#14519 ⟨a, b | aaab=b, babbb=a⟩

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

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 85 elements
Not cancellative, because multiplication by a is not injective:
- a ⋅ a³ = a and a ⋅ 1 = a, however a³ ≠ 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.

5 element center:
- 1, b⁷, b¹⁴, b²¹, b²⁸
1 non-trivial idempotent:
- b²⁸
Order of generators:
- a: index 1, period 3
- b: index 1, period 28

Histogram:

index 1, period 1	1 element	b²⁸
index 1, period 2	1 element	b¹⁴
index 1, period 3	14 elements	a, a², b⁴a, b⁴a², b⁸a, ...
index 1, period 4	2 elements	b⁷, b²¹
index 1, period 6	14 elements	b²a, b²a², b⁶a, b⁶a², b¹⁰a, ...
index 1, period 7	6 elements	b⁴, b⁸, b¹², b¹⁶, b²⁰, ...
index 1, period 12	28 elements	ba, ba², b³a, b³a², b⁵a, ...
index 1, period 14	6 elements	b², b⁶, b¹⁰, b¹⁸, b²², ...
index 1, period 28	12 elements	b, b³, b⁵, b⁹, b¹¹, ...

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

#	Rule	Proof
1.	b²⁹ ⇒ b	[13]
2.	b²⁸a ⇒ a	[11]
3.	ab ⇒ b⁹a	[10]
4.	a³ ⇒ b²⁸	[12]

# ab:aaab=b,babbb=a reversed:b/a
bbbbbbbbbbbbbbbbbbbbbbbbbbbbb=b
bbbbbbbbbbbbbbbbbbbbbbbbbbbba=a
ab=bbbbbbbbba
aaa=bbbbbbbbbbbbbbbbbbbbbbbbbbbb

Anti-isomorphic instances

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

1 total

Σ	#	Presentation	Mapping
11	14531	⟨a, b \| aaab=b, bbbab=a⟩	φ(a) = a, φ(b) = b

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