#24722 ⟨a, b | aa=a, bbbbbb=ab⟩

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

Properties

Sum of relation sides is 11
Finite non-commutative monoid with 22 elements
Not cancellative, because multiplication by a is not injective:
- a ⋅ a = a and a ⋅ 1 = a, however a ≠ 1

Finite structure

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.

1 element center:
- 1
3 non-trivial idempotents:
- a, ab⁵, ab⁵a
Order of generators:
- a: index 1, period 1
- b: index 6, period 5

Histogram:

index 1, period 1	3 elements	a, ab⁵, ab⁵a
index 1, period 5	8 elements	ab, aba, ab², ab²a, ab³, ...
index 2, period 1	2 elements	b⁵, b⁵a
index 2, period 5	6 elements	ba, b²a, b³, b³a, b⁴, ...
index 3, period 5	1 element	b²
index 6, period 5	1 element	b

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

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 ⇒ a	[1]
2.	bab ⇒ ab²	[3]
3.	b⁶ ⇒ ab	[2]

# ab:aa=a,bbbbbb=ab ab - -
aa=a
bab=abb
bbbbbb=ab

Same cardinality

3 unique, 35 total

Σ	#	Presentation	Properties	φ
9	3375	⟨a, b \| aa=1, ababba=b⟩	Noc 22	27
10	9759	⟨a, b \| aa=1, abbbbbb=b⟩	Noc 22	5
11	19756	⟨a, b \| aba=b, baaab=aa⟩	Noc 22

Up:	Monoid enumeration
Prev:	#24720 ⟨a, b \| aa=a, bbabbb=bb⟩
Next:	#24723 ⟨a, b \| aa=a, bbbbbb=bb⟩

#24722 ⟨a, b | aa=a, bbbbbb=ab⟩

Contents

Properties

Finite structure

Cayley table

Right Cayley graph

Left Cayley graph

Rewriting system

Same cardinality