#3914 ⟨a, b | aabb=ba, bbbb=1⟩

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Properties
Elements
Right Cayley graph
Left Cayley graph
Rewriting system
Isomorphic instances
Anti-isomorphic instances

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 44 elements
Not cancellative, because multiplication by a is not injective:
- a ⋅ a⁴ = a and a ⋅ 1 = a, however a⁴ ≠ 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.

1 element center:
- 1
2 non-trivial idempotents:
- a⁴, (a³b)²
Order of generators:
- a: index 1, period 4
- b: order 4

Histogram:

order 2	1 element	b²
order 4	2 elements	b, b³
index 1, period 1	2 elements	a⁴, (a³b)²
index 1, period 2	10 elements	a², a³b, a²ba, aba², ba³, ...
index 1, period 4	20 elements	a, a³, a²b, aba, ba², ...
index 1, period 5	8 elements	ab, ba, (ab)², ba²b, a³ba², ...

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

#	Rule	Proof
1.	a⁵ ⇒ a	[19]
2.	a⁴ba ⇒ ba	[21]
3.	b²a ⇒ a²ba²	[4]
4.	(ba)² ⇒ aba⁴	[15]
5.	ba²ba ⇒ a³ba³	[25]
6.	ba³ba ⇒ a²	[22]
7.	ab² ⇒ a³ba	[24]
8.	b⁴ ⇒ 1	[2]

# ab:aabb=ba,bbbb=1 a/b
aaaaa=a
aaaaba=ba
bba=aabaa
baba=abaaaa
baaba=aaabaaa
baaaba=aa
abb=aaaba
bbbb=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	3992	⟨a, b \| babb=aa, bbbb=1⟩	φ(a) = a, φ(b) = b
10	4090	⟨a, b \| baa=abb, aaaa=1⟩	φ(a) = bbb, φ(b) = a
11	11090	⟨a, b \| babb=aba, bbbb=1⟩	φ(a) = aa, φ(b) = b

Anti-isomorphic instances

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

4 total

Σ	#	Presentation	Mapping
10	5707	⟨a, b \| aaaa=1, aabba=b⟩	φ(a) = bbb, φ(b) = a
11	11005	⟨a, b \| abab=bba, bbbb=1⟩	φ(a) = ab, φ(b) = bbb
11	17107	⟨a, b \| aaaa=1, aababa=b⟩	φ(a) = bbb, φ(b) = ab
11	17636	⟨a, b \| aaaa=1, aabab=ba⟩	φ(a) = b, φ(b) = aa

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Next:	#3931 ⟨a, b \| abab=aa, bbbb=1⟩