#2207 ⟨a, b | ab=aa, bbbb=b⟩

Properties
Finite structure
Cayley table
Right Cayley graph
Left Cayley graph
Rewriting system
Same cardinality
Anti-isomorphic instances

Properties

Sum of relation sides is 9
Finite non-commutative monoid with 20 elements
Not cancellative, because multiplication by a is not injective:
- a ⋅ b = a² and a ⋅ a = a², however b ≠ 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:
- a³, ba², b³, b²a⁴, b³a³
Order of generators:
- a: index 2, period 3
- b: index 1, period 3
Histogram:

index 1, period 1 5 elements a³, ba², b³, b²a⁴, b³a³

index 1, period 3 10 elements b, a², b², a⁴, ba³, ...

index 2, period 1 1 element b²a

index 2, period 3 3 elements a, ba, b³a

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ba	b²	a³	ba²	b²a	b³	a⁴	ba³	b²a²	b³a	ba⁴	b²a³	b³a²	b²a⁴	b³a³	b³a⁴
1	1	a	b	a²	ba	b²	a³	ba²	b²a	b³	a⁴	ba³	b²a²	b³a	ba⁴	b²a³	b³a²	b²a⁴	b³a³	b³a⁴
a	a	a²	a²	a³	a³	a³	a⁴	a⁴	a⁴	a⁴	a²	a²	a²	a²	a³	a³	a³	a⁴	a⁴	a²
b	b	ba	b²	ba²	b²a	b³	ba³	b²a²	b³a	b	ba⁴	b²a³	b³a²	ba	b²a⁴	b³a³	ba²	b³a⁴	ba³	ba⁴
a²	a²	a³	a³	a⁴	a⁴	a⁴	a²	a²	a²	a²	a³	a³	a³	a³	a⁴	a⁴	a⁴	a²	a²	a³
ba	ba	ba²	ba²	ba³	ba³	ba³	ba⁴	ba⁴	ba⁴	ba⁴	ba²	ba²	ba²	ba²	ba³	ba³	ba³	ba⁴	ba⁴	ba²
b²	b²	b²a	b³	b²a²	b³a	b	b²a³	b³a²	ba	b²	b²a⁴	b³a³	ba²	b²a	b³a⁴	ba³	b²a²	ba⁴	b²a³	b²a⁴
a³	a³	a⁴	a⁴	a²	a²	a²	a³	a³	a³	a³	a⁴	a⁴	a⁴	a⁴	a²	a²	a²	a³	a³	a⁴
ba²	ba²	ba³	ba³	ba⁴	ba⁴	ba⁴	ba²	ba²	ba²	ba²	ba³	ba³	ba³	ba³	ba⁴	ba⁴	ba⁴	ba²	ba²	ba³
b²a	b²a	b²a²	b²a²	b²a³	b²a³	b²a³	b²a⁴	b²a⁴	b²a⁴	b²a⁴	b²a²	b²a²	b²a²	b²a²	b²a³	b²a³	b²a³	b²a⁴	b²a⁴	b²a²
b³	b³	b³a	b	b³a²	ba	b²	b³a³	ba²	b²a	b³	b³a⁴	ba³	b²a²	b³a	ba⁴	b²a³	b³a²	b²a⁴	b³a³	b³a⁴
a⁴	a⁴	a²	a²	a³	a³	a³	a⁴	a⁴	a⁴	a⁴	a²	a²	a²	a²	a³	a³	a³	a⁴	a⁴	a²
ba³	ba³	ba⁴	ba⁴	ba²	ba²	ba²	ba³	ba³	ba³	ba³	ba⁴	ba⁴	ba⁴	ba⁴	ba²	ba²	ba²	ba³	ba³	ba⁴
b²a²	b²a²	b²a³	b²a³	b²a⁴	b²a⁴	b²a⁴	b²a²	b²a²	b²a²	b²a²	b²a³	b²a³	b²a³	b²a³	b²a⁴	b²a⁴	b²a⁴	b²a²	b²a²	b²a³
b³a	b³a	b³a²	b³a²	b³a³	b³a³	b³a³	b³a⁴	b³a⁴	b³a⁴	b³a⁴	b³a²	b³a²	b³a²	b³a²	b³a³	b³a³	b³a³	b³a⁴	b³a⁴	b³a²
ba⁴	ba⁴	ba²	ba²	ba³	ba³	ba³	ba⁴	ba⁴	ba⁴	ba⁴	ba²	ba²	ba²	ba²	ba³	ba³	ba³	ba⁴	ba⁴	ba²
b²a³	b²a³	b²a⁴	b²a⁴	b²a²	b²a²	b²a²	b²a³	b²a³	b²a³	b²a³	b²a⁴	b²a⁴	b²a⁴	b²a⁴	b²a²	b²a²	b²a²	b²a³	b²a³	b²a⁴
b³a²	b³a²	b³a³	b³a³	b³a⁴	b³a⁴	b³a⁴	b³a²	b³a²	b³a²	b³a²	b³a³	b³a³	b³a³	b³a³	b³a⁴	b³a⁴	b³a⁴	b³a²	b³a²	b³a³
b²a⁴	b²a⁴	b²a²	b²a²	b²a³	b²a³	b²a³	b²a⁴	b²a⁴	b²a⁴	b²a⁴	b²a²	b²a²	b²a²	b²a²	b²a³	b²a³	b²a³	b²a⁴	b²a⁴	b²a²
b³a³	b³a³	b³a⁴	b³a⁴	b³a²	b³a²	b³a²	b³a³	b³a³	b³a³	b³a³	b³a⁴	b³a⁴	b³a⁴	b³a⁴	b³a²	b³a²	b³a²	b³a³	b³a³	b³a⁴
b³a⁴	b³a⁴	b³a²	b³a²	b³a³	b³a³	b³a³	b³a⁴	b³a⁴	b³a⁴	b³a⁴	b³a²	b³a²	b³a²	b³a²	b³a³	b³a³	b³a³	b³a⁴	b³a⁴	b³a²

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.	ab ⇒ a²	[1]
2.	b⁴ ⇒ b	[2]
3.	a⁵ ⇒ a²	[3]

# ab:ab=aa,bbbb=b ab - -
ab=aa
bbbb=b
aaaaa=aa

Same cardinality

20 unique, 64 total

Σ	#	Presentation	Properties	Description	φ
8	526	⟨a, b \| aab=a, bbbb=1⟩	Noc 20		19
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	16124	⟨a, b \| aab=aa, baba=bb⟩	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

Anti-isomorphic instances

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

1 total

Σ	#	Presentation	Mapping
10	5190	⟨a, b \| aab=bb, aaaa=a⟩	φ(a) = bb, φ(b) = a

Up:	Monoid enumeration
Prev:	#2206 ⟨a, b \| ab=aa, bbbb=a⟩
Next:	#2208 ⟨a, b \| ba=ab, aaaa=a⟩

index 1, period 1	5 elements	a³, ba², b³, b²a⁴, b³a³
index 1, period 3	10 elements	b, a², b², a⁴, ba³, ...
index 2, period 1	1 element	b²a
index 2, period 3	3 elements	a, ba, b³a

#2207 ⟨a, b | ab=aa, bbbb=b⟩

Contents