#16053 ⟨a, b | aaa=bb, aaba=ab⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 19 elements

Element profile

4 element center:
- 1, a³, a⁶, a⁹
2 idempotent elements:
- 1, a⁶

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

a¹⁰ ⇒ a⁴
ba⁹ ⇒ ba³
ab ⇒ ba⁵
b² ⇒ a³

# ab:aaa=bb,aaba=ab a/b
aaaaaaaaaa=aaaa
baaaaaaaaa=baaa
ab=baaaaa
bb=aaa

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸
1	1	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸
a	a	a²	ba⁵	a³	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a⁴	ba⁷
b	b	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a⁴	ba³	a⁵
a²	a²	a³	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba³	a⁹	ba⁴	a⁴	ba⁵	a⁵	ba⁶
ba	ba	ba²	a⁸	ba³	a⁹	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba³	a⁹	ba⁴	a⁴
a³	a³	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵
ba²	ba²	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹
a⁴	a⁴	a⁵	ba⁸	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba³	a⁷	ba⁴
ba³	ba³	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸
a⁵	a⁵	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba³	a⁹	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba³
ba⁴	ba⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba³	a⁹	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷
a⁶	a⁶	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸
ba⁵	ba⁵	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶
a⁷	a⁷	a⁸	ba⁵	a⁹	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a⁴	ba⁷
ba⁶	ba⁶	ba⁷	a⁹	ba⁸	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a⁴	ba³	a⁵
a⁸	a⁸	a⁹	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba³	a⁹	ba⁴	a⁴	ba⁵	a⁵	ba⁶
ba⁷	ba⁷	ba⁸	a⁸	ba³	a⁹	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba³	a⁹	ba⁴	a⁴
a⁹	a⁹	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵
ba⁸	ba⁸	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

5 unique, 29 total

Σ	#	Presentation	Description	Related
10	6263	⟨a, b \| aaa=a, abbba=b⟩	Finite non-commutative monoid with 19 elements
10	6838	⟨a, b \| aba=b, bab=aaa⟩	Finite non-commutative monoid with 19 elements
11	13340	⟨a, b \| aaaab=1, abbbbb=1⟩	Isomorphic to ℤ₁₉	20 iso
11	14501	⟨a, b \| aaab=b, abbba=a⟩	Finite non-commutative monoid with 19 elements	3 iso
11	16036	⟨a, b \| aaa=ab, bbab=bb⟩	Finite non-commutative monoid with 19 elements	1 iso

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