#6263 ⟨a, b | aaa=a, abbba=b⟩

Properties

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

Element profile

4 element center:
- 1, a², b⁴, b⁸
3 idempotent elements:
- 1, a², b⁸

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

b⁹ ⇒ b
ba ⇒ ab³
a²b ⇒ b
a³ ⇒ a

# ab:aaa=a,abbba=b b/a
bbbbbbbbb=b
ba=abbb
aab=b
aaa=a

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	b²	ab²	b³	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶	ab⁶	b⁷	ab⁷	b⁸	ab⁸
1	1	a	b	a²	ab	b²	ab²	b³	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶	ab⁶	b⁷	ab⁷	b⁸	ab⁸
a	a	a²	ab	a	b	ab²	b²	ab³	b³	ab⁴	b⁴	ab⁵	b⁵	ab⁶	b⁶	ab⁷	b⁷	ab⁸	b⁸
b	b	ab³	b²	b	ab⁴	b³	ab⁵	b⁴	ab⁶	b⁵	ab⁷	b⁶	ab⁸	b⁷	ab	b⁸	ab²	b	ab³
a²	a²	a	b	a²	ab	b²	ab²	b³	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶	ab⁶	b⁷	ab⁷	b⁸	ab⁸
ab	ab	b³	ab²	ab	b⁴	ab³	b⁵	ab⁴	b⁶	ab⁵	b⁷	ab⁶	b⁸	ab⁷	b	ab⁸	b²	ab	b³
b²	b²	ab⁶	b³	b²	ab⁷	b⁴	ab⁸	b⁵	ab	b⁶	ab²	b⁷	ab³	b⁸	ab⁴	b	ab⁵	b²	ab⁶
ab²	ab²	b⁶	ab³	ab²	b⁷	ab⁴	b⁸	ab⁵	b	ab⁶	b²	ab⁷	b³	ab⁸	b⁴	ab	b⁵	ab²	b⁶
b³	b³	ab	b⁴	b³	ab²	b⁵	ab³	b⁶	ab⁴	b⁷	ab⁵	b⁸	ab⁶	b	ab⁷	b²	ab⁸	b³	ab
ab³	ab³	b	ab⁴	ab³	b²	ab⁵	b³	ab⁶	b⁴	ab⁷	b⁵	ab⁸	b⁶	ab	b⁷	ab²	b⁸	ab³	b
b⁴	b⁴	ab⁴	b⁵	b⁴	ab⁵	b⁶	ab⁶	b⁷	ab⁷	b⁸	ab⁸	b	ab	b²	ab²	b³	ab³	b⁴	ab⁴
ab⁴	ab⁴	b⁴	ab⁵	ab⁴	b⁵	ab⁶	b⁶	ab⁷	b⁷	ab⁸	b⁸	ab	b	ab²	b²	ab³	b³	ab⁴	b⁴
b⁵	b⁵	ab⁷	b⁶	b⁵	ab⁸	b⁷	ab	b⁸	ab²	b	ab³	b²	ab⁴	b³	ab⁵	b⁴	ab⁶	b⁵	ab⁷
ab⁵	ab⁵	b⁷	ab⁶	ab⁵	b⁸	ab⁷	b	ab⁸	b²	ab	b³	ab²	b⁴	ab³	b⁵	ab⁴	b⁶	ab⁵	b⁷
b⁶	b⁶	ab²	b⁷	b⁶	ab³	b⁸	ab⁴	b	ab⁵	b²	ab⁶	b³	ab⁷	b⁴	ab⁸	b⁵	ab	b⁶	ab²
ab⁶	ab⁶	b²	ab⁷	ab⁶	b³	ab⁸	b⁴	ab	b⁵	ab²	b⁶	ab³	b⁷	ab⁴	b⁸	ab⁵	b	ab⁶	b²
b⁷	b⁷	ab⁵	b⁸	b⁷	ab⁶	b	ab⁷	b²	ab⁸	b³	ab	b⁴	ab²	b⁵	ab³	b⁶	ab⁴	b⁷	ab⁵
ab⁷	ab⁷	b⁵	ab⁸	ab⁷	b⁶	ab	b⁷	ab²	b⁸	ab³	b	ab⁴	b²	ab⁵	b³	ab⁶	b⁴	ab⁷	b⁵
b⁸	b⁸	ab⁸	b	b⁸	ab	b²	ab²	b³	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶	ab⁶	b⁷	ab⁷	b⁸	ab⁸
ab⁸	ab⁸	b⁸	ab	ab⁸	b	ab²	b²	ab³	b³	ab⁴	b⁴	ab⁵	b⁵	ab⁶	b⁶	ab⁷	b⁷	ab⁸	b⁸

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

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