#12633 ⟨a, b | babb=aa, bbbb=b⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 79 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.

4 element center:
- 1, a⁷, a³ba²b², (aba)³
2 non-trivial idempotents:
- b³, a⁷
Order of generators:
- a: index 4, period 7
- b: index 1, period 3

Histogram:

index 1, period 1	2 elements	b³, a⁷
index 1, period 3	16 elements	b, b², a⁴b, a⁵b, a⁴b², ...
index 1, period 7	6 elements	a⁴, a⁵, a⁶, a⁸, a⁹, ...
index 2, period 1	2 elements	a³ba²b², (aba)³
index 2, period 3	22 elements	a³b, a²ba, aba², a³ba, a³b², ...
index 2, period 7	15 elements	a², a³, ab³a, ba²b², b²a²b, ...
index 3, period 7	2 elements	ab³, b³a
index 4, period 3	12 elements	ab, ba, a²b, aba, ab², ...
index 4, period 7	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:
- Right-to-left recursive path with deg(a) = 0; deg(b) = 1

a¹¹ ⇒ a⁴
a⁴ba ⇒ a⁶b
ba³ ⇒ a²ba²
bab ⇒ a²b²
a³b²a ⇒ a(a²b)²
a(a²b)²a ⇒ a⁴b²
a²b²a² ⇒ a³b²
a²(ba²)² ⇒ a(a²b)²
a²b³ ⇒ a²
(a²b)²ba ⇒ a³ba²b²
b³a² ⇒ a²
b(ba²)² ⇒ a³
b⁴ ⇒ b

# ab:babb=aa,bbbb=b reversed:a/b
aaaaaaaaaaa=aaaa
aaaaba=aaaaaab
baaa=aabaa
bab=aabb
aaabba=aaabaab
aaabaaba=aaaabb
aabbaa=aaabb
aabaabaa=aaabaab
aabbb=aa
aabaabba=aaabaabb
bbbaa=aa
bbaabaa=aaa
bbbb=b

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#12633 ⟨a, b | babb=aa, bbbb=b⟩

Properties

Elements

Complete rewriting system

Right Cayley graph

Left Cayley graph