#526 ⟨a, b | aab=a, bbbb=1⟩

Properties

Presentation has sum-of-sides 8
Finite non-commutative monoid with 20 elements

Element profile

1 element center:
- 1
5 idempotent elements:
- 1, ba, a⁴, b²a², 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 recursive path with deg(a) = 0; deg(b) = 1

a⁵ ⇒ a
ab ⇒ a⁴
b⁴ ⇒ 1

# ab:aab=a,bbbb=1 a/b
aaaaa=a
ab=aaaa
bbbb=1

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

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

20 unique, 46 total

Σ	#	Presentation	Description	Related
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	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 isomorphic instances

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

10 total

Σ	#	Presentation	Mapping
8	617	⟨a, b \| ab=aa, bbbb=1⟩	φ(a) = a, φ(b) = bbb
9	1412	⟨a, b \| abab=a, bbbb=1⟩	φ(a) = aa, φ(b) = b
9	1565	⟨a, b \| aba=ab, bbbb=1⟩	φ(a) = ba, φ(b) = b
10	3905	⟨a, b \| aabb=ab, bbbb=1⟩	φ(a) = a, φ(b) = b
10	3967	⟨a, b \| abba=ab, bbbb=1⟩	φ(a) = a, φ(b) = b
10	3978	⟨a, b \| abbb=aa, bbbb=1⟩	φ(a) = a, φ(b) = b
10	4064	⟨a, b \| abb=aba, aaaa=1⟩	φ(a) = bbb, φ(b) = a
11	10987	⟨a, b \| abab=abb, bbbb=1⟩	φ(a) = ba, φ(b) = b
11	11064	⟨a, b \| abbb=aba, bbbb=1⟩	φ(a) = aa, φ(b) = b
11	17651	⟨a, b \| aaaa=1, ababa=ab⟩	φ(a) = b, φ(b) = aa

Other anti-isomorphic instances

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

9 total

Σ	#	Presentation	Mapping
10	3986	⟨a, b \| baab=ab, bbbb=1⟩	φ(a) = a, φ(b) = b
10	4054	⟨a, b \| abb=aab, aaaa=1⟩	φ(a) = b, φ(b) = aaa
10	5701	⟨a, b \| aaaa=1, aaabb=b⟩	φ(a) = b, φ(b) = aaa
10	5711	⟨a, b \| aaaa=1, abaab=b⟩	φ(a) = b, φ(b) = aaa
11	11001	⟨a, b \| abab=bab, bbbb=1⟩	φ(a) = ba, φ(b) = b
11	17099	⟨a, b \| aaaa=1, aaabab=b⟩	φ(a) = b, φ(b) = ba
11	17119	⟨a, b \| aaaa=1, abaaab=b⟩	φ(a) = b, φ(b) = ba
11	17635	⟨a, b \| aaaa=1, aabab=ab⟩	φ(a) = b, φ(b) = aa
11	17669	⟨a, b \| aaaa=1, baaab=ab⟩	φ(a) = b, φ(b) = aa

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Next:	#527 ⟨a, b \| aab=b, aaaa=1⟩