#18727 ⟨a, b | aaa=a, ababba=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 23 elements
Not cancellative, because multiplication by a is not injective:
- a ⋅ a² = a and a ⋅ 1 = a, however a² ≠ 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.

3 element center:
- 1, a², b⁴
2 non-trivial idempotents:
- a², b⁴
Order of generators:
- a: index 1, period 2
- b: index 1, period 4

Histogram:

index 1, period 1	2 elements	a², b⁴
index 1, period 2	6 elements	a, b², (ab)², ab²a, (ba)², ...
index 1, period 4	10 elements	b, ab, ba, aba, b³, ...
index 1, period 5	4 elements	ab², bab, b²a, a(ba)²

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	ba	b²	aba	ab²	bab	b²a	b³	(ab)²	ab²a	ab³	(ba)²	b²ab	b⁴	a(ba)²	ab²ab	ab⁴	b(ab)²	(ab)³
1	1	a	b	a²	ab	ba	b²	aba	ab²	bab	b²a	b³	(ab)²	ab²a	ab³	(ba)²	b²ab	b⁴	a(ba)²	ab²ab	ab⁴	b(ab)²	(ab)³
a	a	a²	ab	a	b	aba	ab²	ba	b²	(ab)²	ab²a	ab³	bab	b²a	b³	a(ba)²	ab²ab	ab⁴	(ba)²	b²ab	b⁴	(ab)³	b(ab)²
b	b	ba	b²	b	bab	b²a	b³	(ba)²	aba	b²ab	(ab)³	b⁴	b(ab)²	ab	(ab)²	ab²ab	ab²a	b	ab³	ab²	ba	a(ba)²	ab⁴
a²	a²	a	b	a²	ab	ba	b²	aba	ab²	bab	b²a	b³	(ab)²	ab²a	ab³	(ba)²	b²ab	b⁴	a(ba)²	ab²ab	ab⁴	b(ab)²	(ab)³
ab	ab	aba	ab²	ab	(ab)²	ab²a	ab³	a(ba)²	ba	ab²ab	b(ab)²	ab⁴	(ab)³	b	bab	b²ab	b²a	ab	b³	b²	aba	(ba)²	b⁴
ba	ba	b	bab	ba	b²	(ba)²	aba	b²a	b³	b(ab)²	ab	(ab)²	b²ab	(ab)³	b⁴	ab³	ab²	ba	ab²ab	ab²a	b	ab⁴	a(ba)²
b²	b²	b²a	b³	b²	b²ab	(ab)³	b⁴	ab²ab	(ba)²	ab²a	ab⁴	b	a(ba)²	bab	b(ab)²	ab²	ab	b²	(ab)²	aba	b²a	ab³	ba
aba	aba	ab	(ab)²	aba	ab²	a(ba)²	ba	ab²a	ab³	(ab)³	b	bab	ab²ab	b(ab)²	ab⁴	b³	b²	aba	b²ab	b²a	ab	b⁴	(ba)²
ab²	ab²	ab²a	ab³	ab²	ab²ab	b(ab)²	ab⁴	b²ab	a(ba)²	b²a	b⁴	ab	(ba)²	(ab)²	(ab)³	b²	b	ab²	bab	ba	ab²a	b³	aba
bab	bab	(ba)²	aba	bab	b(ab)²	ab	(ab)²	ab³	b²a	ab²	a(ba)²	ba	ab⁴	b²	b²ab	ab²a	(ab)³	bab	b⁴	b³	(ba)²	ab²ab	b
b²a	b²a	b²	b²ab	b²a	b³	ab²ab	(ba)²	(ab)³	b⁴	a(ba)²	bab	b(ab)²	ab²a	ab⁴	b	(ab)²	aba	b²a	ab²	ab	b²	ba	ab³
b³	b³	(ab)³	b⁴	b³	ab²a	ab⁴	b	ab²	ab²ab	ab	ba	b²	ab³	b²ab	a(ba)²	aba	bab	b³	b(ab)²	(ba)²	(ab)³	(ab)²	b²a
(ab)²	(ab)²	a(ba)²	ba	(ab)²	(ab)³	b	bab	b³	ab²a	b²	(ba)²	aba	b⁴	ab²	ab²ab	b²a	b(ab)²	(ab)²	ab⁴	ab³	a(ba)²	b²ab	ab
ab²a	ab²a	ab²	ab²ab	ab²a	ab³	b²ab	a(ba)²	b(ab)²	ab⁴	(ba)²	(ab)²	(ab)³	b²a	b⁴	ab	bab	ba	ab²a	b²	b	ab²	aba	b³
ab³	ab³	b(ab)²	ab⁴	ab³	b²a	b⁴	ab	b²	b²ab	b	aba	ab²	b³	ab²ab	(ba)²	ba	(ab)²	ab³	(ab)³	a(ba)²	b(ab)²	bab	ab²a
(ba)²	(ba)²	bab	b(ab)²	(ba)²	aba	ab³	b²a	ab	(ab)²	ab⁴	b²	b²ab	ab²	a(ba)²	ba	b⁴	b³	(ba)²	ab²a	(ab)³	bab	b	ab²ab
b²ab	b²ab	ab²ab	(ba)²	b²ab	a(ba)²	bab	b(ab)²	(ab)²	(ab)³	aba	ab³	b²a	ba	b³	ab²a	ab	ab⁴	b²ab	b	b⁴	ab²ab	ab²	b²
b⁴	b⁴	ab⁴	b	b⁴	ab	ba	b²	aba	ab²	bab	b²a	b³	(ab)²	ab²a	ab³	(ba)²	b²ab	b⁴	a(ba)²	ab²ab	ab⁴	b(ab)²	(ab)³
a(ba)²	a(ba)²	(ab)²	(ab)³	a(ba)²	ba	b³	ab²a	b	bab	b⁴	ab²	ab²ab	b²	(ba)²	aba	ab⁴	ab³	a(ba)²	b²a	b(ab)²	(ab)²	ab	b²ab
ab²ab	ab²ab	b²ab	a(ba)²	ab²ab	(ba)²	(ab)²	(ab)³	bab	b(ab)²	ba	b³	ab²a	aba	ab³	b²a	b	b⁴	ab²ab	ab	ab⁴	b²ab	b²	ab²
ab⁴	ab⁴	b⁴	ab	ab⁴	b	aba	ab²	ba	b²	(ab)²	ab²a	ab³	bab	b²a	b³	a(ba)²	ab²ab	ab⁴	(ba)²	b²ab	b⁴	(ab)³	b(ab)²
b(ab)²	b(ab)²	ab³	b²a	b(ab)²	ab⁴	b²	b²ab	b⁴	ab	b³	ab²ab	(ba)²	b	aba	ab²	(ab)³	a(ba)²	b(ab)²	ba	(ab)²	ab³	ab²a	bab
(ab)³	(ab)³	b³	ab²a	(ab)³	b⁴	ab²	ab²ab	ab⁴	b	ab³	b²ab	a(ba)²	ab	ba	b²	b(ab)²	(ba)²	(ab)³	aba	bab	b³	b²a	(ab)²

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

#	Rule	Proof
1.	a³ ⇒ a	[1]
2.	a²b ⇒ b	[3]
3.	ba² ⇒ b	[4]
4.	bab² ⇒ aba	[6]
5.	b³a ⇒ (ab)³	[12]
6.	b(ba)² ⇒ ab²ab	[8]
7.	(ba)³ ⇒ ab³	[7]
8.	b⁵ ⇒ b	[11]

# ab:aaa=a,ababba=b reversed:a/b
aaa=a
aab=b
baa=b
babb=aba
bbba=ababab
bbaba=abbab
bababa=abbb
bbbbb=b

Same cardinality

1 unique, 1 total

Σ	#	Presentation	Description	Related
11	20300	⟨a, b \| aba=b, baab=aaa⟩	Finite non-commutative monoid with 23 elements

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