#13248 ⟨a, b | bab=aaa, bbb=aa⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 21 elements

Element profile

4 element center:
- 1, b³, b⁶, b⁹
2 idempotent elements:
- 1, b⁹

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

b¹⁰ ⇒ b⁷
ab⁷ ⇒ ab⁴
bab ⇒ ab³
b³a ⇒ ab³
a² ⇒ b³

# ab:bab=aaa,bbb=aa b/a
bbbbbbbbbb=bbbbbbb
abbbbbbb=abbbb
bab=abbb
bbba=abbb
aa=bbb

Cayley table

Idempotents are shown in bold.

	1	a	b	ab	ba	b²	aba	ab²	b²a	b³	ab²a	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶	ab⁶	b⁷	b⁸	b⁹
1	1	a	b	ab	ba	b²	aba	ab²	b²a	b³	ab²a	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶	ab⁶	b⁷	b⁸	b⁹
a	a	b³	ab	b⁴	aba	ab²	ab⁵	b⁵	ab²a	ab³	ab⁴	b⁶	ab⁴	b⁷	ab⁵	b⁸	ab⁶	b⁹	ab⁴	ab⁵	ab⁶
b	b	ba	b²	ab³	b²a	b³	b⁶	ab⁴	ab³	b⁴	b⁸	ab⁵	b⁵	ab⁶	b⁶	ab⁴	b⁷	ab⁵	b⁸	b⁹	b⁷
ab	ab	aba	ab²	b⁶	ab²a	ab³	ab⁶	b⁷	b⁶	ab⁴	ab⁵	b⁸	ab⁵	b⁹	ab⁶	b⁷	ab⁴	b⁸	ab⁵	ab⁶	ab⁴
ba	ba	b⁴	ab³	b⁵	b⁶	ab⁴	ab⁴	b⁶	b⁸	ab⁵	ab⁶	b⁷	ab⁶	b⁸	ab⁴	b⁹	ab⁵	b⁷	ab⁶	ab⁴	ab⁵
b²	b²	b²a	b³	ab⁵	ab³	b⁴	b⁷	ab⁶	ab⁵	b⁵	b⁹	ab⁴	b⁶	ab⁵	b⁷	ab⁶	b⁸	ab⁴	b⁹	b⁷	b⁸
aba	aba	ab⁴	b⁶	ab⁵	ab⁶	b⁷	b⁷	ab⁶	ab⁵	b⁸	b⁹	ab⁴	b⁹	ab⁵	b⁷	ab⁶	b⁸	ab⁴	b⁹	b⁷	b⁸
ab²	ab²	ab²a	ab³	b⁸	b⁶	ab⁴	ab⁴	b⁹	b⁸	ab⁵	ab⁶	b⁷	ab⁶	b⁸	ab⁴	b⁹	ab⁵	b⁷	ab⁶	ab⁴	ab⁵
b²a	b²a	b⁵	ab⁵	b⁶	b⁷	ab⁶	ab⁶	b⁷	b⁹	ab⁴	ab⁵	b⁸	ab⁵	b⁹	ab⁶	b⁷	ab⁴	b⁸	ab⁵	ab⁶	ab⁴
b³	b³	ab³	b⁴	ab⁴	ab⁵	b⁵	b⁸	ab⁵	ab⁴	b⁶	b⁷	ab⁶	b⁷	ab⁴	b⁸	ab⁵	b⁹	ab⁶	b⁷	b⁸	b⁹
ab²a	ab²a	ab⁵	b⁸	ab⁶	ab⁴	b⁹	b⁹	ab⁴	ab⁶	b⁷	b⁸	ab⁵	b⁸	ab⁶	b⁹	ab⁴	b⁷	ab⁵	b⁸	b⁹	b⁷
ab³	ab³	b⁶	ab⁴	b⁷	b⁸	ab⁵	ab⁵	b⁸	b⁷	ab⁶	ab⁴	b⁹	ab⁴	b⁷	ab⁵	b⁸	ab⁶	b⁹	ab⁴	ab⁵	ab⁶
b⁴	b⁴	ab⁵	b⁵	ab⁶	ab⁴	b⁶	b⁹	ab⁴	ab⁶	b⁷	b⁸	ab⁵	b⁸	ab⁶	b⁹	ab⁴	b⁷	ab⁵	b⁸	b⁹	b⁷
ab⁴	ab⁴	b⁸	ab⁵	b⁹	b⁷	ab⁶	ab⁶	b⁷	b⁹	ab⁴	ab⁵	b⁸	ab⁵	b⁹	ab⁶	b⁷	ab⁴	b⁸	ab⁵	ab⁶	ab⁴
b⁵	b⁵	ab⁴	b⁶	ab⁵	ab⁶	b⁷	b⁷	ab⁶	ab⁵	b⁸	b⁹	ab⁴	b⁹	ab⁵	b⁷	ab⁶	b⁸	ab⁴	b⁹	b⁷	b⁸
ab⁵	ab⁵	b⁷	ab⁶	b⁸	b⁹	ab⁴	ab⁴	b⁹	b⁸	ab⁵	ab⁶	b⁷	ab⁶	b⁸	ab⁴	b⁹	ab⁵	b⁷	ab⁶	ab⁴	ab⁵
b⁶	b⁶	ab⁶	b⁷	ab⁴	ab⁵	b⁸	b⁸	ab⁵	ab⁴	b⁹	b⁷	ab⁶	b⁷	ab⁴	b⁸	ab⁵	b⁹	ab⁶	b⁷	b⁸	b⁹
ab⁶	ab⁶	b⁹	ab⁴	b⁷	b⁸	ab⁵	ab⁵	b⁸	b⁷	ab⁶	ab⁴	b⁹	ab⁴	b⁷	ab⁵	b⁸	ab⁶	b⁹	ab⁴	ab⁵	ab⁶
b⁷	b⁷	ab⁵	b⁸	ab⁶	ab⁴	b⁹	b⁹	ab⁴	ab⁶	b⁷	b⁸	ab⁵	b⁸	ab⁶	b⁹	ab⁴	b⁷	ab⁵	b⁸	b⁹	b⁷
b⁸	b⁸	ab⁴	b⁹	ab⁵	ab⁶	b⁷	b⁷	ab⁶	ab⁵	b⁸	b⁹	ab⁴	b⁹	ab⁵	b⁷	ab⁶	b⁸	ab⁴	b⁹	b⁷	b⁸
b⁹	b⁹	ab⁶	b⁷	ab⁴	ab⁵	b⁸	b⁸	ab⁵	ab⁴	b⁹	b⁷	ab⁶	b⁷	ab⁴	b⁸	ab⁵	b⁹	ab⁶	b⁷	b⁸	b⁹

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

8 unique, 64 total

Σ	#	Presentation	Description	Related
8	402	⟨a, b \| aaa=ab, bbb=1⟩	Finite non-commutative monoid with 21 elements	12 iso, 27 anti-iso
9	1599	⟨a, b \| aaa=ab, bbb=b⟩	Finite non-commutative monoid with 21 elements	1 anti-iso
10	5309	⟨a, b \| aaa=aa, bbb=ab⟩	Finite non-commutative monoid with 21 elements
10	6265	⟨a, b \| aaa=a, abbbb=b⟩	Finite non-commutative monoid with 21 elements	2 iso
11	11127	⟨a, b \| aaaaa=b, abbbb=1⟩	Isomorphic to ℤ₂₁	13 iso
11	13091	⟨a, b \| bab=aaa, abba=b⟩	Finite non-commutative monoid with 21 elements
11	15158	⟨a, b \| aab=ba, ababab=1⟩	Finite non-Abelian group with 21 elements	1 iso
11	19212	⟨a, b \| aba=b, baaaab=a⟩	Finite non-commutative monoid with 21 elements

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