#16036 ⟨a, b | aaa=ab, bbab=bb⟩

Properties

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

Element profile

1 element center:
- 1
4 idempotent elements:
- 1, b²a², a⁶, ba⁷

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

a⁸ ⇒ a⁵
ab ⇒ a³
b²a³ ⇒ b²
b³ ⇒ b²a²

# ab:aaa=ab,bbab=bb a/b
aaaaaaaa=aaaaa
ab=aaa
bbaaa=bb
bbb=bbaa

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ba	b²	a³	ba²	b²a	a⁴	ba³	b²a²	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	ba⁷
1	1	a	b	a²	ba	b²	a³	ba²	b²a	a⁴	ba³	b²a²	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	ba⁷
a	a	a²	a³	a³	a⁴	a⁵	a⁴	a⁵	a⁶	a⁵	a⁶	a⁷	a⁶	a⁷	a⁷	a⁵	a⁵	a⁶	a⁷
b	b	ba	b²	ba²	b²a	b²a²	ba³	b²a²	b²	ba⁴	b²	b²a	ba⁵	b²a	ba⁶	b²a²	ba⁷	b²	b²a
a²	a²	a³	a⁴	a⁴	a⁵	a⁶	a⁵	a⁶	a⁷	a⁶	a⁷	a⁵	a⁷	a⁵	a⁵	a⁶	a⁶	a⁷	a⁵
ba	ba	ba²	ba³	ba³	ba⁴	ba⁵	ba⁴	ba⁵	ba⁶	ba⁵	ba⁶	ba⁷	ba⁶	ba⁷	ba⁷	ba⁵	ba⁵	ba⁶	ba⁷
b²	b²	b²a	b²a²	b²a²	b²	b²a	b²	b²a	b²a²	b²a	b²a²	b²	b²a²	b²	b²	b²a	b²a	b²a²	b²
a³	a³	a⁴	a⁵	a⁵	a⁶	a⁷	a⁶	a⁷	a⁵	a⁷	a⁵	a⁶	a⁵	a⁶	a⁶	a⁷	a⁷	a⁵	a⁶
ba²	ba²	ba³	ba⁴	ba⁴	ba⁵	ba⁶	ba⁵	ba⁶	ba⁷	ba⁶	ba⁷	ba⁵	ba⁷	ba⁵	ba⁵	ba⁶	ba⁶	ba⁷	ba⁵
b²a	b²a	b²a²	b²	b²	b²a	b²a²	b²a	b²a²	b²	b²a²	b²	b²a	b²	b²a	b²a	b²a²	b²a²	b²	b²a
a⁴	a⁴	a⁵	a⁶	a⁶	a⁷	a⁵	a⁷	a⁵	a⁶	a⁵	a⁶	a⁷	a⁶	a⁷	a⁷	a⁵	a⁵	a⁶	a⁷
ba³	ba³	ba⁴	ba⁵	ba⁵	ba⁶	ba⁷	ba⁶	ba⁷	ba⁵	ba⁷	ba⁵	ba⁶	ba⁵	ba⁶	ba⁶	ba⁷	ba⁷	ba⁵	ba⁶
b²a²	b²a²	b²	b²a	b²a	b²a²	b²	b²a²	b²	b²a	b²	b²a	b²a²	b²a	b²a²	b²a²	b²	b²	b²a	b²a²
a⁵	a⁵	a⁶	a⁷	a⁷	a⁵	a⁶	a⁵	a⁶	a⁷	a⁶	a⁷	a⁵	a⁷	a⁵	a⁵	a⁶	a⁶	a⁷	a⁵
ba⁴	ba⁴	ba⁵	ba⁶	ba⁶	ba⁷	ba⁵	ba⁷	ba⁵	ba⁶	ba⁵	ba⁶	ba⁷	ba⁶	ba⁷	ba⁷	ba⁵	ba⁵	ba⁶	ba⁷
a⁶	a⁶	a⁷	a⁵	a⁵	a⁶	a⁷	a⁶	a⁷	a⁵	a⁷	a⁵	a⁶	a⁵	a⁶	a⁶	a⁷	a⁷	a⁵	a⁶
ba⁵	ba⁵	ba⁶	ba⁷	ba⁷	ba⁵	ba⁶	ba⁵	ba⁶	ba⁷	ba⁶	ba⁷	ba⁵	ba⁷	ba⁵	ba⁵	ba⁶	ba⁶	ba⁷	ba⁵
a⁷	a⁷	a⁵	a⁶	a⁶	a⁷	a⁵	a⁷	a⁵	a⁶	a⁵	a⁶	a⁷	a⁶	a⁷	a⁷	a⁵	a⁵	a⁶	a⁷
ba⁶	ba⁶	ba⁷	ba⁵	ba⁵	ba⁶	ba⁷	ba⁶	ba⁷	ba⁵	ba⁷	ba⁵	ba⁶	ba⁵	ba⁶	ba⁶	ba⁷	ba⁷	ba⁵	ba⁶
ba⁷	ba⁷	ba⁵	ba⁶	ba⁶	ba⁷	ba⁵	ba⁷	ba⁵	ba⁶	ba⁵	ba⁶	ba⁷	ba⁶	ba⁷	ba⁷	ba⁵	ba⁵	ba⁶	ba⁷

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

5 unique, 28 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	14501	⟨a, b \| aaab=b, abbba=a⟩	Finite non-commutative monoid with 19 elements	3 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.

1 total

Σ	#	Presentation	Mapping
11	16040	⟨a, b \| aaa=ab, bbba=bb⟩	φ(a) = a, φ(b) = b

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