#19054 ⟨a, b | aab=b, babbbb=a⟩

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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 31 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.

6 element center:
- 1, b³, b⁶, b⁹, b¹², b¹⁵
1 non-trivial idempotent:
- b¹⁵
Order of generators:
- a: index 1, period 2
- b: index 1, period 15

Histogram:

index 1, period 1	1 element	b¹⁵
index 1, period 2	3 elements	a, ab⁵, ab¹⁰
index 1, period 3	2 elements	b⁵, b¹⁰
index 1, period 5	4 elements	b³, b⁶, b⁹, b¹²
index 1, period 10	12 elements	ab, ab², ab³, ab⁴, ab⁶, ...
index 1, period 15	8 elements	b, b², b⁴, b⁷, b⁸, ...

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	[11]
2.	ab¹⁵ ⇒ a	[12]
3.	ba ⇒ ab¹¹	[13]
4.	a² ⇒ b¹⁵	[10]

# ab:aab=b,babbbb=a b/a
bbbbbbbbbbbbbbbb=b
abbbbbbbbbbbbbbb=a
ba=abbbbbbbbbbb
aa=bbbbbbbbbbbbbbb

Isomorphic instances

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

1 total

Σ	#	Presentation	Mapping
11	19082	⟨a, b \| aab=b, bbbbab=a⟩	φ(a) = a, φ(b) = b

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