#4594 ⟨a, b | abba=b, aaaaa=1⟩

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Properties
Elements
Right Cayley graph
Left Cayley graph
Rewriting system
Isomorphic instances

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 605 elements
Not cancellative, because multiplication by b is not injective:
- b ⋅ b³⁰ = b and b ⋅ 1 = b, however b³⁰ ≠ 1

Elements in the center commute with all other elements.
An idempotent element x satisfies x² = x.
The order of x is the least n (if it exists) such that xⁿ = 1.
The index and period of x is the least m (index) and n (period) such that x^(m+n) = x^m.

11 element center:
- 1, b³, b⁶, b⁹, b¹², b¹⁵, b¹⁸, b²¹, b²⁴, b²⁷, b³⁰
1 non-trivial idempotent:
- b³⁰
Order of generators:
- a: order 5
- b: index 1, period 30

Histogram:

order 5	4 elements	a, a², a³, a⁴
index 1, period 1	1 element	b³⁰
index 1, period 2	1 element	b¹⁵
index 1, period 3	20 elements	b¹⁰, a²b¹¹, ba²b¹⁰, b²a²b⁹, a²bab¹³, ...
index 1, period 4	30 elements	ab⁸, bab⁷, b²ab⁶, (a²b)²ab³, a²ba²b⁶, ...
index 1, period 5	124 elements	ab, ba, (ab)², (ba)², (a²b)², ...
index 1, period 6	20 elements	b⁵, a²bab³, aba²b³, ba²bab², baba²b², ...
index 1, period 10	124 elements	a²b, aba, ba², b³, ab³, ...
index 1, period 15	80 elements	b², a²ba, aba², b⁴, a(ab)², ...
index 1, period 20	120 elements	ab², bab, b²a, a²ba², a²bab², ...
index 1, period 30	80 elements	b, a²b², ba²b, b²a², b⁷, ...

Right Cayley graph

Left Cayley graph

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(b) = 0; deg(a) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	b³¹ ⇒ b	[21]
2.	bab³⁰ ⇒ ba	[22]
3.	b³a ⇒ ab³	[3]
4.	ab²a ⇒ b	[1]
5.	ba²b³⁰ ⇒ ba²	[23]
6.	a³b ⇒ b(ba)²	[6]
7.	a(ba)² ⇒ ba²b²⁸	[24]
8.	ba³ ⇒ abab²	[7]
9.	(ba²)² ⇒ (aba)²b⁵	[10]
10.	b²(aba)² ⇒ a²ba²b²⁸	[17]
11.	a⁵ ⇒ 1	[2]

# ab:abba=b,aaaaa=1 b/a
bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=b
babbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=ba
bbba=abbb
abba=b
baabbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=baa
aaab=bbaba
ababa=baabbbbbbbbbbbbbbbbbbbbbbbbbbbb
baaa=ababb
baabaa=abaababbbbb
bbabaaba=aabaabbbbbbbbbbbbbbbbbbbbbbbbbbbb
aaaaa=1

Isomorphic instances

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

3 total

Σ	#	Presentation	Mapping
10	5002	⟨a, b \| aba=bb, aaaaa=1⟩	φ(a) = aaaa, φ(b) = b
11	11612	⟨a, b \| babab=a, bbbbb=1⟩	φ(a) = abb, φ(b) = a
11	12860	⟨a, b \| bab=aba, aaaaa=1⟩	φ(a) = aaaa, φ(b) = ab

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