#6728 ⟨a, b | aba=a, aaaa=bb⟩

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 20 elements

Element profile

7 element center:
- 1, b², b³, b⁴, b⁵, b⁶, b⁷
4 idempotent elements:
- 1, ab, ba, b⁶

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

b⁸ ⇒ b²
ab² ⇒ b⁷
b²a ⇒ b⁷
aba ⇒ a
a⁴ ⇒ b²

# ab:aba=a,aaaa=bb b/a
bbbbbbbb=bb
abb=bbbbbbb
bba=bbbbbbb
aba=a
aaaa=bb

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	ba	b²	a³	a²b	ba²	bab	b³	a³b	ba³	ba²b	b⁴	ba³b	b⁵	b⁶	b⁷
1	1	a	b	a²	ab	ba	b²	a³	a²b	ba²	bab	b³	a³b	ba³	ba²b	b⁴	ba³b	b⁵	b⁶	b⁷
a	a	a²	ab	a³	a²b	a	b⁷	b²	a³b	a²	ab	b²	b³	a³	a²b	b³	a³b	b⁴	b⁵	b⁶
b	b	ba	b²	ba²	bab	b⁷	b³	ba³	ba²b	b⁶	b²	b⁴	ba³b	b⁵	b⁷	b⁵	b⁶	b⁶	b⁷	b²
a²	a²	a³	a²b	b²	a³b	a²	b⁶	b⁷	b³	a³	a²b	b⁷	b²	b²	a³b	b²	b³	b³	b⁴	b⁵
ab	ab	a	b⁷	a²	ab	b⁶	b²	a³	a²b	b⁵	b⁷	b³	a³b	b⁴	b⁶	b⁴	b⁵	b⁵	b⁶	b⁷
ba	ba	ba²	bab	ba³	ba²b	ba	b²	b³	ba³b	ba²	bab	b³	b⁴	ba³	ba²b	b⁴	ba³b	b⁵	b⁶	b⁷
b²	b²	b⁷	b³	b⁶	b²	b²	b⁴	b⁵	b⁷	b⁷	b³	b⁵	b⁶	b⁶	b²	b⁶	b⁷	b⁷	b²	b³
a³	a³	b²	a³b	b⁷	b³	a³	b⁵	b⁶	b²	b²	a³b	b⁶	b⁷	b⁷	b³	b⁷	b²	b²	b³	b⁴
a²b	a²b	a²	b⁶	a³	a²b	b⁵	b⁷	b²	a³b	b⁴	b⁶	b²	b³	b³	b⁵	b³	b⁴	b⁴	b⁵	b⁶
ba²	ba²	ba³	ba²b	b³	ba³b	ba²	b⁷	b²	b⁴	ba³	ba²b	b²	b³	b³	ba³b	b³	b⁴	b⁴	b⁵	b⁶
bab	bab	ba	b²	ba²	bab	b⁷	b³	ba³	ba²b	b⁶	b²	b⁴	ba³b	b⁵	b⁷	b⁵	b⁶	b⁶	b⁷	b²
b³	b³	b²	b⁴	b⁷	b³	b³	b⁵	b⁶	b²	b²	b⁴	b⁶	b⁷	b⁷	b³	b⁷	b²	b²	b³	b⁴
a³b	a³b	a³	b⁵	b²	a³b	b⁴	b⁶	b⁷	b³	b³	b⁵	b⁷	b²	b²	b⁴	b²	b³	b³	b⁴	b⁵
ba³	ba³	b³	ba³b	b²	b⁴	ba³	b⁶	b⁷	b³	b³	ba³b	b⁷	b²	b²	b⁴	b²	b³	b³	b⁴	b⁵
ba²b	ba²b	ba²	b⁷	ba³	ba²b	b⁶	b²	b³	ba³b	b⁵	b⁷	b³	b⁴	b⁴	b⁶	b⁴	b⁵	b⁵	b⁶	b⁷
b⁴	b⁴	b³	b⁵	b²	b⁴	b⁴	b⁶	b⁷	b³	b³	b⁵	b⁷	b²	b²	b⁴	b²	b³	b³	b⁴	b⁵
ba³b	ba³b	ba³	b⁶	b³	ba³b	b⁵	b⁷	b²	b⁴	b⁴	b⁶	b²	b³	b³	b⁵	b³	b⁴	b⁴	b⁵	b⁶
b⁵	b⁵	b⁴	b⁶	b³	b⁵	b⁵	b⁷	b²	b⁴	b⁴	b⁶	b²	b³	b³	b⁵	b³	b⁴	b⁴	b⁵	b⁶
b⁶	b⁶	b⁵	b⁷	b⁴	b⁶	b⁶	b²	b³	b⁵	b⁵	b⁷	b³	b⁴	b⁴	b⁶	b⁴	b⁵	b⁵	b⁶	b⁷
b⁷	b⁷	b⁶	b²	b⁵	b⁷	b⁷	b³	b⁴	b⁶	b⁶	b²	b⁴	b⁵	b⁵	b⁷	b⁵	b⁶	b⁶	b⁷	b²

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	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	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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