#17679 ⟨a, b | aaaa=1, babbb=ab⟩

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

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 244 elements
Not cancellative, because multiplication by b is not injective:
- b ⋅ b¹⁵ = b and b ⋅ 1 = b, however b¹⁵ ≠ 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
4 non-trivial idempotents:
- b¹⁵, a³b¹⁵a, a²b¹⁵a², ab¹⁵a³
Order of generators:
- a: order 4
- b: index 1, period 15

Histogram:

order 2	1 element	a²
order 4	2 elements	a, a³
index 1, period 1	4 elements	b¹⁵, a³b¹⁵a, a²b¹⁵a², ab¹⁵a³
index 1, period 2	20 elements	a²b³, ab³a, b³a², a²b⁶, ab⁶a, ...
index 1, period 3	8 elements	b⁵, a³b⁵a, a²b⁵a², ab⁵a³, b¹⁰, ...
index 1, period 4	40 elements	ab³, b³a, a³b³, a²b³a, ab³a², ...
index 1, period 5	16 elements	b³, b⁶, a³b³a, a²b³a², ab³a³, ...
index 1, period 6	40 elements	a²b, aba, ba², a²b², ab²a, ...
index 1, period 12	80 elements	ab, ba, ab², b²a, a³b, ...
index 1, period 15	32 elements	b, b², b⁴, a³ba, a²ba², ...

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.	b¹⁶ ⇒ b	[14]
2.	bab ⇒ ab¹⁴	[15]
3.	ba²b ⇒ a²b⁵	[7]
4.	ba³b ⇒ a³b⁸	[13]
5.	a⁴ ⇒ 1	[1]

# ab:aaaa=1,babbb=ab b/a
bbbbbbbbbbbbbbbb=b
bab=abbbbbbbbbbbbbb
baab=aabbbbb
baaab=aaabbbbbbbb
aaaa=1

Isomorphic instances

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

1 total

Σ	#	Presentation	Mapping
11	17683	⟨a, b \| aaaa=1, bbabb=ab⟩	φ(a) = aaa, φ(b) = b

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