#16339 ⟨a, b | aba=aa, aaaa=bb⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 20 elements

Element profile

8 element center:
- 1, a⁴, a⁵, ba⁴, ba³b, a⁶, a⁷, a⁸
2 idempotent elements:
- 1, a⁸

Complete rewriting system

Format:	Pretty Plain
Word to reduce:
	Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Reduction strategy:	Leftmost Rightmost
Path to normal form:	1 1

Reduction order:
- Left-to-right recursive path with deg(a) = 0; deg(b) = 1

a⁹ ⇒ a⁵
ba⁵ ⇒ a⁵
aba ⇒ a²
a⁴b ⇒ ba⁴
b² ⇒ a⁴

# ab:aba=aa,aaaa=bb a/b
aaaaaaaaa=aaaaa
baaaaa=aaaaa
aba=aa
aaaab=baaaa
bb=aaaa

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	ba	a³	a²b	ba²	bab	a⁴	a³b	ba³	ba²b	a⁵	ba⁴	ba³b	a⁶	a⁷	a⁸
1	1	a	b	a²	ab	ba	a³	a²b	ba²	bab	a⁴	a³b	ba³	ba²b	a⁵	ba⁴	ba³b	a⁶	a⁷	a⁸
a	a	a²	ab	a³	a²b	a²	a⁴	a³b	a³	a²b	a⁵	ba⁴	a⁴	a³b	a⁶	a⁵	ba⁴	a⁷	a⁸	a⁵
b	b	ba	a⁴	ba²	bab	a⁵	ba³	ba²b	a⁶	a⁵	ba⁴	ba³b	a⁷	a⁶	a⁵	a⁸	a⁷	a⁶	a⁷	a⁸
a²	a²	a³	a²b	a⁴	a³b	a³	a⁵	ba⁴	a⁴	a³b	a⁶	a⁵	a⁵	ba⁴	a⁷	a⁶	a⁵	a⁸	a⁵	a⁶
ab	ab	a²	a⁵	a³	a²b	a⁶	a⁴	a³b	a⁷	a⁶	a⁵	ba⁴	a⁸	a⁷	a⁶	a⁵	a⁸	a⁷	a⁸	a⁵
ba	ba	ba²	bab	ba³	ba²b	ba²	ba⁴	ba³b	ba³	ba²b	a⁵	a⁸	ba⁴	ba³b	a⁶	a⁵	a⁸	a⁷	a⁸	a⁵
a³	a³	a⁴	a³b	a⁵	ba⁴	a⁴	a⁶	a⁵	a⁵	ba⁴	a⁷	a⁶	a⁶	a⁵	a⁸	a⁷	a⁶	a⁵	a⁶	a⁷
a²b	a²b	a³	a⁶	a⁴	a³b	a⁷	a⁵	ba⁴	a⁸	a⁷	a⁶	a⁵	a⁵	a⁸	a⁷	a⁶	a⁵	a⁸	a⁵	a⁶
ba²	ba²	ba³	ba²b	ba⁴	ba³b	ba³	a⁵	a⁸	ba⁴	ba³b	a⁶	a⁵	a⁵	a⁸	a⁷	a⁶	a⁵	a⁸	a⁵	a⁶
bab	bab	ba²	a⁵	ba³	ba²b	a⁶	ba⁴	ba³b	a⁷	a⁶	a⁵	a⁸	a⁸	a⁷	a⁶	a⁵	a⁸	a⁷	a⁸	a⁵
a⁴	a⁴	a⁵	ba⁴	a⁶	a⁵	a⁵	a⁷	a⁶	a⁶	a⁵	a⁸	a⁷	a⁷	a⁶	a⁵	a⁸	a⁷	a⁶	a⁷	a⁸
a³b	a³b	a⁴	a⁷	a⁵	ba⁴	a⁸	a⁶	a⁵	a⁵	a⁸	a⁷	a⁶	a⁶	a⁵	a⁸	a⁷	a⁶	a⁵	a⁶	a⁷
ba³	ba³	ba⁴	ba³b	a⁵	a⁸	ba⁴	a⁶	a⁵	a⁵	a⁸	a⁷	a⁶	a⁶	a⁵	a⁸	a⁷	a⁶	a⁵	a⁶	a⁷
ba²b	ba²b	ba³	a⁶	ba⁴	ba³b	a⁷	a⁵	a⁸	a⁸	a⁷	a⁶	a⁵	a⁵	a⁸	a⁷	a⁶	a⁵	a⁸	a⁵	a⁶
a⁵	a⁵	a⁶	a⁵	a⁷	a⁶	a⁶	a⁸	a⁷	a⁷	a⁶	a⁵	a⁸	a⁸	a⁷	a⁶	a⁵	a⁸	a⁷	a⁸	a⁵
ba⁴	ba⁴	a⁵	a⁸	a⁶	a⁵	a⁵	a⁷	a⁶	a⁶	a⁵	a⁸	a⁷	a⁷	a⁶	a⁵	a⁸	a⁷	a⁶	a⁷	a⁸
ba³b	ba³b	ba⁴	a⁷	a⁵	a⁸	a⁸	a⁶	a⁵	a⁵	a⁸	a⁷	a⁶	a⁶	a⁵	a⁸	a⁷	a⁶	a⁵	a⁶	a⁷
a⁶	a⁶	a⁷	a⁶	a⁸	a⁷	a⁷	a⁵	a⁸	a⁸	a⁷	a⁶	a⁵	a⁵	a⁸	a⁷	a⁶	a⁵	a⁸	a⁵	a⁶
a⁷	a⁷	a⁸	a⁷	a⁵	a⁸	a⁸	a⁶	a⁵	a⁵	a⁸	a⁷	a⁶	a⁶	a⁵	a⁸	a⁷	a⁶	a⁵	a⁶	a⁷
a⁸	a⁸	a⁵	a⁸	a⁶	a⁵	a⁵	a⁷	a⁶	a⁶	a⁵	a⁸	a⁷	a⁷	a⁶	a⁵	a⁸	a⁷	a⁶	a⁷	a⁸

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

20 unique, 65 total

Σ	#	Presentation	Description	Related
8	526	⟨a, b \| aab=a, bbbb=1⟩	Finite non-commutative monoid with 20 elements	10 iso, 9 anti-iso
9	2207	⟨a, b \| ab=aa, bbbb=b⟩	Finite non-commutative monoid with 20 elements	1 anti-iso
10	4095	⟨a, b \| baa=abb, abab=1⟩	Finite non-Abelian group with 20 elements	4 iso, 1 anti-iso
10	4212	⟨a, b \| aaaaa=1, abbbb=1⟩	Isomorphic to ℤ₂₀	16 iso
10	5349	⟨a, b \| aaa=bb, bab=aa⟩	Finite non-commutative monoid with 20 elements	1 iso
10	6728	⟨a, b \| aba=a, aaaa=bb⟩	Finite non-commutative monoid with 20 elements
10	6764	⟨a, b \| aba=b, aaaa=bb⟩	Finite non-commutative monoid with 20 elements
10	7117	⟨a, b \| ab=aa, bbbb=bb⟩	Finite non-commutative monoid with 20 elements
11	12159	⟨a, b \| aaaa=aa, abbb=b⟩	Finite non-commutative monoid with 20 elements
11	12322	⟨a, b \| aaab=bb, bbba=a⟩	Finite non-commutative monoid with 20 elements
11	14367	⟨a, b \| aaaa=a, bbbbb=a⟩	Isomorphic to ℕ_{(20 = 5)}
11	14407	⟨a, b \| aaaa=b, bbbbb=a⟩	Isomorphic to ℕ_{(20 = 1)}
11	14408	⟨a, b \| aaaa=b, bbbbb=b⟩	Isomorphic to ℕ_{(20 = 4)}
11	15933	⟨a, b \| aba=bb, baaab=a⟩	Finite non-commutative monoid with 20 elements
11	16124	⟨a, b \| aab=aa, baba=bb⟩	Finite non-commutative monoid with 20 elements
11	16343	⟨a, b \| aba=aa, aaab=bb⟩	Finite non-commutative monoid with 20 elements
11	16545	⟨a, b \| aba=bb, abb=aaa⟩	Finite non-commutative monoid with 20 elements
11	18619	⟨a, b \| aba=b, aaaaabb=1⟩	Finite non-Abelian group with 20 elements	3 iso
11	19503	⟨a, b \| aab=a, bbbbb=bb⟩	Finite non-commutative monoid with 20 elements
11	21047	⟨a, b \| ab=aa, bbbb=bbb⟩	Finite non-commutative monoid with 20 elements

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