#6838 ⟨a, b | aba=b, bab=aaa⟩

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 19 elements

Element profile

3 element center:
- 1, a⁴, a⁸
2 idempotent elements:
- 1, a⁸

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

a¹¹ ⇒ a³
ba⁸ ⇒ b
ab ⇒ ba⁷
b² ⇒ a⁴

# ab:aba=b,bab=aaa a/b
aaaaaaaaaaa=aaa
baaaaaaaa=b
ab=baaaaaaa
bb=aaaa

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	a¹⁰
1	1	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	a¹⁰
a	a	a²	ba⁷	a³	b	a⁴	ba	a⁵	ba²	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a¹⁰	a³
b	b	ba	a⁴	ba²	a⁵	ba³	a⁶	ba⁴	a⁷	ba⁵	a⁸	ba⁶	a⁹	ba⁷	a¹⁰	b	a³	ba	ba²
a²	a²	a³	ba⁶	a⁴	ba⁷	a⁵	b	a⁶	ba	a⁷	ba²	a⁸	ba³	a⁹	ba⁴	a¹⁰	ba⁵	a³	a⁴
ba	ba	ba²	a³	ba³	a⁴	ba⁴	a⁵	ba⁵	a⁶	ba⁶	a⁷	ba⁷	a⁸	b	a⁹	ba	a¹⁰	ba²	ba³
a³	a³	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	b	a⁸	ba	a⁹	ba²	a¹⁰	ba³	a³	ba⁴	a⁴	a⁵
ba²	ba²	ba³	a¹⁰	ba⁴	a³	ba⁵	a⁴	ba⁶	a⁵	ba⁷	a⁶	b	a⁷	ba	a⁸	ba²	a⁹	ba³	ba⁴
a⁴	a⁴	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	b	a¹⁰	ba	a³	ba²	a⁴	ba³	a⁵	a⁶
ba³	ba³	ba⁴	a⁹	ba⁵	a¹⁰	ba⁶	a³	ba⁷	a⁴	b	a⁵	ba	a⁶	ba²	a⁷	ba³	a⁸	ba⁴	ba⁵
a⁵	a⁵	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a¹⁰	ba⁷	a³	b	a⁴	ba	a⁵	ba²	a⁶	a⁷
ba⁴	ba⁴	ba⁵	a⁸	ba⁶	a⁹	ba⁷	a¹⁰	b	a³	ba	a⁴	ba²	a⁵	ba³	a⁶	ba⁴	a⁷	ba⁵	ba⁶
a⁶	a⁶	a⁷	ba²	a⁸	ba³	a⁹	ba⁴	a¹⁰	ba⁵	a³	ba⁶	a⁴	ba⁷	a⁵	b	a⁶	ba	a⁷	a⁸
ba⁵	ba⁵	ba⁶	a⁷	ba⁷	a⁸	b	a⁹	ba	a¹⁰	ba²	a³	ba³	a⁴	ba⁴	a⁵	ba⁵	a⁶	ba⁶	ba⁷
a⁷	a⁷	a⁸	ba	a⁹	ba²	a¹⁰	ba³	a³	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	b	a⁸	a⁹
ba⁶	ba⁶	ba⁷	a⁶	b	a⁷	ba	a⁸	ba²	a⁹	ba³	a¹⁰	ba⁴	a³	ba⁵	a⁴	ba⁶	a⁵	ba⁷	b
a⁸	a⁸	a⁹	b	a¹⁰	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	a¹⁰
ba⁷	ba⁷	b	a⁵	ba	a⁶	ba²	a⁷	ba³	a⁸	ba⁴	a⁹	ba⁵	a¹⁰	ba⁶	a³	ba⁷	a⁴	b	ba
a⁹	a⁹	a¹⁰	ba⁷	a³	b	a⁴	ba	a⁵	ba²	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a¹⁰	a³
a¹⁰	a¹⁰	a³	ba⁶	a⁴	ba⁷	a⁵	b	a⁶	ba	a⁷	ba²	a⁸	ba³	a⁹	ba⁴	a¹⁰	ba⁵	a³	a⁴

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
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
11	16053	⟨a, b \| aaa=bb, aaba=ab⟩	Finite non-commutative monoid with 19 elements

Up:	Monoids with two generators and two relations
Prev:	#6834 ⟨a, b \| aba=b, abb=aaa⟩
Next:	#6845 ⟨a, b \| bab=b, bbb=bab⟩