#13119 ⟨a, b | bbb=aaa, abba=b⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 75 elements

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.

7 element center:
- 1, a³, a⁶, a⁹, a¹², a¹⁵, a¹⁸
1 non-trivial idempotent:
- a¹⁸
Order of generators:
- a: index 3, period 18
- b: index 1, period 18

Histogram:

index 1, period 1	1 element	a¹⁸
index 1, period 2	1 element	a⁹
index 1, period 3	2 elements	a⁶, a¹²
index 1, period 4	6 elements	a²b, aba, ba², a²ba⁹, aba¹⁰, ...
index 1, period 6	2 elements	a³, a¹⁵
index 1, period 9	24 elements	a⁴, a²ba, aba², ba³, a²ba², ...
index 1, period 12	12 elements	a²ba³, aba⁴, ba⁵, a²ba⁶, aba⁷, ...
index 1, period 18	24 elements	b, ab, ba, a⁵, a⁷, ...
index 2, period 9	1 element	a²
index 3, period 18	1 element	a

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(a) = 0; deg(b) = 1

a²¹ ⇒ a³
ba¹⁸ ⇒ b
a³b ⇒ ba³
b² ⇒ a²ba¹⁴
bab ⇒ aba⁷
ba²b ⇒ a⁷

# ab:bbb=aaa,abba=b a/b
aaaaaaaaaaaaaaaaaaaaa=aaa
baaaaaaaaaaaaaaaaaa=b
aaab=baaa
bb=aabaaaaaaaaaaaaaa
bab=abaaaaaaa
baab=aaaaaaa

Right Cayley graph

Left Cayley graph

Others with same cardinality

1 unique, 1 total

Σ	#	Presentation	Description	Related
11	16549	⟨a, b \| aba=bb, bab=aaa⟩	Finite non-commutative monoid with 75 elements

Other isomorphic instances

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

1 total

Σ	#	Presentation	Mapping
11	15524	⟨a, b \| aaa=bb, ababa=b⟩	φ(a) = a, φ(b) = aabaaaaaaaaaaaaaaa

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