⟨a, b | aaa=bb, abb=b⟩

Up:

Monoids with two generators and two relations

#1604 ⟨a, b | aaa=bb, abb=a⟩

#1606 ⟨a, b | aaa=bb, bab=a⟩

Properties

Presentation has sum-of-sides 9

Isomorphic to ℕ_{(8 = 3)}

Finite commutative monoid with 8 elements

Not cancellative, because multiplication by a³ is not injective:

a³ ⋅ a⁵ = a³ and a³ ⋅ 1 = a³, however a⁵ ≠ 1

Commutative Gröbner basis: ⟨a, b | a⁸=a³, b=a⁴⟩

Cancellative quotient is isomorphic to ℤ₅

Enveloping group is isomorphic to ℤ₅

Group of units is isomorphic to ℤ₁

2 Archimedian components:

1, b

	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷
1	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷
a	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a³
a²	a²	a³	a⁴	a⁵	a⁶	a⁷	a³	a⁴
a³	a³	a⁴	a⁵	a⁶	a⁷	a³	a⁴	a⁵
a⁴	a⁴	a⁵	a⁶	a⁷	a³	a⁴	a⁵	a⁶
a⁵	a⁵	a⁶	a⁷	a³	a⁴	a⁵	a⁶	a⁷
a⁶	a⁶	a⁷	a³	a⁴	a⁵	a⁶	a⁷	a³
a⁷	a⁷	a³	a⁴	a⁵	a⁶	a⁷	a³	a⁴

a²

a³

a⁴

a⁵

a⁶

a⁷

a²

a³

a⁴

a⁵

a⁶

a⁷

a²

a³

a⁴

a⁵

a⁶

a⁷

a³

a²

a³

a⁴

a⁵

a⁶

a⁷

a³

a⁴

a³

a⁴

a⁵

a⁶

a⁷

a³

a⁴

a⁵

a⁴

a⁵

a⁶

a⁷

a³

a⁴

a⁵

a⁶

a⁵

a⁶

a⁷

a³

a⁴

a⁵

a⁶

a⁷

a⁶

a⁷

a³

a⁴

a⁵

a⁶

a⁷

a³

a⁷

a³

a⁴

a⁵

a⁶

a⁷

a³

a⁴

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
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	a⁸ ⇒ a³	[3]
2.	b ⇒ a⁴	[2]

# ab:aaa=bb,abb=b a/b aaaaaaaa=aaa b=aaaa

Same cardinality

Σ	#	Presentation	Description	Related
7	188	⟨a, b \| aab=1, bbbb=1⟩	Isomorphic to ℤ₈	727 iso
7	330	⟨a, b \| aa=1, abba=b⟩	Finite non-commutative monoid with 8 elements	60 iso
8	556	⟨a, b \| aba=b, aabb=1⟩	Finite non-Abelian group with 8 elements	28 iso
8	898	⟨a, b \| aa=a, bbbb=a⟩	Isomorphic to ℕ_{(8 = 4)}	48 iso
8	918	⟨a, b \| aa=b, bbbb=a⟩	Isomorphic to ℕ_{(8 = 1)}	34 iso
8	919	⟨a, b \| aa=b, bbbb=b⟩	Isomorphic to ℕ_{(8 = 2)}	46 iso
8	961	⟨a, b \| aa=a, aba=bb⟩	Finite non-commutative monoid with 8 elements	5 iso
9	1606	⟨a, b \| aaa=bb, bab=a⟩	Finite commutative monoid with 8 elements	14 iso
9	1615	⟨a, b \| aab=aa, baa=b⟩	Finite non-commutative monoid with 8 elements	9 iso, 5 anti-iso
9	1650	⟨a, b \| aab=bb, baa=a⟩	Finite non-commutative monoid with 8 elements	4 iso, 11 anti-iso
9	2206	⟨a, b \| ab=aa, bbbb=a⟩	Isomorphic to ℕ_{(8 = 5)}	32 iso
9	2220	⟨a, b \| bb=aa, aaab=a⟩	Finite commutative monoid with 8 elements	22 iso
9	2247	⟨a, b \| ab=aa, baa=bb⟩	Finite non-commutative monoid with 8 elements	1 iso
9	2256	⟨a, b \| ab=aa, bbb=aa⟩	Finite non-commutative monoid with 8 elements	1 iso
9	2258	⟨a, b \| ab=aa, bbb=ba⟩	Finite non-commutative monoid with 8 elements
9	2883	⟨a, b \| aa=a, abbbb=b⟩	Finite non-commutative monoid with 8 elements	14 iso
9	3107	⟨a, b \| ab=a, baaa=bb⟩	Finite non-commutative monoid with 8 elements	9 iso
9	3123	⟨a, b \| ab=a, bbaa=bb⟩	Finite non-commutative monoid with 8 elements	10 iso
10	5191	⟨a, b \| aab=bb, aaaa=b⟩	Isomorphic to ℕ_{(8 = 6)}	18 iso
10	6664	⟨a, b \| aab=b, aaaa=ba⟩	Finite non-commutative monoid with 8 elements	1 iso
10	7057	⟨a, b \| ab=aa, aaaa=bb⟩	Finite non-commutative monoid with 8 elements	7 iso
10	9380	⟨a, b \| ab=a, bbbb=baa⟩	Finite non-commutative monoid with 8 elements	2 iso
11	12606	⟨a, b \| abbb=aa, bbbb=a⟩	Isomorphic to ℕ_{(8 = 7)}	14 iso
11	20047	⟨a, b \| aab=a, bbbb=bbb⟩	Finite non-commutative monoid with 8 elements
11	20051	⟨a, b \| aab=b, aaaa=abb⟩	Finite commutative monoid with 8 elements	1 iso
11	24863	⟨a, b \| ab=a, aaaaaa=bb⟩	Finite commutative monoid with 8 elements
11	25112	⟨a, b \| ab=a, bbbbbb=aa⟩	Finite commutative monoid with 8 elements
11	25114	⟨a, b \| ab=a, bbbbbb=ba⟩	Finite non-commutative monoid with 8 elements
11	25411	⟨a, b \| ab=a, aaaaa=bbb⟩	Finite commutative monoid with 8 elements
11	25652	⟨a, b \| ab=a, bbbbb=aaa⟩	Finite commutative monoid with 8 elements
11	25658	⟨a, b \| ab=a, bbbbb=bba⟩	Finite non-commutative monoid with 8 elements
11	25897	⟨a, b \| ab=a, bbbb=aaaa⟩	Finite commutative monoid with 8 elements
11	25911	⟨a, b \| ab=a, bbbb=bbba⟩	Finite non-commutative monoid with 8 elements

Presentation

Description

188

⟨a, b | aab=1, bbbb=1⟩

Isomorphic to ℤ₈

727 iso

330

⟨a, b | aa=1, abba=b⟩

Finite non-commutative monoid with 8 elements

60 iso

556

⟨a, b | aba=b, aabb=1⟩

Finite non-Abelian group with 8 elements

28 iso

898

⟨a, b | aa=a, bbbb=a⟩

Isomorphic to ℕ_{(8 = 4)}

48 iso

918

⟨a, b | aa=b, bbbb=a⟩

Isomorphic to ℕ_{(8 = 1)}

34 iso

919

⟨a, b | aa=b, bbbb=b⟩

Isomorphic to ℕ_{(8 = 2)}

46 iso

961

⟨a, b | aa=a, aba=bb⟩

Finite non-commutative monoid with 8 elements

5 iso

1606

⟨a, b | aaa=bb, bab=a⟩

Finite commutative monoid with 8 elements

14 iso

1615

⟨a, b | aab=aa, baa=b⟩

Finite non-commutative monoid with 8 elements

9 iso, 5 anti-iso

1650

⟨a, b | aab=bb, baa=a⟩

Finite non-commutative monoid with 8 elements

4 iso, 11 anti-iso

2206

⟨a, b | ab=aa, bbbb=a⟩

Isomorphic to ℕ_{(8 = 5)}

32 iso

2220

⟨a, b | bb=aa, aaab=a⟩

Finite commutative monoid with 8 elements

22 iso

2247

⟨a, b | ab=aa, baa=bb⟩

Finite non-commutative monoid with 8 elements

1 iso

2256

⟨a, b | ab=aa, bbb=aa⟩

Finite non-commutative monoid with 8 elements

1 iso

2258

⟨a, b | ab=aa, bbb=ba⟩

Finite non-commutative monoid with 8 elements

2883

⟨a, b | aa=a, abbbb=b⟩

Finite non-commutative monoid with 8 elements

14 iso

3107

⟨a, b | ab=a, baaa=bb⟩

Finite non-commutative monoid with 8 elements

9 iso

3123

⟨a, b | ab=a, bbaa=bb⟩

Finite non-commutative monoid with 8 elements

10 iso

5191

⟨a, b | aab=bb, aaaa=b⟩

Isomorphic to ℕ_{(8 = 6)}

18 iso

6664

⟨a, b | aab=b, aaaa=ba⟩

Finite non-commutative monoid with 8 elements

1 iso

7057

⟨a, b | ab=aa, aaaa=bb⟩

Finite non-commutative monoid with 8 elements

7 iso

9380

⟨a, b | ab=a, bbbb=baa⟩

Finite non-commutative monoid with 8 elements

2 iso

12606

⟨a, b | abbb=aa, bbbb=a⟩

Isomorphic to ℕ_{(8 = 7)}

14 iso

20047

⟨a, b | aab=a, bbbb=bbb⟩

Finite non-commutative monoid with 8 elements

20051

⟨a, b | aab=b, aaaa=abb⟩

Finite commutative monoid with 8 elements

1 iso

24863

⟨a, b | ab=a, aaaaaa=bb⟩

Finite commutative monoid with 8 elements

25112

⟨a, b | ab=a, bbbbbb=aa⟩

Finite commutative monoid with 8 elements

25114

⟨a, b | ab=a, bbbbbb=ba⟩

Finite non-commutative monoid with 8 elements

25411

⟨a, b | ab=a, aaaaa=bbb⟩

Finite commutative monoid with 8 elements

25652

⟨a, b | ab=a, bbbbb=aaa⟩

Finite commutative monoid with 8 elements

25658

⟨a, b | ab=a, bbbbb=bba⟩

Finite non-commutative monoid with 8 elements

25897

⟨a, b | ab=a, bbbb=aaaa⟩

Finite commutative monoid with 8 elements

25911

⟨a, b | ab=a, bbbb=bbba⟩

Finite non-commutative monoid with 8 elements

Isomorphic instances

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

Σ	#	Presentation	Mapping
9	2007	⟨a, b \| aaa=b, aabb=b⟩	φ(a) = a, φ(b) = aaa
9	2009	⟨a, b \| aaa=b, abab=b⟩	φ(a) = a, φ(b) = aaa
9	2011	⟨a, b \| aaa=b, abba=b⟩	φ(a) = a, φ(b) = aaa
9	2015	⟨a, b \| aaa=b, baab=b⟩	φ(a) = a, φ(b) = aaa
9	3070	⟨a, b \| aa=b, bbbb=ab⟩	φ(a) = a, φ(b) = aa
10	5057	⟨a, b \| aaa=ab, abbb=b⟩	φ(a) = a, φ(b) = aaaaaaa
10	5075	⟨a, b \| aaa=bb, aaaa=b⟩	φ(a) = a, φ(b) = aaaa
10	8939	⟨a, b \| aa=b, aabbb=ab⟩	φ(a) = a, φ(b) = aa
10	8940	⟨a, b \| aa=b, aabbb=ba⟩	φ(a) = a, φ(b) = aa
10	8950	⟨a, b \| aa=b, ababb=ab⟩	φ(a) = a, φ(b) = aa
10	8951	⟨a, b \| aa=b, ababb=ba⟩	φ(a) = a, φ(b) = aa
10	8954	⟨a, b \| aa=b, abbab=ab⟩	φ(a) = a, φ(b) = aa
10	8955	⟨a, b \| aa=b, abbab=ba⟩	φ(a) = a, φ(b) = aa
10	8958	⟨a, b \| aa=b, abbba=ab⟩	φ(a) = a, φ(b) = aa
10	8968	⟨a, b \| aa=b, baabb=ab⟩	φ(a) = a, φ(b) = aa
10	8969	⟨a, b \| aa=b, baabb=ba⟩	φ(a) = a, φ(b) = aa
10	8972	⟨a, b \| aa=b, babab=ab⟩	φ(a) = a, φ(b) = aa
10	9250	⟨a, b \| aa=b, bbbb=aaa⟩	φ(a) = a, φ(b) = aa
11	15448	⟨a, b \| aaa=ab, aaabb=b⟩	φ(a) = a, φ(b) = aaaaaaa
11	15452	⟨a, b \| aaa=ab, aabab=b⟩	φ(a) = a, φ(b) = aaaaaaa
11	15454	⟨a, b \| aaa=ab, aabba=b⟩	φ(a) = a, φ(b) = aaaaaaa
11	15460	⟨a, b \| aaa=ab, abaab=b⟩	φ(a) = a, φ(b) = aaaaaaa
11	15462	⟨a, b \| aaa=ab, ababa=b⟩	φ(a) = a, φ(b) = aaaaaaa
11	15466	⟨a, b \| aaa=ab, abbaa=b⟩	φ(a) = a, φ(b) = aaaaaaa
11	18763	⟨a, b \| aaa=b, aaaaab=b⟩	φ(a) = a, φ(b) = aaa
11	18765	⟨a, b \| aaa=b, aaaaba=b⟩	φ(a) = a, φ(b) = aaa
11	18769	⟨a, b \| aaa=b, aaabaa=b⟩	φ(a) = a, φ(b) = aaa
11	19870	⟨a, b \| aaa=b, aabb=aaa⟩	φ(a) = a, φ(b) = aaa
11	19878	⟨a, b \| aaa=b, abab=aaa⟩	φ(a) = a, φ(b) = aaa
11	19886	⟨a, b \| aaa=b, abba=aaa⟩	φ(a) = a, φ(b) = aaa
11	19900	⟨a, b \| aaa=b, baab=aaa⟩	φ(a) = a, φ(b) = aaa
11	19935	⟨a, b \| aab=a, aaab=bbb⟩	φ(a) = aaaa, φ(b) = a
11	19943	⟨a, b \| aab=a, aaba=bbb⟩	φ(a) = aaaa, φ(b) = a
11	19959	⟨a, b \| aab=a, abaa=bbb⟩	φ(a) = aaaa, φ(b) = a
11	24736	⟨a, b \| aa=b, aaaabb=ab⟩	φ(a) = a, φ(b) = aa
11	24737	⟨a, b \| aa=b, aaaabb=ba⟩	φ(a) = a, φ(b) = aa
11	24744	⟨a, b \| aa=b, aaabab=ab⟩	φ(a) = a, φ(b) = aa
11	24745	⟨a, b \| aa=b, aaabab=ba⟩	φ(a) = a, φ(b) = aa
11	24748	⟨a, b \| aa=b, aaabba=ab⟩	φ(a) = a, φ(b) = aa
11	24749	⟨a, b \| aa=b, aaabba=ba⟩	φ(a) = a, φ(b) = aa
11	24756	⟨a, b \| aa=b, aabaab=ab⟩	φ(a) = a, φ(b) = aa
11	24757	⟨a, b \| aa=b, aabaab=ba⟩	φ(a) = a, φ(b) = aa
11	24760	⟨a, b \| aa=b, aababa=ab⟩	φ(a) = a, φ(b) = aa
11	24761	⟨a, b \| aa=b, aababa=ba⟩	φ(a) = a, φ(b) = aa
11	24768	⟨a, b \| aa=b, aabbaa=ab⟩	φ(a) = a, φ(b) = aa
11	24783	⟨a, b \| aa=b, abaaab=ab⟩	φ(a) = a, φ(b) = aa
11	24784	⟨a, b \| aa=b, abaaab=ba⟩	φ(a) = a, φ(b) = aa
11	24787	⟨a, b \| aa=b, abaaba=ab⟩	φ(a) = a, φ(b) = aa
11	24825	⟨a, b \| aa=b, baaaab=ab⟩	φ(a) = a, φ(b) = aa
11	25312	⟨a, b \| aa=b, aabbb=aaa⟩	φ(a) = a, φ(b) = aa
11	25334	⟨a, b \| aa=b, ababb=aaa⟩	φ(a) = a, φ(b) = aa
11	25342	⟨a, b \| aa=b, abbab=aaa⟩	φ(a) = a, φ(b) = aa
11	25350	⟨a, b \| aa=b, abbba=aaa⟩	φ(a) = a, φ(b) = aa
11	25370	⟨a, b \| aa=b, baabb=aaa⟩	φ(a) = a, φ(b) = aa
11	25378	⟨a, b \| aa=b, babab=aaa⟩	φ(a) = a, φ(b) = aa

Presentation

Mapping

2007

⟨a, b | aaa=b, aabb=b⟩

φ(a) = a, φ(b) = aaa

2009

⟨a, b | aaa=b, abab=b⟩

φ(a) = a, φ(b) = aaa

2011

⟨a, b | aaa=b, abba=b⟩

φ(a) = a, φ(b) = aaa

2015

⟨a, b | aaa=b, baab=b⟩

φ(a) = a, φ(b) = aaa

3070

⟨a, b | aa=b, bbbb=ab⟩

φ(a) = a, φ(b) = aa

5057

⟨a, b | aaa=ab, abbb=b⟩

φ(a) = a, φ(b) = aaaaaaa

5075

⟨a, b | aaa=bb, aaaa=b⟩

φ(a) = a, φ(b) = aaaa

8939

⟨a, b | aa=b, aabbb=ab⟩

φ(a) = a, φ(b) = aa

8940

⟨a, b | aa=b, aabbb=ba⟩

φ(a) = a, φ(b) = aa

8950

⟨a, b | aa=b, ababb=ab⟩

φ(a) = a, φ(b) = aa

8951

⟨a, b | aa=b, ababb=ba⟩

φ(a) = a, φ(b) = aa

8954

⟨a, b | aa=b, abbab=ab⟩

φ(a) = a, φ(b) = aa

8955

⟨a, b | aa=b, abbab=ba⟩

φ(a) = a, φ(b) = aa

8958

⟨a, b | aa=b, abbba=ab⟩

φ(a) = a, φ(b) = aa

8968

⟨a, b | aa=b, baabb=ab⟩

φ(a) = a, φ(b) = aa

8969

⟨a, b | aa=b, baabb=ba⟩

φ(a) = a, φ(b) = aa

8972

⟨a, b | aa=b, babab=ab⟩

φ(a) = a, φ(b) = aa

9250

⟨a, b | aa=b, bbbb=aaa⟩

φ(a) = a, φ(b) = aa

15448

⟨a, b | aaa=ab, aaabb=b⟩

φ(a) = a, φ(b) = aaaaaaa

15452

⟨a, b | aaa=ab, aabab=b⟩

φ(a) = a, φ(b) = aaaaaaa

15454

⟨a, b | aaa=ab, aabba=b⟩

φ(a) = a, φ(b) = aaaaaaa

15460

⟨a, b | aaa=ab, abaab=b⟩

φ(a) = a, φ(b) = aaaaaaa

15462

⟨a, b | aaa=ab, ababa=b⟩

φ(a) = a, φ(b) = aaaaaaa

15466

⟨a, b | aaa=ab, abbaa=b⟩

φ(a) = a, φ(b) = aaaaaaa

18763

⟨a, b | aaa=b, aaaaab=b⟩

φ(a) = a, φ(b) = aaa

18765

⟨a, b | aaa=b, aaaaba=b⟩

φ(a) = a, φ(b) = aaa

18769

⟨a, b | aaa=b, aaabaa=b⟩

φ(a) = a, φ(b) = aaa

19870

⟨a, b | aaa=b, aabb=aaa⟩

φ(a) = a, φ(b) = aaa

19878

⟨a, b | aaa=b, abab=aaa⟩

φ(a) = a, φ(b) = aaa

19886

⟨a, b | aaa=b, abba=aaa⟩

φ(a) = a, φ(b) = aaa

19900

⟨a, b | aaa=b, baab=aaa⟩

φ(a) = a, φ(b) = aaa

19935

⟨a, b | aab=a, aaab=bbb⟩

φ(a) = aaaa, φ(b) = a

19943

⟨a, b | aab=a, aaba=bbb⟩

φ(a) = aaaa, φ(b) = a

19959

⟨a, b | aab=a, abaa=bbb⟩

φ(a) = aaaa, φ(b) = a

24736

⟨a, b | aa=b, aaaabb=ab⟩

φ(a) = a, φ(b) = aa

24737

⟨a, b | aa=b, aaaabb=ba⟩

φ(a) = a, φ(b) = aa

24744

⟨a, b | aa=b, aaabab=ab⟩

φ(a) = a, φ(b) = aa

24745

⟨a, b | aa=b, aaabab=ba⟩

φ(a) = a, φ(b) = aa

24748

⟨a, b | aa=b, aaabba=ab⟩

φ(a) = a, φ(b) = aa

24749

⟨a, b | aa=b, aaabba=ba⟩

φ(a) = a, φ(b) = aa

24756

⟨a, b | aa=b, aabaab=ab⟩

φ(a) = a, φ(b) = aa

24757

⟨a, b | aa=b, aabaab=ba⟩

φ(a) = a, φ(b) = aa

24760

⟨a, b | aa=b, aababa=ab⟩

φ(a) = a, φ(b) = aa

24761

⟨a, b | aa=b, aababa=ba⟩

φ(a) = a, φ(b) = aa

24768

⟨a, b | aa=b, aabbaa=ab⟩

φ(a) = a, φ(b) = aa

24783

⟨a, b | aa=b, abaaab=ab⟩

φ(a) = a, φ(b) = aa

24784

⟨a, b | aa=b, abaaab=ba⟩

φ(a) = a, φ(b) = aa

24787

⟨a, b | aa=b, abaaba=ab⟩

φ(a) = a, φ(b) = aa

24825

⟨a, b | aa=b, baaaab=ab⟩

φ(a) = a, φ(b) = aa

25312

⟨a, b | aa=b, aabbb=aaa⟩

φ(a) = a, φ(b) = aa

25334

⟨a, b | aa=b, ababb=aaa⟩

φ(a) = a, φ(b) = aa

25342

⟨a, b | aa=b, abbab=aaa⟩

φ(a) = a, φ(b) = aa

25350

⟨a, b | aa=b, abbba=aaa⟩

φ(a) = a, φ(b) = aa

25370

⟨a, b | aa=b, baabb=aaa⟩

φ(a) = a, φ(b) = aa

25378

⟨a, b | aa=b, babab=aaa⟩

φ(a) = a, φ(b) = aa

index 1, period 1	1 element	a⁵
index 1, period 5	4 elements	a³, a⁴, a⁶, a⁷
index 2, period 5	1 element	a²
index 3, period 5	1 element	a

#1605 ⟨a, b | aaa=bb, abb=b⟩

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Properties

Elements

Staircase diagram

Cayley table

Right Cayley graph

Rewriting system

Same cardinality

Isomorphic instances

Up:	Monoids with two generators and two relations
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