#26619 ⟨a, b | aa=1, abbabbba=b⟩

Properties
Finite structure
Right Cayley graph
Left Cayley graph
Rewriting system
Isomorphic instances
Anti-isomorphic instances

Properties

Sum of relation sides is 11
Finite non-commutative monoid with 74 elements
Not cancellative, because multiplication by b is not injective:
- b ⋅ b⁸ = b and b ⋅ 1 = b, however b⁸ ≠ 1

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, b⁸
1 non-trivial idempotent:
- b⁸
Order of generators:
- a: order 2
- b: index 1, period 8

Histogram:

order 2	1 element	a
index 1, period 1	1 element	b⁸
index 1, period 2	9 elements	b⁴, abab³, (ab²)², ab³ab, (ba)²b², ...
index 1, period 3	8 elements	ab⁴, bab³, b²ab², b³ab, (ab)³b, ...
index 1, period 4	18 elements	b², ab², bab, b²a, (ab)², ...
index 1, period 8	36 elements	b, ab, ba, aba, b³, ...

Right Cayley graph

Left Cayley graph

Rewriting system

Format:	Pretty Plain
Word:
	Enter a word above to compute its normal form. Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Strategy:	Leftmost Rightmost
Result:	1 1

Reduction order:
- Right-to-left recursive path with deg(a) = 0; deg(b) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	a² ⇒ 1	[1]
2.	bab²a ⇒ abab²	[12]
3.	b⁴a ⇒ ab²(ab)²	[10]
4.	b³aba ⇒ (ab)⁴	[21]
5.	b(ba)³ ⇒ ab⁴	[5]
6.	(ba)⁴ ⇒ ab³ab	[7]
7.	b²ab³ ⇒ aba	[4]
8.	(ba)²b³a ⇒ (ab)³b²	[19]
9.	bab⁵ ⇒ ab²a	[17]
10.	b(ba)²b⁴ ⇒ ab(ba)²	[8]
11.	b⁹ ⇒ b	[18]

# ab:aa=1,abbabbba=b reversed:a/b - -
aa=1
babba=ababb
bbbba=abbabab
bbbaba=abababab
bbababa=abbbb
babababa=abbbab
bbabbb=aba
bababbba=abababbb
babbbbb=abba
bbababbbb=abbaba
bbbbbbbbb=b

Isomorphic instances

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

2 total

Σ	#	Presentation	Mapping
11	27147	⟨a, b \| aa=1, abbabbb=ba⟩	φ(a) = a, φ(b) = b
11	27756	⟨a, b \| aa=1, bbabbb=aba⟩	φ(a) = a, φ(b) = b

Anti-isomorphic instances