#14501 ⟨a, b | aaab=b, abbba=a⟩

Properties

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

Element profile

1 element center:
- 1
3 idempotent elements:
- 1, b³a, 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 ⇒ b
ab ⇒ b⁷
b⁹a ⇒ a
a² ⇒ b⁶a

# ab:aaab=b,abbba=a b/a
bbbbbbbbbb=b
ab=bbbbbbb
bbbbbbbbba=a
aa=bbbbbba

Cayley table

Idempotents are shown in bold.

	1	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹
1	1	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹
a	a	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶
b	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	a	b
ba	ba	b⁷a	b⁸	b⁸a	b⁹	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷
b²	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	a	b	ba	b²
b²a	b²a	b⁸a	b⁹	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸
b³	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	a	b	ba	b²	b²a	b³
b³a	b³a	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹
b⁴	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	a	b	ba	b²	b²a	b³	b³a	b⁴
b⁴a	b⁴a	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	a	b
b⁵	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵
b⁵a	b⁵a	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	a	b	ba	b²
b⁶	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶
b⁶a	b⁶a	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	a	b	ba	b²	b²a	b³
b⁷	b⁷	b⁷a	b⁸	b⁸a	b⁹	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷
b⁷a	b⁷a	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	a	b	ba	b²	b²a	b³	b³a	b⁴
b⁸	b⁸	b⁸a	b⁹	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸
b⁸a	b⁸a	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵
b⁹	b⁹	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

5 unique, 26 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	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

Other isomorphic instances

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

3 total

Σ	#	Presentation	Mapping
11	14517	⟨a, b \| aaab=b, babba=a⟩	φ(a) = a, φ(b) = b
11	14525	⟨a, b \| aaab=b, bbaba=a⟩	φ(a) = a, φ(b) = b
11	14529	⟨a, b \| aaab=b, bbbaa=a⟩	φ(a) = a, φ(b) = b