#1599 ⟨a, b | aaa=ab, bbb=b⟩

Properties

Presentation has sum-of-sides 9
Finite non-commutative monoid with 21 elements

Element profile

1 element center:
- 1
5 idempotent elements:
- 1, b², a⁴, b²a⁴, ba⁶

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

a⁷ ⇒ a³
ab ⇒ a³
b³ ⇒ b

# ab:aaa=ab,bbb=b a/b
aaaaaaa=aaa
ab=aaa
bbb=b

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ba	b²	a³	ba²	b²a	a⁴	ba³	b²a²	a⁵	ba⁴	b²a³	a⁶	ba⁵	b²a⁴	ba⁶	b²a⁵	b²a⁶
1	1	a	b	a²	ba	b²	a³	ba²	b²a	a⁴	ba³	b²a²	a⁵	ba⁴	b²a³	a⁶	ba⁵	b²a⁴	ba⁶	b²a⁵	b²a⁶
a	a	a²	a³	a³	a⁴	a⁵	a⁴	a⁵	a⁶	a⁵	a⁶	a³	a⁶	a³	a⁴	a³	a⁴	a⁵	a⁵	a⁶	a³
b	b	ba	b²	ba²	b²a	b	ba³	b²a²	ba	ba⁴	b²a³	ba²	ba⁵	b²a⁴	ba³	ba⁶	b²a⁵	ba⁴	b²a⁶	ba⁵	ba⁶
a²	a²	a³	a⁴	a⁴	a⁵	a⁶	a⁵	a⁶	a³	a⁶	a³	a⁴	a³	a⁴	a⁵	a⁴	a⁵	a⁶	a⁶	a³	a⁴
ba	ba	ba²	ba³	ba³	ba⁴	ba⁵	ba⁴	ba⁵	ba⁶	ba⁵	ba⁶	ba³	ba⁶	ba³	ba⁴	ba³	ba⁴	ba⁵	ba⁵	ba⁶	ba³
b²	b²	b²a	b	b²a²	ba	b²	b²a³	ba²	b²a	b²a⁴	ba³	b²a²	b²a⁵	ba⁴	b²a³	b²a⁶	ba⁵	b²a⁴	ba⁶	b²a⁵	b²a⁶
a³	a³	a⁴	a⁵	a⁵	a⁶	a³	a⁶	a³	a⁴	a³	a⁴	a⁵	a⁴	a⁵	a⁶	a⁵	a⁶	a³	a³	a⁴	a⁵
ba²	ba²	ba³	ba⁴	ba⁴	ba⁵	ba⁶	ba⁵	ba⁶	ba³	ba⁶	ba³	ba⁴	ba³	ba⁴	ba⁵	ba⁴	ba⁵	ba⁶	ba⁶	ba³	ba⁴
b²a	b²a	b²a²	b²a³	b²a³	b²a⁴	b²a⁵	b²a⁴	b²a⁵	b²a⁶	b²a⁵	b²a⁶	b²a³	b²a⁶	b²a³	b²a⁴	b²a³	b²a⁴	b²a⁵	b²a⁵	b²a⁶	b²a³
a⁴	a⁴	a⁵	a⁶	a⁶	a³	a⁴	a³	a⁴	a⁵	a⁴	a⁵	a⁶	a⁵	a⁶	a³	a⁶	a³	a⁴	a⁴	a⁵	a⁶
ba³	ba³	ba⁴	ba⁵	ba⁵	ba⁶	ba³	ba⁶	ba³	ba⁴	ba³	ba⁴	ba⁵	ba⁴	ba⁵	ba⁶	ba⁵	ba⁶	ba³	ba³	ba⁴	ba⁵
b²a²	b²a²	b²a³	b²a⁴	b²a⁴	b²a⁵	b²a⁶	b²a⁵	b²a⁶	b²a³	b²a⁶	b²a³	b²a⁴	b²a³	b²a⁴	b²a⁵	b²a⁴	b²a⁵	b²a⁶	b²a⁶	b²a³	b²a⁴
a⁵	a⁵	a⁶	a³	a³	a⁴	a⁵	a⁴	a⁵	a⁶	a⁵	a⁶	a³	a⁶	a³	a⁴	a³	a⁴	a⁵	a⁵	a⁶	a³
ba⁴	ba⁴	ba⁵	ba⁶	ba⁶	ba³	ba⁴	ba³	ba⁴	ba⁵	ba⁴	ba⁵	ba⁶	ba⁵	ba⁶	ba³	ba⁶	ba³	ba⁴	ba⁴	ba⁵	ba⁶
b²a³	b²a³	b²a⁴	b²a⁵	b²a⁵	b²a⁶	b²a³	b²a⁶	b²a³	b²a⁴	b²a³	b²a⁴	b²a⁵	b²a⁴	b²a⁵	b²a⁶	b²a⁵	b²a⁶	b²a³	b²a³	b²a⁴	b²a⁵
a⁶	a⁶	a³	a⁴	a⁴	a⁵	a⁶	a⁵	a⁶	a³	a⁶	a³	a⁴	a³	a⁴	a⁵	a⁴	a⁵	a⁶	a⁶	a³	a⁴
ba⁵	ba⁵	ba⁶	ba³	ba³	ba⁴	ba⁵	ba⁴	ba⁵	ba⁶	ba⁵	ba⁶	ba³	ba⁶	ba³	ba⁴	ba³	ba⁴	ba⁵	ba⁵	ba⁶	ba³
b²a⁴	b²a⁴	b²a⁵	b²a⁶	b²a⁶	b²a³	b²a⁴	b²a³	b²a⁴	b²a⁵	b²a⁴	b²a⁵	b²a⁶	b²a⁵	b²a⁶	b²a³	b²a⁶	b²a³	b²a⁴	b²a⁴	b²a⁵	b²a⁶
ba⁶	ba⁶	ba³	ba⁴	ba⁴	ba⁵	ba⁶	ba⁵	ba⁶	ba³	ba⁶	ba³	ba⁴	ba³	ba⁴	ba⁵	ba⁴	ba⁵	ba⁶	ba⁶	ba³	ba⁴
b²a⁵	b²a⁵	b²a⁶	b²a³	b²a³	b²a⁴	b²a⁵	b²a⁴	b²a⁵	b²a⁶	b²a⁵	b²a⁶	b²a³	b²a⁶	b²a³	b²a⁴	b²a³	b²a⁴	b²a⁵	b²a⁵	b²a⁶	b²a³
b²a⁶	b²a⁶	b²a³	b²a⁴	b²a⁴	b²a⁵	b²a⁶	b²a⁵	b²a⁶	b²a³	b²a⁶	b²a³	b²a⁴	b²a³	b²a⁴	b²a⁵	b²a⁴	b²a⁵	b²a⁶	b²a⁶	b²a³	b²a⁴

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

8 unique, 63 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
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	13248	⟨a, b \| bab=aaa, bbb=aa⟩	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

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
11	19789	⟨a, b \| aaa=a, aaab=bbb⟩	φ(a) = b, φ(b) = a

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