#1991 ⟨a, b | aaa=a, abba=b⟩

Properties

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

Element profile

3 element center:
- 1, a², b³
3 idempotent elements:
- 1, a², b³

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

b⁴ ⇒ b
ba ⇒ ab²
a²b ⇒ b
a³ ⇒ a

# ab:aaa=a,abba=b b/a
bbbb=b
ba=abb
aab=b
aaa=a

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	b²	ab²	b³	ab³
1	1	a	b	a²	ab	b²	ab²	b³	ab³
a	a	a²	ab	a	b	ab²	b²	ab³	b³
b	b	ab²	b²	b	ab³	b³	ab	b	ab²
a²	a²	a	b	a²	ab	b²	ab²	b³	ab³
ab	ab	b²	ab²	ab	b³	ab³	b	ab	b²
b²	b²	ab	b³	b²	ab²	b	ab³	b²	ab
ab²	ab²	b	ab³	ab²	b²	ab	b³	ab²	b
b³	b³	ab³	b	b³	ab	b²	ab²	b³	ab³
ab³	ab³	b³	ab	ab³	b	ab²	b²	ab³	b³

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

26 unique, 939 total

Σ	#	Presentation	Description	Related
7	118	⟨a, b \| aaa=b, bbb=1⟩	Isomorphic to ℤ₉	562 iso
8	568	⟨a, b \| aaa=a, abb=b⟩	Finite non-commutative monoid with 9 elements	5 iso
8	571	⟨a, b \| aaa=a, bbb=a⟩	Isomorphic to ℕ_{(9 = 3)}	48 iso
8	581	⟨a, b \| aaa=b, bbb=a⟩	Isomorphic to ℕ_{(9 = 1)}	62 iso
8	601	⟨a, b \| aba=b, bab=a⟩	Finite non-commutative monoid with 9 elements	3 iso
9	1584	⟨a, b \| aaa=aa, bbb=a⟩	Isomorphic to ℕ_{(9 = 6)}	16 iso
9	1595	⟨a, b \| aaa=ab, bab=b⟩	Finite non-commutative monoid with 9 elements	4 iso, 5 anti-iso
9	1598	⟨a, b \| aaa=ab, bbb=a⟩	Isomorphic to ℕ_{(9 = 4)}	40 iso
9	1608	⟨a, b \| aaa=bb, bbb=a⟩	Isomorphic to ℕ_{(9 = 2)}	70 iso
9	2064	⟨a, b \| aab=b, abba=a⟩	Finite non-commutative monoid with 9 elements	14 iso, 24 anti-iso
9	2255	⟨a, b \| ab=aa, bba=bb⟩	Finite non-commutative monoid with 9 elements
10	4125	⟨a, b \| aab=aaa, baa=b⟩	Finite non-commutative monoid with 9 elements	1 iso, 1 anti-iso
10	4130	⟨a, b \| aab=aaa, bbb=a⟩	Isomorphic to ℕ_{(9 = 7)}	1 iso
10	4156	⟨a, b \| abb=aaa, bbb=a⟩	Isomorphic to ℕ_{(9 = 5)}	12 iso
10	5035	⟨a, b \| aaa=aa, abbb=b⟩	Finite non-commutative monoid with 9 elements	3 iso
10	5079	⟨a, b \| aaa=bb, aaba=b⟩	Finite commutative monoid with 9 elements	8 iso
10	5124	⟨a, b \| aab=aa, bbbb=a⟩	Isomorphic to ℕ_{(9 = 8)}	5 iso
10	5339	⟨a, b \| aaa=bb, aab=ab⟩	Finite non-commutative monoid with 9 elements	1 iso
10	6824	⟨a, b \| aab=b, bbb=aaa⟩	Finite commutative monoid with 9 elements	5 iso
10	7139	⟨a, b \| bb=aa, aaaa=ab⟩	Finite non-commutative monoid with 9 elements	3 iso
10	9327	⟨a, b \| ab=a, baaa=bbb⟩	Finite non-commutative monoid with 9 elements	3 iso
10	9359	⟨a, b \| ab=a, bbaa=bbb⟩	Finite non-commutative monoid with 9 elements	2 iso
11	19506	⟨a, b \| aab=b, aaaaa=ba⟩	Finite non-commutative monoid with 9 elements
11	19507	⟨a, b \| aab=b, aaaaa=bb⟩	Finite commutative monoid with 9 elements
11	20723	⟨a, b \| ab=aa, aaaaa=bb⟩	Finite non-commutative monoid with 9 elements	15 iso
11	25656	⟨a, b \| ab=a, bbbbb=baa⟩	Finite non-commutative monoid with 9 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
10	6257	⟨a, b \| aaa=a, ababa=b⟩	φ(a) = a, φ(b) = ab
11	18701	⟨a, b \| aaa=a, aaabba=b⟩	φ(a) = a, φ(b) = b
11	18721	⟨a, b \| aaa=a, abaaba=b⟩	φ(a) = a, φ(b) = b

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