#22279 ⟨a, b | aaa=1, abbabb=ba⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 174 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.

2 element center:
- 1, b⁹
1 non-trivial idempotent:
- b⁹
Order of generators:
- a: order 3
- b: index 1, period 9

Histogram:

order 3	2 elements	a, a²
index 1, period 1	1 element	b⁹
index 1, period 3	38 elements	b³, ab³a, b³a², b⁶, ab⁶, ...
index 1, period 9	114 elements	b, ab, ba, b², ab², ...
index 1, period 19	18 elements	ab³, bab², b²ab, b³a, ab⁶a, ...

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

b¹⁰ ⇒ b
b⁹ab ⇒ ab
ab⁹ ⇒ b⁹a
aba ⇒ b⁴ab³
ab²a ⇒ bab⁷
ab⁴a ⇒ b²ab⁸
ab⁵a ⇒ b⁶ab⁵
a²b ⇒ b³ab⁷a
ab³ab ⇒ b(b⁶a)²
ab⁷ab ⇒ b⁵ab³a
ab⁶ab³ ⇒ bab⁸a
ab⁸ab³ ⇒ b²a²
a³ ⇒ 1
ab³a² ⇒ b⁵ab
ab⁶a² ⇒ (b⁵ab)²
ab⁷a² ⇒ b⁸ab²
ab⁸a² ⇒ b⁷ab⁴

# ab:aaa=1,abbabb=ba reversed:b/a
bbbbbbbbbb=b
bbbbbbbbbab=ab
abbbbbbbbb=bbbbbbbbba
aba=bbbbabbb
abba=babbbbbbb
abbbba=bbabbbbbbbb
abbbbba=bbbbbbabbbbb
aab=bbbabbbbbbba
abbbab=bbbbbbbabbbbbba
abbbbbbbab=bbbbbabbba
abbbbbbabbb=babbbbbbbba
abbbbbbbbabbb=bbaa
aaa=1
abbbaa=bbbbbab
abbbbbbaa=bbbbbabbbbbbab
abbbbbbbaa=bbbbbbbbabb
abbbbbbbbaa=bbbbbbbabbbb

Up:	Monoids with two generators and two relations
Prev:	#22276 ⟨a, b \| aaa=1, abbaab=bb⟩
Next:	#22283 ⟨a, b \| aaa=1, abbbab=ba⟩

#22279 ⟨a, b | aaa=1, abbabb=ba⟩

Properties

Elements

Complete rewriting system

Right Cayley graph

Left Cayley graph