#15757 ⟨a, b | aab=bb, ababa=a⟩

Properties
Finite structure
Right Cayley graph
Left Cayley graph
Rewriting system
Same cardinality

Properties

Sum of relation sides is 11
Finite non-commutative monoid with 34 elements
Not cancellative, because multiplication by a is not injective:
- a ⋅ (ba)² = a and a ⋅ 1 = a, however (ba)² ≠ 1

Finite structure

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.

1 element center:
- 1
6 non-trivial idempotents:
- (ab)², (ba)², a⁴b, a³ba, a⁶, a⁷bab
Order of generators:
- a: index 2, period 6
- b: index 2, period 3

Histogram:

index 1, period 1	6 elements	(ab)², (ba)², a⁴b, a³ba, a⁶, ...
index 1, period 2	6 elements	ab, ba, a³, a⁴bab, a⁷b, ...
index 1, period 3	8 elements	a², a²b, a⁴, a³bab, a⁶b, ...
index 1, period 6	8 elements	a³b, a²ba, a⁵, a(ab)², a⁵b, ...
index 2, period 3	3 elements	b, aba, b(ab)²
index 2, period 6	2 elements	a, bab

Right Cayley graph

Left Cayley graph

Rewriting system

Format:	Pretty Plain
Word:
	Enter a word above to compute its normal form. Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Strategy:	Leftmost Rightmost
Result:	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.	a⁸ ⇒ a²	[5]
2.	ba² ⇒ a⁴	[4]
3.	b² ⇒ a²b	[1]
4.	a(ba)² ⇒ a	[2]

# ab:aab=bb,ababa=a a/b - -
aaaaaaaa=aa
baa=aaaa
bb=aab
ababa=a

Same cardinality

1 unique, 4 total

Σ	#	Presentation	Properties	φ
10	9739	⟨a, b \| aa=1, ababbbb=b⟩	Noc 34	3

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