#9739 ⟨a, b | aa=1, ababbbb=b⟩

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 34 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
2 non-trivial idempotents:
- b⁸, ab⁸a
Order of generators:
- a: order 2
- b: index 1, period 8

Histogram:

order 2	1 element	a
index 1, period 1	2 elements	b⁸, ab⁸a
index 1, period 2	6 elements	b⁴, ab⁴, b⁴a, ab⁴a, ab⁸, ...
index 1, period 4	8 elements	b², ab², b²a, ab²a, b⁶, ...
index 1, period 8	16 elements	b, ab, ba, aba, b³, ...

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

b⁹ ⇒ b
bab ⇒ ab⁶
a² ⇒ 1

# ab:aa=1,ababbbb=b b/a
bbbbbbbbb=b
bab=abbbbbb
aa=1

Right Cayley graph

Left Cayley graph

Others with same cardinality

1 unique, 1 total

Σ	#	Presentation	Description	Related
11	15757	⟨a, b \| aab=bb, ababa=a⟩	Finite non-commutative monoid with 34 elements

Other isomorphic instances

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

2 total

Σ	#	Presentation	Mapping
10	9753	⟨a, b \| aa=1, abbbabb=b⟩	φ(a) = a, φ(b) = b
10	10051	⟨a, b \| aa=1, babbbb=ab⟩	φ(a) = a, φ(b) = b

Other anti-isomorphic instances

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

1 total

Σ	#	Presentation	Mapping
10	10059	⟨a, b \| aa=1, bbabbb=ba⟩	φ(a) = a, φ(b) = b

Up:	Monoids with two generators and two relations
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