#16124 ⟨a, b | aab=aa, baba=bb⟩

Properties

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

Element profile

1 element center:
- 1
6 idempotent elements:
- 1, a², ba², b³, aba³, abab²

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(b) = 0; deg(a) = 1

b⁴ ⇒ b³
bab³ ⇒ bab²
b²a ⇒ bab²
a²b ⇒ a²
(ba)² ⇒ b²
a⁴ ⇒ a²

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

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.

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	16339	⟨a, b \| aba=aa, aaaa=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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