#16124 ⟨a, b | aab=aa, baba=bb⟩

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 20 elements
Not cancellative, because multiplication by a is not injective:
- a ⋅ ab = a² and a ⋅ a = a², however ab ≠ a

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
5 non-trivial idempotents:
- a², ba², b³, aba³, abab²
Order of generators:
- a: index 2, period 2
- b: index 3, period 1

Histogram:

index 1, period 1	5 elements	a², ba², b³, aba³, abab²
index 1, period 2	5 elements	a³, aba², ab³, ba³, bab²
index 2, period 1	3 elements	b², aba, (ab)²
index 2, period 2	3 elements	a, ab², bab
index 3, period 1	1 element	b
index 3, period 2	2 elements	ab, ba

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	ba	b²	a³	aba	ab²	ba²	bab	b³	aba²	(ab)²	ab³	ba³	bab²	aba³	abab²
1	1	a	b	a²	ab	ba	b²	a³	aba	ab²	ba²	bab	b³	aba²	(ab)²	ab³	ba³	bab²	aba³	abab²
a	a	a²	ab	a³	a²	aba	ab²	a²	a³	a²	aba²	(ab)²	ab³	a²	a³	a²	aba³	abab²	a³	a³
b	b	ba	b²	ba²	bab	bab²	b³	ba³	b²	bab²	b³	bab²	b³	bab²	b³	bab²	bab²	bab²	b³	b³
a²	a²	a³	a²	a²	a³	a³	a²	a³	a²	a³	a²	a³	a²	a³	a²	a³	a³	a³	a²	a²
ab	ab	aba	ab²	aba²	(ab)²	abab²	ab³	aba³	ab²	abab²	ab³	abab²	ab³	abab²	ab³	abab²	abab²	abab²	ab³	ab³
ba	ba	ba²	bab	ba³	ba²	b²	bab²	ba²	ba³	ba²	bab²	b³	bab²	ba²	ba³	ba²	b³	b³	ba³	ba³
b²	b²	bab²	b³	b³	bab²	bab²	b³	bab²	b³	bab²	b³	bab²	b³	bab²	b³	bab²	bab²	bab²	b³	b³
a³	a³	a²	a³	a³	a²	a²	a³	a²	a³	a²	a³	a²	a³	a²	a³	a²	a²	a²	a³	a³
aba	aba	aba²	(ab)²	aba³	aba²	ab²	abab²	aba²	aba³	aba²	abab²	ab³	abab²	aba²	aba³	aba²	ab³	ab³	aba³	aba³
ab²	ab²	abab²	ab³	ab³	abab²	abab²	ab³	abab²	ab³	abab²	ab³	abab²	ab³	abab²	ab³	abab²	abab²	abab²	ab³	ab³
ba²	ba²	ba³	ba²	ba²	ba³	ba³	ba²	ba³	ba²	ba³	ba²	ba³	ba²	ba³	ba²	ba³	ba³	ba³	ba²	ba²
bab	bab	b²	bab²	bab²	b³	b³	bab²	b³	bab²	b³	bab²	b³	bab²	b³	bab²	b³	b³	b³	bab²	bab²
b³	b³	bab²	b³	b³	bab²	bab²	b³	bab²	b³	bab²	b³	bab²	b³	bab²	b³	bab²	bab²	bab²	b³	b³
aba²	aba²	aba³	aba²	aba²	aba³	aba³	aba²	aba³	aba²	aba³	aba²	aba³	aba²	aba³	aba²	aba³	aba³	aba³	aba²	aba²
(ab)²	(ab)²	ab²	abab²	abab²	ab³	ab³	abab²	ab³	abab²	ab³	abab²	ab³	abab²	ab³	abab²	ab³	ab³	ab³	abab²	abab²
ab³	ab³	abab²	ab³	ab³	abab²	abab²	ab³	abab²	ab³	abab²	ab³	abab²	ab³	abab²	ab³	abab²	abab²	abab²	ab³	ab³
ba³	ba³	ba²	ba³	ba³	ba²	ba²	ba³	ba²	ba³	ba²	ba³	ba²	ba³	ba²	ba³	ba²	ba²	ba²	ba³	ba³
bab²	bab²	b³	bab²	bab²	b³	b³	bab²	b³	bab²	b³	bab²	b³	bab²	b³	bab²	b³	b³	b³	bab²	bab²
aba³	aba³	aba²	aba³	aba³	aba²	aba²	aba³	aba²	aba³	aba²	aba³	aba²	aba³	aba²	aba³	aba²	aba²	aba²	aba³	aba³
abab²	abab²	ab³	abab²	abab²	ab³	ab³	abab²	ab³	abab²	ab³	abab²	ab³	abab²	ab³	abab²	ab³	ab³	ab³	abab²	abab²

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

#	Rule	Proof
1.	b⁴ ⇒ b³	[7]
2.	bab³ ⇒ bab²	[8]
3.	b²a ⇒ bab²	[9]
4.	a²b ⇒ a²	[1]
5.	(ba)² ⇒ b²	[2]
6.	a⁴ ⇒ a²	[3]

# ab:aab=aa,baba=bb b/a - -
bbbb=bbb
babbb=babb
bba=babb
aab=aa
baba=bb
aaaa=aa

Same cardinality

20 unique, 65 total

Σ	#	Presentation	Properties	Description	φ
8	526	⟨a, b \| aab=a, bbbb=1⟩	Noc 20		19
9	2207	⟨a, b \| ab=aa, bbbb=b⟩	Noc 20		1
10	4095	⟨a, b \| baa=abb, abab=1⟩	Grp 20		5
10	4212	⟨a, b \| aaaaa=1, abbbb=1⟩	Grp Com 20	ℤ₂₀	16
10	5349	⟨a, b \| aaa=bb, bab=aa⟩	Noc 20		1
10	6728	⟨a, b \| aba=a, aaaa=bb⟩	Noc 20
10	6764	⟨a, b \| aba=b, aaaa=bb⟩	Noc 20
10	7117	⟨a, b \| ab=aa, bbbb=bb⟩	Noc 20
11	12159	⟨a, b \| aaaa=aa, abbb=b⟩	Noc 20
11	12322	⟨a, b \| aaab=bb, bbba=a⟩	Noc 20
11	14367	⟨a, b \| aaaa=a, bbbbb=a⟩	Noc Com 20	ℕ_{(20 = 5)}
11	14407	⟨a, b \| aaaa=b, bbbbb=a⟩	Noc Com 20	ℕ_{(20 = 1)}
11	14408	⟨a, b \| aaaa=b, bbbbb=b⟩	Noc Com 20	ℕ_{(20 = 4)}
11	15933	⟨a, b \| aba=bb, baaab=a⟩	Noc 20
11	16339	⟨a, b \| aba=aa, aaaa=bb⟩	Noc 20
11	16343	⟨a, b \| aba=aa, aaab=bb⟩	Noc 20
11	16545	⟨a, b \| aba=bb, abb=aaa⟩	Noc 20
11	18619	⟨a, b \| aba=b, aaaaabb=1⟩	Grp 20		3
11	19503	⟨a, b \| aab=a, bbbbb=bb⟩	Noc 20
11	21047	⟨a, b \| ab=aa, bbbb=bbb⟩	Noc 20

Up:	Monoid enumeration
Prev:	#16123 ⟨a, b \| aab=aa, baba=ba⟩
Next:	#16125 ⟨a, b \| aab=aa, babb=aa⟩

#16124 ⟨a, b | aab=aa, baba=bb⟩

Contents

Properties

Finite structure

Cayley table

Right Cayley graph

Left Cayley graph

Rewriting system

Same cardinality