba⁴	a⁴
a⁹	a⁹	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵
ba⁸	ba⁸	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹

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.	a¹⁰ ⇒ a⁴	[6]
2.	ba⁹ ⇒ ba³	[7]
3.	ab ⇒ ba⁵	[5]
4.	b² ⇒ a³	[1]

# ab:aaa=bb,aaba=ab a/b
aaaaaaaaaa=aaaa
baaaaaaaaa=baaa
ab=baaaaa
bb=aaa

Same cardinality

5 unique, 29 total

Σ	#	Presentation	Description	Related
10	6263	⟨a, b \| aaa=a, abbba=b⟩	Finite non-commutative monoid with 19 elements
10	6838	⟨a, b \| aba=b, bab=aaa⟩	Finite non-commutative monoid with 19 elements
11	13340	⟨a, b \| aaaab=1, abbbbb=1⟩	Isomorphic to ℤ₁₉	20 iso
11	14501	⟨a, b \| aaab=b, abbba=a⟩	Finite non-commutative monoid with 19 elements	3 iso
11	16036	⟨a, b \| aaa=ab, bbab=bb⟩	Finite non-commutative monoid with 19 elements	1 iso

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