#2207 ⟨a, b | ab=aa, bbbb=b⟩

Properties

Presentation has sum-of-sides 9
Finite non-commutative monoid with 20 elements

Element profile

1 element center:
- 1
6 idempotent elements:
- 1, a³, ba², b³, b²a⁴, b³a³

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 shortlex with a < b

ab ⇒ a²
b⁴ ⇒ b
a⁵ ⇒ a²

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

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.

Others with same cardinality

20 unique, 64 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
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	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

Other 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:	Monoids with two generators and two relations
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Next:	#2208 ⟨a, b \| ba=ab, aaaa=a⟩