#27686 ⟨a, b | aa=1, abbbba=bab⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 500 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.

10 element center:
- 1, b⁵, b¹⁰, b¹⁵, b²⁰, b²⁵, b³⁰, b³⁵, b⁴⁰, b⁴⁵
1 non-trivial idempotent:
- b⁴⁰
Order of generators:
- a: order 2
- b: index 6, period 40

Histogram:

order 2	1 element	a
index 1, period 1	1 element	b⁴⁰
index 1, period 2	31 elements	ab²(ab³)², ab²ab³ab⁷, ab²ab³ab¹¹, ab²ab¹⁶, b²⁰, ...
index 1, period 3	20 elements	ab¹⁴, bab¹³, b²ab¹², b³ab¹¹, b⁴ab¹⁰, ...
index 1, period 4	32 elements	a(b²ab)², ab²ab⁶, b¹⁰, ab¹⁰, bab⁹, ...
index 1, period 5	24 elements	(ab²)², b⁸, ab³ab⁵, bab³ab⁴, b²(ab³)², ...
index 1, period 6	20 elements	ab³ab, bab³a, ab⁶, bab⁵, b²ab⁴, ...
index 1, period 8	64 elements	ab²ab, bab⁴, b²ab³, b³ab², b⁴ab, ...
index 1, period 10	24 elements	b¹², ab²ab¹⁰, ab³ab¹³, bab³ab¹², b²ab³ab¹¹, ...
index 1, period 12	40 elements	bab³, b²ab², b³ab, ab²ab⁴, ab²ab⁸, ...
index 1, period 20	48 elements	b⁶, (ab³)², (bab²)², (b²ab)², (b³a)², ...
index 1, period 24	80 elements	ab²ab³, ab³ab², bab³ab, b²ab³a, ab⁹, ...
index 1, period 40	96 elements	bab², b²ab, b⁷, ab⁷, bab⁶, ...
index 2, period 8	2 elements	b⁵, ab⁵
index 2, period 10	4 elements	ab², bab, b²a, b⁴
index 2, period 12	2 elements	ab⁴, b⁴a
index 2, period 40	4 elements	b³, ab³, b³a, ab³a
index 3, period 20	2 elements	b², ab²a
index 4, period 24	2 elements	ab, ba
index 6, period 40	2 elements	b, aba

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

b⁴⁶ ⇒ b⁶
bab⁴² ⇒ bab²
b²ab⁴¹ ⇒ b²ab
b⁵a ⇒ ab⁵
a² ⇒ 1
(ab)² ⇒ b⁴a
ab⁴a ⇒ bab
(ba)² ⇒ ab⁴
bab²a ⇒ ab³ab⁴
bab³ab⁴⁰ ⇒ bab³a
b(b³a)² ⇒ ab²ab
a(b³a)² ⇒ bab³ab³⁶

# ab:aa=1,abbbba=bab b/a
bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=bbbbbb
babbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=babb
bbabbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=bbab
bbbbba=abbbbb
aa=1
abab=bbbba
abbbba=bab
baba=abbbb
babba=abbbabbbb
babbbabbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=babbba
bbbbabbba=abbab
abbbabbba=babbbabbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

Right Cayley graph

Left Cayley graph

Other isomorphic instances

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

2 total

Σ	#	Presentation	Mapping
11	28165	⟨a, b \| aa=1, ababa=bbbb⟩	φ(a) = a, φ(b) = b
11	28218	⟨a, b \| aa=1, abbbb=baba⟩	φ(a) = a, φ(b) = b

Up:	Monoids with two generators and two relations
Prev:	#27679 ⟨a, b \| aa=1, abbbab=bab⟩
Next:	#27687 ⟨a, b \| aa=1, abbbba=bbb⟩