#7531 ⟨a, b | aaa=1, abbbbb=b⟩

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 39 elements

Elements

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
3 non-trivial idempotents:
- b⁴a, b⁸a², b¹²
Order of generators:
- a: order 3
- b: index 1, period 12

Histogram:

order 3	2 elements	a, a²
index 1, period 1	3 elements	b⁴a, b⁸a², b¹²
index 1, period 2	3 elements	b²a², b⁶, b¹⁰a
index 1, period 3	6 elements	b⁴, b⁴a², b⁸, b⁸a, b¹²a, ...
index 1, period 4	6 elements	ba, b³, b⁵a², b⁷a, b⁹, ...
index 1, period 6	6 elements	b², b²a, b⁶a, b⁶a², b¹⁰, ...
index 1, period 12	12 elements	b, ba², b³a, b³a², b⁵, ...

Complete 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

b¹³ ⇒ b
ab ⇒ b⁹
a³ ⇒ 1

# ab:aaa=1,abbbbb=b b/a
bbbbbbbbbbbbb=b
ab=bbbbbbbbb
aaa=1

Right Cayley graph

Left Cayley graph

Others with same cardinality

1 unique, 1 total

Σ	#	Presentation	Description	Related
11	19302	⟨a, b \| aaa=a, bbbbb=ab⟩	Finite non-commutative monoid with 39 elements

Other isomorphic instances

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

4 total

Σ	#	Presentation	Mapping
10	7822	⟨a, b \| aaa=1, bbbbb=ab⟩	φ(a) = aa, φ(b) = b
11	21707	⟨a, b \| aaa=1, aabbbbb=b⟩	φ(a) = aa, φ(b) = b
11	21747	⟨a, b \| aaa=1, abbbbab=b⟩	φ(a) = aa, φ(b) = b
11	22859	⟨a, b \| aaa=1, bbbbb=aab⟩	φ(a) = a, φ(b) = b

Up:	Monoids with two generators and two relations
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Next:	#7537 ⟨a, b \| aaa=1, baabab=b⟩