#12981 ⟨a, b | abb=aaa, baaa=b⟩

Properties

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

Element profile

1 element center:
- 1
3 idempotent elements:
- 1, ab², 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 ⇒ b
ba ⇒ b⁵
a²b ⇒ ab⁵
a³ ⇒ ab²

# ab:abb=aaa,baaa=b b/a
bbbbbbb=b
ba=bbbbb
aab=abbbbb
aaa=abb

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	b²	ab²	b³	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶	ab⁶
1	1	a	b	a²	ab	b²	ab²	b³	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶	ab⁶
a	a	a²	ab	ab²	ab⁵	ab²	ab⁶	ab³	ab	ab⁴	ab²	ab⁵	ab³	ab⁶	ab⁴
b	b	b⁵	b²	b³	b⁶	b³	b	b⁴	b²	b⁵	b³	b⁶	b⁴	b	b⁵
a²	a²	ab²	ab⁵	ab⁶	ab³	ab⁶	ab⁴	ab	ab⁵	ab²	ab⁶	ab³	ab	ab⁴	ab²
ab	ab	ab⁵	ab²	ab³	ab⁶	ab³	ab	ab⁴	ab²	ab⁵	ab³	ab⁶	ab⁴	ab	ab⁵
b²	b²	b⁶	b³	b⁴	b	b⁴	b²	b⁵	b³	b⁶	b⁴	b	b⁵	b²	b⁶
ab²	ab²	ab⁶	ab³	ab⁴	ab	ab⁴	ab²	ab⁵	ab³	ab⁶	ab⁴	ab	ab⁵	ab²	ab⁶
b³	b³	b	b⁴	b⁵	b²	b⁵	b³	b⁶	b⁴	b	b⁵	b²	b⁶	b³	b
ab³	ab³	ab	ab⁴	ab⁵	ab²	ab⁵	ab³	ab⁶	ab⁴	ab	ab⁵	ab²	ab⁶	ab³	ab
b⁴	b⁴	b²	b⁵	b⁶	b³	b⁶	b⁴	b	b⁵	b²	b⁶	b³	b	b⁴	b²
ab⁴	ab⁴	ab²	ab⁵	ab⁶	ab³	ab⁶	ab⁴	ab	ab⁵	ab²	ab⁶	ab³	ab	ab⁴	ab²
b⁵	b⁵	b³	b⁶	b	b⁴	b	b⁵	b²	b⁶	b³	b	b⁴	b²	b⁵	b³
ab⁵	ab⁵	ab³	ab⁶	ab	ab⁴	ab	ab⁵	ab²	ab⁶	ab³	ab	ab⁴	ab²	ab⁵	ab³
b⁶	b⁶	b⁴	b	b²	b⁵	b²	b⁶	b³	b	b⁴	b²	b⁵	b³	b⁶	b⁴
ab⁶	ab⁶	ab⁴	ab	ab²	ab⁵	ab²	ab⁶	ab³	ab	ab⁴	ab²	ab⁵	ab³	ab⁶	ab⁴

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

22 unique, 187 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	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.

3 total

Σ	#	Presentation	Mapping
11	12987	⟨a, b \| abb=aaa, babb=b⟩	φ(a) = a, φ(b) = b
11	12991	⟨a, b \| abb=aaa, bbab=b⟩	φ(a) = a, φ(b) = b
11	12993	⟨a, b \| abb=aaa, bbba=b⟩	φ(a) = a, φ(b) = b

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