#13192 ⟨a, b | abb=aaa, bbb=ba⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 15 elements

Element profile

1 element center:
- 1
3 idempotent elements:
- 1, ab⁴, b⁶

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

b⁷ ⇒ b⁵
ab⁶ ⇒ ab⁴
ba ⇒ b³
a²b² ⇒ ab⁴
a³ ⇒ ab²

# ab:abb=aaa,bbb=ba b/a
bbbbbbb=bbbbb
abbbbbb=abbbb
ba=bbb
aabb=abbbb
aaa=abb

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	b²	a²b	ab²	b³	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶
1	1	a	b	a²	ab	b²	a²b	ab²	b³	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶
a	a	a²	ab	ab²	a²b	ab²	ab³	ab⁴	ab³	ab⁵	ab⁴	ab⁴	ab⁵	ab⁵	ab⁴
b	b	b³	b²	b⁵	b⁴	b³	b⁶	b⁵	b⁴	b⁶	b⁵	b⁵	b⁶	b⁶	b⁵
a²	a²	ab²	a²b	ab⁴	ab³	ab⁴	ab⁵	ab⁴	ab⁵	ab⁵	ab⁴	ab⁴	ab⁵	ab⁵	ab⁴
ab	ab	ab³	ab²	ab⁵	ab⁴	ab³	ab⁴	ab⁵	ab⁴	ab⁴	ab⁵	ab⁵	ab⁴	ab⁴	ab⁵
b²	b²	b⁴	b³	b⁶	b⁵	b⁴	b⁵	b⁶	b⁵	b⁵	b⁶	b⁶	b⁵	b⁵	b⁶
a²b	a²b	ab⁵	ab⁴	ab⁵	ab⁴	ab⁵	ab⁴	ab⁵	ab⁴	ab⁴	ab⁵	ab⁵	ab⁴	ab⁴	ab⁵
ab²	ab²	ab⁴	ab³	ab⁴	ab⁵	ab⁴	ab⁵	ab⁴	ab⁵	ab⁵	ab⁴	ab⁴	ab⁵	ab⁵	ab⁴
b³	b³	b⁵	b⁴	b⁵	b⁶	b⁵	b⁶	b⁵	b⁶	b⁶	b⁵	b⁵	b⁶	b⁶	b⁵
ab³	ab³	ab⁵	ab⁴	ab⁵	ab⁴	ab⁵	ab⁴	ab⁵	ab⁴	ab⁴	ab⁵	ab⁵	ab⁴	ab⁴	ab⁵
b⁴	b⁴	b⁶	b⁵	b⁶	b⁵	b⁶	b⁵	b⁶	b⁵	b⁵	b⁶	b⁶	b⁵	b⁵	b⁶
ab⁴	ab⁴	ab⁴	ab⁵	ab⁴	ab⁵	ab⁴	ab⁵	ab⁴	ab⁵	ab⁵	ab⁴	ab⁴	ab⁵	ab⁵	ab⁴
b⁵	b⁵	b⁵	b⁶	b⁵	b⁶	b⁵	b⁶	b⁵	b⁶	b⁶	b⁵	b⁵	b⁶	b⁶	b⁵
ab⁵	ab⁵	ab⁵	ab⁴	ab⁵	ab⁴	ab⁵	ab⁴	ab⁵	ab⁴	ab⁴	ab⁵	ab⁵	ab⁴	ab⁴	ab⁵
b⁶	b⁶	b⁶	b⁵	b⁶	b⁵	b⁶	b⁵	b⁶	b⁵	b⁵	b⁶	b⁶	b⁵	b⁵	b⁶

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

22 unique, 190 total

Σ	#	Presentation	Description	Related
9	1602	⟨a, b \| aaa=bb, aba=a⟩	Finite non-commutative monoid with 15 elements	2 iso
9	1603	⟨a, b \| aaa=bb, aba=b⟩	Finite non-commutative monoid with 15 elements	1 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	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

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