#1603 ⟨a, b | aaa=bb, aba=b⟩

Properties

Presentation has sum-of-sides 9
Finite non-commutative monoid with 15 elements

Element profile

3 element center:
- 1, a³, a⁶
2 idempotent elements:
- 1, a⁶

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

a⁹ ⇒ a³
ba⁶ ⇒ b
ab ⇒ ba⁵
b² ⇒ a³

# ab:aaa=bb,aba=b a/b
aaaaaaaaa=aaa
baaaaaa=b
ab=baaaaa
bb=aaa

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	a⁸
1	1	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	a⁸
a	a	a²	ba⁵	a³	b	a⁴	ba	a⁵	ba²	a⁶	ba³	a⁷	ba⁴	a⁸	a³
b	b	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	b	a⁸	ba	ba²
a²	a²	a³	ba⁴	a⁴	ba⁵	a⁵	b	a⁶	ba	a⁷	ba²	a⁸	ba³	a³	a⁴
ba	ba	ba²	a⁸	ba³	a³	ba⁴	a⁴	ba⁵	a⁵	b	a⁶	ba	a⁷	ba²	ba³
a³	a³	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	b	a⁸	ba	a³	ba²	a⁴	a⁵
ba²	ba²	ba³	a⁷	ba⁴	a⁸	ba⁵	a³	b	a⁴	ba	a⁵	ba²	a⁶	ba³	ba⁴
a⁴	a⁴	a⁵	ba²	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a³	b	a⁴	ba	a⁵	a⁶
ba³	ba³	ba⁴	a⁶	ba⁵	a⁷	b	a⁸	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	ba⁵
a⁵	a⁵	a⁶	ba	a⁷	ba²	a⁸	ba³	a³	ba⁴	a⁴	ba⁵	a⁵	b	a⁶	a⁷
ba⁴	ba⁴	ba⁵	a⁵	b	a⁶	ba	a⁷	ba²	a⁸	ba³	a³	ba⁴	a⁴	ba⁵	b
a⁶	a⁶	a⁷	b	a⁸	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	a⁸
ba⁵	ba⁵	b	a⁴	ba	a⁵	ba²	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a³	b	ba
a⁷	a⁷	a⁸	ba⁵	a³	b	a⁴	ba	a⁵	ba²	a⁶	ba³	a⁷	ba⁴	a⁸	a³
a⁸	a⁸	a³	ba⁴	a⁴	ba⁵	a⁵	b	a⁶	ba	a⁷	ba²	a⁸	ba³	a³	a⁴

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

22 unique, 189 total

Σ	#	Presentation	Description	Related
9	1602	⟨a, b \| aaa=bb, aba=a⟩	Finite non-commutative monoid with 15 elements	2 iso
9	1751	⟨a, b \| aaab=1, bbbbb=1⟩	Isomorphic to ℤ₁₅	138 iso
9	1993	⟨a, b \| aaa=a, abbb=b⟩	Finite non-commutative monoid with 15 elements	4 iso
10	5342	⟨a, b \| aaa=bb, aba=aa⟩	Finite non-commutative monoid with 15 elements	1 iso
10	5344	⟨a, b \| aaa=bb, aba=bb⟩	Finite non-commutative monoid with 15 elements	2 iso
10	6276	⟨a, b \| aaa=a, bbbbb=a⟩	Isomorphic to ℕ_{(15 = 5)}
10	6316	⟨a, b \| aaa=b, bbbbb=a⟩	Isomorphic to ℕ_{(15 = 1)}
10	6317	⟨a, b \| aaa=b, bbbbb=b⟩	Isomorphic to ℕ_{(15 = 3)}
11	12981	⟨a, b \| abb=aaa, baaa=b⟩	Finite non-commutative monoid with 15 elements	3 iso
11	12985	⟨a, b \| abb=aaa, baba=b⟩	Finite non-commutative monoid with 15 elements	7 iso
11	13123	⟨a, b \| aab=aaa, aba=bb⟩	Finite non-commutative monoid with 15 elements
11	13191	⟨a, b \| abb=aaa, bbb=ab⟩	Finite non-commutative monoid with 15 elements	1 iso
11	13192	⟨a, b \| abb=aaa, bbb=ba⟩	Finite non-commutative monoid with 15 elements
11	15439	⟨a, b \| aaa=aa, bbbbb=a⟩	Isomorphic to ℕ_{(15 = 10)}
11	15503	⟨a, b \| aaa=ab, bbbbb=a⟩	Isomorphic to ℕ_{(15 = 6)}	1 iso
11	15543	⟨a, b \| aaa=bb, bbbbb=a⟩	Isomorphic to ℕ_{(15 = 2)}	1 iso
11	16041	⟨a, b \| aaa=ab, bbbb=aa⟩	Finite non-commutative monoid with 15 elements
11	16142	⟨a, b \| aab=aa, bbbb=ab⟩	Finite non-commutative monoid with 15 elements
11	19374	⟨a, b \| aaa=b, bbbbb=ab⟩	Isomorphic to ℕ_{(15 = 4)}
11	19502	⟨a, b \| aab=a, bbbbb=ba⟩	Finite non-commutative monoid with 15 elements	1 anti-iso
11	20144	⟨a, b \| aab=b, bbaa=aaa⟩	Finite non-commutative monoid with 15 elements
11	20819	⟨a, b \| ab=aa, bbaaa=bb⟩	Finite non-commutative monoid with 15 elements	6 iso

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	20278	⟨a, b \| aba=b, abab=aaa⟩	φ(a) = a, φ(b) = b

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