#9759 ⟨a, b | aa=1, abbbbbb=b⟩

Properties
Finite structure
Cayley table
Right Cayley graph
Left Cayley graph
Rewriting system
Same cardinality
Isomorphic instances

Properties

Sum of relation sides is 10
Finite non-commutative monoid with 22 elements
Not cancellative, because multiplication by b is not injective:
- b ⋅ ab⁵ = b and b ⋅ 1 = b, however ab⁵ ≠ 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.

1 element center:
- 1
2 non-trivial idempotents:
- ab⁵, b⁵a
Order of generators:
- a: order 2
- b: index 1, period 10

Histogram:

order 2	1 element	a
index 1, period 1	2 elements	ab⁵, b⁵a
index 1, period 2	2 elements	b⁵, ab⁵a
index 1, period 5	8 elements	ab, ba, b², ab²a, ab³, ...
index 1, period 10	8 elements	b, aba, ab², b²a, b³, ...

Cayley table

Idempotents are shown in bold.

	1	a	b	ab	ba	b²	aba	ab²	b²a	b³	ab²a	ab³	b³a	b⁴	ab³a	ab⁴	b⁴a	b⁵	ab⁴a	ab⁵	b⁵a	ab⁵a
1	1	a	b	ab	ba	b²	aba	ab²	b²a	b³	ab²a	ab³	b³a	b⁴	ab³a	ab⁴	b⁴a	b⁵	ab⁴a	ab⁵	b⁵a	ab⁵a
a	a	1	ab	b	aba	ab²	ba	b²	ab²a	ab³	b²a	b³	ab³a	ab⁴	b³a	b⁴	ab⁴a	ab⁵	b⁴a	b⁵	ab⁵a	b⁵a
b	b	ba	b²	ab²	b²a	b³	ab²a	ab³	b³a	b⁴	ab³a	ab⁴	b⁴a	b⁵	ab⁴a	ab⁵	b⁵a	ab	ab⁵a	b	aba	ba
ab	ab	aba	ab²	b²	ab²a	ab³	b²a	b³	ab³a	ab⁴	b³a	b⁴	ab⁴a	ab⁵	b⁴a	b⁵	ab⁵a	b	b⁵a	ab	ba	aba
ba	ba	b	ab²	b²	ab²a	ab³	b²a	b³	ab³a	ab⁴	b³a	b⁴	ab⁴a	ab⁵	b⁴a	b⁵	ab⁵a	b	b⁵a	ab	ba	aba
b²	b²	b²a	b³	ab³	b³a	b⁴	ab³a	ab⁴	b⁴a	b⁵	ab⁴a	ab⁵	b⁵a	ab	ab⁵a	b	aba	ab²	ba	b²	ab²a	b²a
aba	aba	ab	b²	ab²	b²a	b³	ab²a	ab³	b³a	b⁴	ab³a	ab⁴	b⁴a	b⁵	ab⁴a	ab⁵	b⁵a	ab	ab⁵a	b	aba	ba
ab²	ab²	ab²a	ab³	b³	ab³a	ab⁴	b³a	b⁴	ab⁴a	ab⁵	b⁴a	b⁵	ab⁵a	b	b⁵a	ab	ba	b²	aba	ab²	b²a	ab²a
b²a	b²a	b²	ab³	b³	ab³a	ab⁴	b³a	b⁴	ab⁴a	ab⁵	b⁴a	b⁵	ab⁵a	b	b⁵a	ab	ba	b²	aba	ab²	b²a	ab²a
b³	b³	b³a	b⁴	ab⁴	b⁴a	b⁵	ab⁴a	ab⁵	b⁵a	ab	ab⁵a	b	aba	ab²	ba	b²	ab²a	ab³	b²a	b³	ab³a	b³a
ab²a	ab²a	ab²	b³	ab³	b³a	b⁴	ab³a	ab⁴	b⁴a	b⁵	ab⁴a	ab⁵	b⁵a	ab	ab⁵a	b	aba	ab²	ba	b²	ab²a	b²a
ab³	ab³	ab³a	ab⁴	b⁴	ab⁴a	ab⁵	b⁴a	b⁵	ab⁵a	b	b⁵a	ab	ba	b²	aba	ab²	b²a	b³	ab²a	ab³	b³a	ab³a
b³a	b³a	b³	ab⁴	b⁴	ab⁴a	ab⁵	b⁴a	b⁵	ab⁵a	b	b⁵a	ab	ba	b²	aba	ab²	b²a	b³	ab²a	ab³	b³a	ab³a
b⁴	b⁴	b⁴a	b⁵	ab⁵	b⁵a	ab	ab⁵a	b	aba	ab²	ba	b²	ab²a	ab³	b²a	b³	ab³a	ab⁴	b³a	b⁴	ab⁴a	b⁴a
ab³a	ab³a	ab³	b⁴	ab⁴	b⁴a	b⁵	ab⁴a	ab⁵	b⁵a	ab	ab⁵a	b	aba	ab²	ba	b²	ab²a	ab³	b²a	b³	ab³a	b³a
ab⁴	ab⁴	ab⁴a	ab⁵	b⁵	ab⁵a	b	b⁵a	ab	ba	b²	aba	ab²	b²a	b³	ab²a	ab³	b³a	b⁴	ab³a	ab⁴	b⁴a	ab⁴a
b⁴a	b⁴a	b⁴	ab⁵	b⁵	ab⁵a	b	b⁵a	ab	ba	b²	aba	ab²	b²a	b³	ab²a	ab³	b³a	b⁴	ab³a	ab⁴	b⁴a	ab⁴a
b⁵	b⁵	b⁵a	ab	b	aba	ab²	ba	b²	ab²a	ab³	b²a	b³	ab³a	ab⁴	b³a	b⁴	ab⁴a	ab⁵	b⁴a	b⁵	ab⁵a	b⁵a
ab⁴a	ab⁴a	ab⁴	b⁵	ab⁵	b⁵a	ab	ab⁵a	b	aba	ab²	ba	b²	ab²a	ab³	b²a	b³	ab³a	ab⁴	b³a	b⁴	ab⁴a	b⁴a
ab⁵	ab⁵	ab⁵a	b	ab	ba	b²	aba	ab²	b²a	b³	ab²a	ab³	b³a	b⁴	ab³a	ab⁴	b⁴a	b⁵	ab⁴a	ab⁵	b⁵a	ab⁵a
b⁵a	b⁵a	b⁵	b	ab	ba	b²	aba	ab²	b²a	b³	ab²a	ab³	b³a	b⁴	ab³a	ab⁴	b⁴a	b⁵	ab⁴a	ab⁵	b⁵a	ab⁵a
ab⁵a	ab⁵a	ab⁵	ab	b	aba	ab²	ba	b²	ab²a	ab³	b²a	b³	ab³a	ab⁴	b³a	b⁴	ab⁴a	ab⁵	b⁴a	b⁵	ab⁵a	b⁵a

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

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:
- Left-to-right shortlex with a < b
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	a² ⇒ 1	[1]
2.	bab ⇒ ab²	[4]
3.	b⁶ ⇒ ab	[3]

# ab:aa=1,abbbbbb=b ab - -
aa=1
bab=abb
bbbbbb=ab

Same cardinality

3 unique, 30 total

Σ	#	Presentation	Properties	φ
9	3375	⟨a, b \| aa=1, ababba=b⟩	Noc 22	27
11	19756	⟨a, b \| aba=b, baaab=aa⟩	Noc 22
11	24722	⟨a, b \| aa=a, bbbbbb=ab⟩	Noc 22

Isomorphic instances

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

5 total

Σ	#	Presentation	Mapping
10	10062	⟨a, b \| aa=1, bbbbbb=ab⟩	φ(a) = a, φ(b) = b
11	26621	⟨a, b \| aa=1, abbabbbb=b⟩	φ(a) = a, φ(b) = ab
11	26629	⟨a, b \| aa=1, abbbabbb=b⟩	φ(a) = a, φ(b) = ab
11	27231	⟨a, b \| aa=1, bbabbbb=ab⟩	φ(a) = a, φ(b) = ab
11	27235	⟨a, b \| aa=1, bbbabbb=ab⟩	φ(a) = a, φ(b) = ab

Up:	Monoid enumeration
Prev:	#9757 ⟨a, b \| aa=1, abbbbba=b⟩
Next:	#9779 ⟨a, b \| aa=1, bababab=b⟩

#9759 ⟨a, b | aa=1, abbbbbb=b⟩

Contents