⟨a, b | aa=b, bbb=a⟩

Up:

Monoids with two generators and two relations

#254 ⟨a, b | aa=b, abb=b⟩

#258 ⟨a, b | aa=b, bbb=b⟩

Properties

Presentation has sum-of-sides 7

Isomorphic to ℕ_{(6 = 1)}

Finite commutative monoid with 6 elements

Not cancellative, because multiplication by a is not injective:

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

Commutative Gröbner basis: ⟨a, b | b⁶=b, a=b³⟩

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
1	1	a	a²	a³	a⁴	a⁵
a	a	a²	a³	a⁴	a⁵	a
a²	a²	a³	a⁴	a⁵	a	a²
a³	a³	a⁴	a⁵	a	a²	a³
a⁴	a⁴	a⁵	a	a²	a³	a⁴
a⁵	a⁵	a	a²	a³	a⁴	a⁵

a²

a³

a⁴

a⁵

a²

a³

a⁴

a⁵

a²

a³

a⁴

a⁵

a²

a³

a⁴

a⁵

a²

a³

a⁴

a⁵

a²

a³

a⁴

a⁵

a²

a³

a⁴

a⁵

a²

a³

a⁴

a⁵

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.	a⁶ ⇒ a	[2]
2.	b ⇒ a²	[1]

# ab:aa=b,bbb=a a/b aaaaaa=a b=aa

Same cardinality

Σ	#	Presentation	Description	Related
6	22	⟨a, b \| ab=aa, bb=1⟩	Finite non-commutative monoid with 6 elements	44 iso, 63 anti-iso
6	26	⟨a, b \| aaa=1, abb=1⟩	Isomorphic to ℤ₆	1373 iso
7	160	⟨a, b \| ab=aa, bb=b⟩	Finite non-commutative monoid with 6 elements	4 anti-iso
7	245	⟨a, b \| aa=a, bbb=a⟩	Isomorphic to ℕ_{(6 = 3)}	49 iso
7	258	⟨a, b \| aa=b, bbb=b⟩	Isomorphic to ℕ_{(6 = 2)}	55 iso
8	639	⟨a, b \| ab=aa, baa=b⟩	Finite non-commutative monoid with 6 elements	5 iso, 8 anti-iso
8	644	⟨a, b \| ab=aa, bbb=a⟩	Isomorphic to ℕ_{(6 = 4)}	46 iso
8	893	⟨a, b \| aa=a, abbb=b⟩	Finite non-commutative monoid with 6 elements	19 iso
8	1011	⟨a, b \| ab=a, baa=bb⟩	Finite non-commutative monoid with 6 elements	12 iso, 1 anti-iso
9	1427	⟨a, b \| abba=b, baba=1⟩	Finite non-Abelian group with 6 elements	66 iso
9	1581	⟨a, b \| aaa=aa, abb=b⟩	Finite non-commutative monoid with 6 elements	5 iso
9	1648	⟨a, b \| aab=bb, abb=a⟩	Finite commutative monoid with 6 elements	13 iso
9	1686	⟨a, b \| abb=aa, bbb=a⟩	Isomorphic to ℕ_{(6 = 5)}	49 iso
9	3075	⟨a, b \| ab=a, aaaa=bb⟩	Finite commutative monoid with 6 elements	14 iso
9	3132	⟨a, b \| ab=a, bbbb=aa⟩	Finite commutative monoid with 6 elements	5 iso
9	3134	⟨a, b \| ab=a, bbbb=ba⟩	Finite non-commutative monoid with 6 elements	2 iso
9	3193	⟨a, b \| ab=a, bbb=aaa⟩	Finite commutative monoid with 6 elements	9 iso
9	3199	⟨a, b \| ab=a, bbb=bba⟩	Finite non-commutative monoid with 6 elements	3 iso
11	24145	⟨a, b \| aa=a, abbbbba=b⟩	Finite commutative monoid with 6 elements

Presentation

Description

⟨a, b | ab=aa, bb=1⟩

Finite non-commutative monoid with 6 elements

44 iso, 63 anti-iso

⟨a, b | aaa=1, abb=1⟩

Isomorphic to ℤ₆

1373 iso

160

⟨a, b | ab=aa, bb=b⟩

Finite non-commutative monoid with 6 elements

4 anti-iso

245

⟨a, b | aa=a, bbb=a⟩

Isomorphic to ℕ_{(6 = 3)}

49 iso

258

⟨a, b | aa=b, bbb=b⟩

Isomorphic to ℕ_{(6 = 2)}

55 iso

639

⟨a, b | ab=aa, baa=b⟩

Finite non-commutative monoid with 6 elements

5 iso, 8 anti-iso

644

⟨a, b | ab=aa, bbb=a⟩

Isomorphic to ℕ_{(6 = 4)}

46 iso

893

⟨a, b | aa=a, abbb=b⟩

Finite non-commutative monoid with 6 elements

19 iso

1011

⟨a, b | ab=a, baa=bb⟩

Finite non-commutative monoid with 6 elements

12 iso, 1 anti-iso

1427

⟨a, b | abba=b, baba=1⟩

Finite non-Abelian group with 6 elements

66 iso

1581

⟨a, b | aaa=aa, abb=b⟩

Finite non-commutative monoid with 6 elements

5 iso

1648

⟨a, b | aab=bb, abb=a⟩

Finite commutative monoid with 6 elements

13 iso

1686

⟨a, b | abb=aa, bbb=a⟩

Isomorphic to ℕ_{(6 = 5)}

49 iso

3075

⟨a, b | ab=a, aaaa=bb⟩

Finite commutative monoid with 6 elements

14 iso

3132

⟨a, b | ab=a, bbbb=aa⟩

Finite commutative monoid with 6 elements

5 iso

3134

⟨a, b | ab=a, bbbb=ba⟩

Finite non-commutative monoid with 6 elements

2 iso

3193

⟨a, b | ab=a, bbb=aaa⟩

Finite commutative monoid with 6 elements

9 iso

3199

⟨a, b | ab=a, bbb=bba⟩

Finite non-commutative monoid with 6 elements

3 iso

24145

⟨a, b | aa=a, abbbbba=b⟩

Finite commutative monoid with 6 elements

Isomorphic instances

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

Σ	#	Presentation	Mapping
8	906	⟨a, b \| aa=b, aabb=a⟩	φ(a) = bbb, φ(b) = b
8	908	⟨a, b \| aa=b, abab=a⟩	φ(a) = bbb, φ(b) = b
8	910	⟨a, b \| aa=b, abba=a⟩	φ(a) = bbb, φ(b) = b
8	914	⟨a, b \| aa=b, baab=a⟩	φ(a) = bbb, φ(b) = b
9	2002	⟨a, b \| aaa=b, aaab=a⟩	φ(a) = bb, φ(b) = b
9	2004	⟨a, b \| aaa=b, aaba=a⟩	φ(a) = bb, φ(b) = b
9	2021	⟨a, b \| aab=a, aaaa=b⟩	φ(a) = bbbb, φ(b) = b
9	2085	⟨a, b \| aba=a, aaaa=b⟩	φ(a) = bbbb, φ(b) = b
9	2898	⟨a, b \| aa=b, aaaab=a⟩	φ(a) = bbb, φ(b) = b
9	2900	⟨a, b \| aa=b, aaaba=a⟩	φ(a) = bbb, φ(b) = b
9	2904	⟨a, b \| aa=b, aabaa=a⟩	φ(a) = bbb, φ(b) = b
9	2937	⟨a, b \| ab=a, aaaaa=b⟩	φ(a) = b, φ(b) = bbbbb
10	8640	⟨a, b \| aa=b, aaaaaa=a⟩	φ(a) = bbb, φ(b) = b
10	8715	⟨a, b \| ab=a, aaaaab=b⟩	φ(a) = b, φ(b) = bbbbb
10	8717	⟨a, b \| ab=a, aaaaba=b⟩	φ(a) = b, φ(b) = bbbbb
10	8721	⟨a, b \| ab=a, aaabaa=b⟩	φ(a) = b, φ(b) = bbbbb
10	8729	⟨a, b \| ab=a, aabaaa=b⟩	φ(a) = b, φ(b) = bbbbb
10	8745	⟨a, b \| ab=a, abaaaa=b⟩	φ(a) = b, φ(b) = bbbbb
11	14440	⟨a, b \| aaab=a, abbbb=b⟩	φ(a) = bb, φ(b) = b
11	14672	⟨a, b \| aabb=a, aaabb=b⟩	φ(a) = bbb, φ(b) = b
11	14678	⟨a, b \| aabb=a, aabba=b⟩	φ(a) = bbb, φ(b) = b
11	14684	⟨a, b \| aabb=a, abaab=b⟩	φ(a) = bbb, φ(b) = b
11	14686	⟨a, b \| aabb=a, ababa=b⟩	φ(a) = bbb, φ(b) = b
11	14690	⟨a, b \| aabb=a, abbaa=b⟩	φ(a) = bbb, φ(b) = b
11	14736	⟨a, b \| abab=a, aaabb=b⟩	φ(a) = bbb, φ(b) = b
11	14740	⟨a, b \| abab=a, aabab=b⟩	φ(a) = bbb, φ(b) = b
11	14742	⟨a, b \| abab=a, aabba=b⟩	φ(a) = bbb, φ(b) = b
11	14748	⟨a, b \| abab=a, abaab=b⟩	φ(a) = bbb, φ(b) = b
11	14750	⟨a, b \| abab=a, ababa=b⟩	φ(a) = bbb, φ(b) = b
11	14754	⟨a, b \| abab=a, abbaa=b⟩	φ(a) = bbb, φ(b) = b
11	14806	⟨a, b \| abba=a, aabba=b⟩	φ(a) = bbb, φ(b) = b
11	14812	⟨a, b \| abba=a, ababa=b⟩	φ(a) = bbb, φ(b) = b
11	14847	⟨a, b \| abba=b, aabbb=a⟩	φ(a) = bb, φ(b) = b
11	14853	⟨a, b \| abba=b, ababb=a⟩	φ(a) = bb, φ(b) = b
11	14855	⟨a, b \| abba=b, abbab=a⟩	φ(a) = bb, φ(b) = b
11	14857	⟨a, b \| abba=b, abbba=a⟩	φ(a) = bb, φ(b) = b
11	14863	⟨a, b \| abba=b, baabb=a⟩	φ(a) = bb, φ(b) = b
11	14865	⟨a, b \| abba=b, babab=a⟩	φ(a) = bb, φ(b) = b
11	18760	⟨a, b \| aaa=b, aaaaaa=a⟩	φ(a) = bb, φ(b) = b
11	18835	⟨a, b \| aab=a, aaaaab=b⟩	φ(a) = bbbb, φ(b) = b
11	18837	⟨a, b \| aab=a, aaaaba=b⟩	φ(a) = bbbb, φ(b) = b
11	18841	⟨a, b \| aab=a, aaabaa=b⟩	φ(a) = bbbb, φ(b) = b
11	18849	⟨a, b \| aab=a, aabaaa=b⟩	φ(a) = bbbb, φ(b) = b
11	18865	⟨a, b \| aab=a, abaaaa=b⟩	φ(a) = bbbb, φ(b) = b
11	19093	⟨a, b \| aba=a, aaaaba=b⟩	φ(a) = bbbb, φ(b) = b
11	19097	⟨a, b \| aba=a, aaabaa=b⟩	φ(a) = bbbb, φ(b) = b
11	24339	⟨a, b \| ab=a, aaaaabb=b⟩	φ(a) = b, φ(b) = bbbbb
11	24343	⟨a, b \| ab=a, aaaabab=b⟩	φ(a) = b, φ(b) = bbbbb
11	24345	⟨a, b \| ab=a, aaaabba=b⟩	φ(a) = b, φ(b) = bbbbb
11	24351	⟨a, b \| ab=a, aaabaab=b⟩	φ(a) = b, φ(b) = bbbbb
11	24353	⟨a, b \| ab=a, aaababa=b⟩	φ(a) = b, φ(b) = bbbbb
11	24357	⟨a, b \| ab=a, aaabbaa=b⟩	φ(a) = b, φ(b) = bbbbb
11	24367	⟨a, b \| ab=a, aabaaab=b⟩	φ(a) = b, φ(b) = bbbbb
11	24369	⟨a, b \| ab=a, aabaaba=b⟩	φ(a) = b, φ(b) = bbbbb
11	24373	⟨a, b \| ab=a, aababaa=b⟩	φ(a) = b, φ(b) = bbbbb
11	24381	⟨a, b \| ab=a, aabbaaa=b⟩	φ(a) = b, φ(b) = bbbbb
11	24399	⟨a, b \| ab=a, abaaaab=b⟩	φ(a) = b, φ(b) = bbbbb
11	24401	⟨a, b \| ab=a, abaaaba=b⟩	φ(a) = b, φ(b) = bbbbb
11	24405	⟨a, b \| ab=a, abaabaa=b⟩	φ(a) = b, φ(b) = bbbbb
11	24413	⟨a, b \| ab=a, ababaaa=b⟩	φ(a) = b, φ(b) = bbbbb
11	24429	⟨a, b \| ab=a, abbaaaa=b⟩	φ(a) = b, φ(b) = bbbbb

Presentation

Mapping

906

⟨a, b | aa=b, aabb=a⟩

φ(a) = bbb, φ(b) = b

908

⟨a, b | aa=b, abab=a⟩

φ(a) = bbb, φ(b) = b

910

⟨a, b | aa=b, abba=a⟩

φ(a) = bbb, φ(b) = b

914

⟨a, b | aa=b, baab=a⟩

φ(a) = bbb, φ(b) = b

2002

⟨a, b | aaa=b, aaab=a⟩

φ(a) = bb, φ(b) = b

2004

⟨a, b | aaa=b, aaba=a⟩

φ(a) = bb, φ(b) = b

2021

⟨a, b | aab=a, aaaa=b⟩

φ(a) = bbbb, φ(b) = b

2085

⟨a, b | aba=a, aaaa=b⟩

φ(a) = bbbb, φ(b) = b

2898

⟨a, b | aa=b, aaaab=a⟩

φ(a) = bbb, φ(b) = b

2900

⟨a, b | aa=b, aaaba=a⟩

φ(a) = bbb, φ(b) = b

2904

⟨a, b | aa=b, aabaa=a⟩

φ(a) = bbb, φ(b) = b

2937

⟨a, b | ab=a, aaaaa=b⟩

φ(a) = b, φ(b) = bbbbb

8640

⟨a, b | aa=b, aaaaaa=a⟩

φ(a) = bbb, φ(b) = b

8715

⟨a, b | ab=a, aaaaab=b⟩

φ(a) = b, φ(b) = bbbbb

8717

⟨a, b | ab=a, aaaaba=b⟩

φ(a) = b, φ(b) = bbbbb

8721

⟨a, b | ab=a, aaabaa=b⟩

φ(a) = b, φ(b) = bbbbb

8729

⟨a, b | ab=a, aabaaa=b⟩

φ(a) = b, φ(b) = bbbbb

8745

⟨a, b | ab=a, abaaaa=b⟩

φ(a) = b, φ(b) = bbbbb

14440

⟨a, b | aaab=a, abbbb=b⟩

φ(a) = bb, φ(b) = b

14672

⟨a, b | aabb=a, aaabb=b⟩

φ(a) = bbb, φ(b) = b

14678

⟨a, b | aabb=a, aabba=b⟩

φ(a) = bbb, φ(b) = b

14684

⟨a, b | aabb=a, abaab=b⟩

φ(a) = bbb, φ(b) = b

14686

⟨a, b | aabb=a, ababa=b⟩

φ(a) = bbb, φ(b) = b

14690

⟨a, b | aabb=a, abbaa=b⟩

φ(a) = bbb, φ(b) = b

14736

⟨a, b | abab=a, aaabb=b⟩

φ(a) = bbb, φ(b) = b

14740

⟨a, b | abab=a, aabab=b⟩

φ(a) = bbb, φ(b) = b

14742

⟨a, b | abab=a, aabba=b⟩

φ(a) = bbb, φ(b) = b

14748

⟨a, b | abab=a, abaab=b⟩

φ(a) = bbb, φ(b) = b

14750

⟨a, b | abab=a, ababa=b⟩

φ(a) = bbb, φ(b) = b

14754

⟨a, b | abab=a, abbaa=b⟩

φ(a) = bbb, φ(b) = b

14806

⟨a, b | abba=a, aabba=b⟩

φ(a) = bbb, φ(b) = b

14812

⟨a, b | abba=a, ababa=b⟩

φ(a) = bbb, φ(b) = b

14847

⟨a, b | abba=b, aabbb=a⟩

φ(a) = bb, φ(b) = b

14853

⟨a, b | abba=b, ababb=a⟩

φ(a) = bb, φ(b) = b

14855

⟨a, b | abba=b, abbab=a⟩

φ(a) = bb, φ(b) = b

14857

⟨a, b | abba=b, abbba=a⟩

φ(a) = bb, φ(b) = b

14863

⟨a, b | abba=b, baabb=a⟩

φ(a) = bb, φ(b) = b

14865

⟨a, b | abba=b, babab=a⟩

φ(a) = bb, φ(b) = b

18760

⟨a, b | aaa=b, aaaaaa=a⟩

φ(a) = bb, φ(b) = b

18835

⟨a, b | aab=a, aaaaab=b⟩

φ(a) = bbbb, φ(b) = b

18837

⟨a, b | aab=a, aaaaba=b⟩

φ(a) = bbbb, φ(b) = b

18841

⟨a, b | aab=a, aaabaa=b⟩

φ(a) = bbbb, φ(b) = b

18849

⟨a, b | aab=a, aabaaa=b⟩

φ(a) = bbbb, φ(b) = b

18865

⟨a, b | aab=a, abaaaa=b⟩

φ(a) = bbbb, φ(b) = b

19093

⟨a, b | aba=a, aaaaba=b⟩

φ(a) = bbbb, φ(b) = b

19097

⟨a, b | aba=a, aaabaa=b⟩

φ(a) = bbbb, φ(b) = b

24339

⟨a, b | ab=a, aaaaabb=b⟩

φ(a) = b, φ(b) = bbbbb

24343

⟨a, b | ab=a, aaaabab=b⟩

φ(a) = b, φ(b) = bbbbb

24345

⟨a, b | ab=a, aaaabba=b⟩

φ(a) = b, φ(b) = bbbbb

24351

⟨a, b | ab=a, aaabaab=b⟩

φ(a) = b, φ(b) = bbbbb

24353

⟨a, b | ab=a, aaababa=b⟩

φ(a) = b, φ(b) = bbbbb

24357

⟨a, b | ab=a, aaabbaa=b⟩

φ(a) = b, φ(b) = bbbbb

24367

⟨a, b | ab=a, aabaaab=b⟩

φ(a) = b, φ(b) = bbbbb

24369

⟨a, b | ab=a, aabaaba=b⟩

φ(a) = b, φ(b) = bbbbb

24373

⟨a, b | ab=a, aababaa=b⟩

φ(a) = b, φ(b) = bbbbb

24381

⟨a, b | ab=a, aabbaaa=b⟩

φ(a) = b, φ(b) = bbbbb

24399

⟨a, b | ab=a, abaaaab=b⟩

φ(a) = b, φ(b) = bbbbb

24401

⟨a, b | ab=a, abaaaba=b⟩

φ(a) = b, φ(b) = bbbbb

24405

⟨a, b | ab=a, abaabaa=b⟩

φ(a) = b, φ(b) = bbbbb

24413

⟨a, b | ab=a, ababaaa=b⟩

φ(a) = b, φ(b) = bbbbb

24429

⟨a, b | ab=a, abbaaaa=b⟩

φ(a) = b, φ(b) = bbbbb

index 1, period 1	1 element	a⁵
index 1, period 5	4 elements	a, a², a³, a⁴

#257 ⟨a, b | aa=b, bbb=a⟩

Quick links

Properties

Elements

Staircase diagram

Cayley table

Right Cayley graph

Rewriting system

Same cardinality

Isomorphic instances

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