⟨a, b | aaa=b, bbb=a⟩

Up:

Monoids with two generators and two relations

#578 ⟨a, b | aaa=b, abb=b⟩

#582 ⟨a, b | aab=a, aba=a⟩

Properties

Presentation has sum-of-sides 8

Isomorphic to ℕ_{(9 = 1)}

Finite commutative monoid with 9 elements

Not cancellative, because multiplication by a is not injective:

a ⋅ a⁸ = a and a ⋅ 1 = a, however a⁸ ≠ 1

Commutative Gröbner basis: ⟨a, b | b⁹=b, a=b³⟩

Cancellative quotient is isomorphic to ℤ₈

Enveloping group is isomorphic to ℤ₈

Group of units is isomorphic to ℤ₁

2 Archimedian components:

1, b

	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸
1	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸
a	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a
a²	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a	a²
a³	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a	a²	a³
a⁴	a⁴	a⁵	a⁶	a⁷	a⁸	a	a²	a³	a⁴
a⁵	a⁵	a⁶	a⁷	a⁸	a	a²	a³	a⁴	a⁵
a⁶	a⁶	a⁷	a⁸	a	a²	a³	a⁴	a⁵	a⁶
a⁷	a⁷	a⁸	a	a²	a³	a⁴	a⁵	a⁶	a⁷
a⁸	a⁸	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸

a²

a³

a⁴

a⁵

a⁶

a⁷

a⁸

a²

a³

a⁴

a⁵

a⁶

a⁷

a⁸

a²

a³

a⁴

a⁵

a⁶

a⁷

a⁸

a²

a³

a⁴

a⁵

a⁶

a⁷

a⁸

a²

a³

a⁴

a⁵

a⁶

a⁷

a⁸

a²

a³

a⁴

a⁵

a⁶

a⁷

a⁸

a²

a³

a⁴

a⁵

a⁶

a⁷

a⁸

a²

a³

a⁴

a⁵

a⁶

a⁷

a⁸

a²

a³

a⁴

a⁵

a⁶

a⁷

a⁸

a²

a³

a⁴

a⁵

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a²

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a⁸

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.	a⁹ ⇒ a	[2]
2.	b ⇒ a³	[1]

# ab:aaa=b,bbb=a a/b aaaaaaaaa=a b=aaa

Same cardinality

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	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	1991	⟨a, b \| aaa=a, abba=b⟩	Finite non-commutative monoid with 9 elements	3 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

Presentation

Description

118

⟨a, b | aaa=b, bbb=1⟩

Isomorphic to ℤ₉

562 iso

568

⟨a, b | aaa=a, abb=b⟩

Finite non-commutative monoid with 9 elements

5 iso

571

⟨a, b | aaa=a, bbb=a⟩

Isomorphic to ℕ_{(9 = 3)}

48 iso

601

⟨a, b | aba=b, bab=a⟩

Finite non-commutative monoid with 9 elements

3 iso

1584

⟨a, b | aaa=aa, bbb=a⟩

Isomorphic to ℕ_{(9 = 6)}

16 iso

1595

⟨a, b | aaa=ab, bab=b⟩

Finite non-commutative monoid with 9 elements

4 iso, 5 anti-iso

1598

⟨a, b | aaa=ab, bbb=a⟩

Isomorphic to ℕ_{(9 = 4)}

40 iso

1608

⟨a, b | aaa=bb, bbb=a⟩

Isomorphic to ℕ_{(9 = 2)}

70 iso

1991

⟨a, b | aaa=a, abba=b⟩

Finite non-commutative monoid with 9 elements

3 iso

2064

⟨a, b | aab=b, abba=a⟩

Finite non-commutative monoid with 9 elements

14 iso, 24 anti-iso

2255

⟨a, b | ab=aa, bba=bb⟩

Finite non-commutative monoid with 9 elements

4125

⟨a, b | aab=aaa, baa=b⟩

Finite non-commutative monoid with 9 elements

1 iso, 1 anti-iso

4130

⟨a, b | aab=aaa, bbb=a⟩

Isomorphic to ℕ_{(9 = 7)}

1 iso

4156

⟨a, b | abb=aaa, bbb=a⟩

Isomorphic to ℕ_{(9 = 5)}

12 iso

5035

⟨a, b | aaa=aa, abbb=b⟩

Finite non-commutative monoid with 9 elements

3 iso

5079

⟨a, b | aaa=bb, aaba=b⟩

Finite commutative monoid with 9 elements

8 iso

5124

⟨a, b | aab=aa, bbbb=a⟩

Isomorphic to ℕ_{(9 = 8)}

5 iso

5339

⟨a, b | aaa=bb, aab=ab⟩

Finite non-commutative monoid with 9 elements

1 iso

6824

⟨a, b | aab=b, bbb=aaa⟩

Finite commutative monoid with 9 elements

5 iso

7139

⟨a, b | bb=aa, aaaa=ab⟩

Finite non-commutative monoid with 9 elements

3 iso

9327

⟨a, b | ab=a, baaa=bbb⟩

Finite non-commutative monoid with 9 elements

3 iso

9359

⟨a, b | ab=a, bbaa=bbb⟩

Finite non-commutative monoid with 9 elements

2 iso

19506

⟨a, b | aab=b, aaaaa=ba⟩

Finite non-commutative monoid with 9 elements

19507

⟨a, b | aab=b, aaaaa=bb⟩

Finite commutative monoid with 9 elements

20723

⟨a, b | ab=aa, aaaaa=bb⟩

Finite non-commutative monoid with 9 elements

15 iso

25656

⟨a, b | ab=a, bbbbb=baa⟩

Finite non-commutative monoid with 9 elements

Isomorphic instances

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

Σ	#	Presentation	Mapping
9	2082	⟨a, b \| aab=b, bbbb=a⟩	φ(a) = bbbb, φ(b) = b
9	2122	⟨a, b \| aba=b, bbbb=a⟩	φ(a) = bbbb, φ(b) = b
9	2922	⟨a, b \| aa=b, abbbb=a⟩	φ(a) = b, φ(b) = bb
9	2930	⟨a, b \| aa=b, babbb=a⟩	φ(a) = b, φ(b) = bb
9	2932	⟨a, b \| aa=b, bbabb=a⟩	φ(a) = b, φ(b) = bb
10	6284	⟨a, b \| aaa=b, aaabb=a⟩	φ(a) = bbb, φ(b) = b
10	6288	⟨a, b \| aaa=b, aabab=a⟩	φ(a) = bbb, φ(b) = b
10	6290	⟨a, b \| aaa=b, aabba=a⟩	φ(a) = bbb, φ(b) = b
10	6294	⟨a, b \| aaa=b, abaab=a⟩	φ(a) = bbb, φ(b) = b
10	6296	⟨a, b \| aaa=b, ababa=a⟩	φ(a) = bbb, φ(b) = b
10	6306	⟨a, b \| aaa=b, baaab=a⟩	φ(a) = bbb, φ(b) = b
10	8654	⟨a, b \| aa=b, aaabbb=a⟩	φ(a) = b, φ(b) = bb
10	8660	⟨a, b \| aa=b, aababb=a⟩	φ(a) = b, φ(b) = bb
10	8664	⟨a, b \| aa=b, aabbab=a⟩	φ(a) = b, φ(b) = bb
10	8666	⟨a, b \| aa=b, aabbba=a⟩	φ(a) = b, φ(b) = bb
10	8674	⟨a, b \| aa=b, abaabb=a⟩	φ(a) = b, φ(b) = bb
10	8676	⟨a, b \| aa=b, ababab=a⟩	φ(a) = b, φ(b) = bb
10	8678	⟨a, b \| aa=b, ababba=a⟩	φ(a) = b, φ(b) = bb
10	8682	⟨a, b \| aa=b, abbaab=a⟩	φ(a) = b, φ(b) = bb
10	8694	⟨a, b \| aa=b, baaabb=a⟩	φ(a) = b, φ(b) = bb
10	8696	⟨a, b \| aa=b, baabab=a⟩	φ(a) = b, φ(b) = bb
11	14668	⟨a, b \| aabb=a, aaaab=b⟩	φ(a) = bbbbbb, φ(b) = b
11	14670	⟨a, b \| aabb=a, aaaba=b⟩	φ(a) = bbbbbb, φ(b) = b
11	14674	⟨a, b \| aabb=a, aabaa=b⟩	φ(a) = bbbbbb, φ(b) = b
11	14682	⟨a, b \| aabb=a, abaaa=b⟩	φ(a) = bbbbbb, φ(b) = b
11	14732	⟨a, b \| abab=a, aaaab=b⟩	φ(a) = bbbbbb, φ(b) = b
11	14734	⟨a, b \| abab=a, aaaba=b⟩	φ(a) = bbbbbb, φ(b) = b
11	14738	⟨a, b \| abab=a, aabaa=b⟩	φ(a) = bbbbbb, φ(b) = b
11	14746	⟨a, b \| abab=a, abaaa=b⟩	φ(a) = bbbbbb, φ(b) = b
11	14798	⟨a, b \| abba=a, aaaba=b⟩	φ(a) = bbbbbb, φ(b) = b
11	14802	⟨a, b \| abba=a, aabaa=b⟩	φ(a) = bbbbbb, φ(b) = b
11	14859	⟨a, b \| abba=b, abbbb=a⟩	φ(a) = b, φ(b) = bbbbbb
11	14867	⟨a, b \| abba=b, babbb=a⟩	φ(a) = b, φ(b) = bbbbbb
11	18990	⟨a, b \| aab=b, aabbbb=a⟩	φ(a) = bbbb, φ(b) = b
11	19006	⟨a, b \| aab=b, ababbb=a⟩	φ(a) = bbbb, φ(b) = b
11	19014	⟨a, b \| aab=b, abbabb=a⟩	φ(a) = bbbb, φ(b) = b
11	19018	⟨a, b \| aab=b, abbbab=a⟩	φ(a) = bbbb, φ(b) = b
11	19038	⟨a, b \| aab=b, baabbb=a⟩	φ(a) = bbbb, φ(b) = b
11	19046	⟨a, b \| aab=b, bababb=a⟩	φ(a) = bbbb, φ(b) = b
11	19050	⟨a, b \| aab=b, babbab=a⟩	φ(a) = bbbb, φ(b) = b
11	19062	⟨a, b \| aab=b, bbaabb=a⟩	φ(a) = bbbb, φ(b) = b
11	19066	⟨a, b \| aab=b, bbabab=a⟩	φ(a) = bbbb, φ(b) = b
11	19074	⟨a, b \| aab=b, bbbaab=a⟩	φ(a) = bbbb, φ(b) = b
11	19188	⟨a, b \| aba=b, aabbbb=a⟩	φ(a) = bbbb, φ(b) = b
11	19200	⟨a, b \| aba=b, ababbb=a⟩	φ(a) = bbbb, φ(b) = b
11	19204	⟨a, b \| aba=b, abbabb=a⟩	φ(a) = bbbb, φ(b) = b
11	19206	⟨a, b \| aba=b, abbbab=a⟩	φ(a) = bbbb, φ(b) = b
11	19208	⟨a, b \| aba=b, abbbba=a⟩	φ(a) = bbbb, φ(b) = b
11	19220	⟨a, b \| aba=b, bababb=a⟩	φ(a) = bbbb, φ(b) = b
11	19226	⟨a, b \| aba=b, bbaabb=a⟩	φ(a) = bbbb, φ(b) = b
11	24194	⟨a, b \| aa=b, aaaaabb=a⟩	φ(a) = b, φ(b) = bb
11	24198	⟨a, b \| aa=b, aaaabab=a⟩	φ(a) = b, φ(b) = bb
11	24200	⟨a, b \| aa=b, aaaabba=a⟩	φ(a) = b, φ(b) = bb
11	24206	⟨a, b \| aa=b, aaabaab=a⟩	φ(a) = b, φ(b) = bb
11	24208	⟨a, b \| aa=b, aaababa=a⟩	φ(a) = b, φ(b) = bb
11	24212	⟨a, b \| aa=b, aaabbaa=a⟩	φ(a) = b, φ(b) = bb
11	24220	⟨a, b \| aa=b, aabaaab=a⟩	φ(a) = b, φ(b) = bb
11	24222	⟨a, b \| aa=b, aabaaba=a⟩	φ(a) = b, φ(b) = bb
11	24226	⟨a, b \| aa=b, aababaa=a⟩	φ(a) = b, φ(b) = bb
11	24248	⟨a, b \| aa=b, abaaaab=a⟩	φ(a) = b, φ(b) = bb
11	24250	⟨a, b \| aa=b, abaaaba=a⟩	φ(a) = b, φ(b) = bb
11	24292	⟨a, b \| aa=b, baaaaab=a⟩	φ(a) = b, φ(b) = bb

Presentation

Mapping

2082

⟨a, b | aab=b, bbbb=a⟩

φ(a) = bbbb, φ(b) = b

2122

⟨a, b | aba=b, bbbb=a⟩

φ(a) = bbbb, φ(b) = b

2922

⟨a, b | aa=b, abbbb=a⟩

φ(a) = b, φ(b) = bb

2930

⟨a, b | aa=b, babbb=a⟩

φ(a) = b, φ(b) = bb

2932

⟨a, b | aa=b, bbabb=a⟩

φ(a) = b, φ(b) = bb

6284

⟨a, b | aaa=b, aaabb=a⟩

φ(a) = bbb, φ(b) = b

6288

⟨a, b | aaa=b, aabab=a⟩

φ(a) = bbb, φ(b) = b

6290

⟨a, b | aaa=b, aabba=a⟩

φ(a) = bbb, φ(b) = b

6294

⟨a, b | aaa=b, abaab=a⟩

φ(a) = bbb, φ(b) = b

6296

⟨a, b | aaa=b, ababa=a⟩

φ(a) = bbb, φ(b) = b

6306

⟨a, b | aaa=b, baaab=a⟩

φ(a) = bbb, φ(b) = b

8654

⟨a, b | aa=b, aaabbb=a⟩

φ(a) = b, φ(b) = bb

8660

⟨a, b | aa=b, aababb=a⟩

φ(a) = b, φ(b) = bb

8664

⟨a, b | aa=b, aabbab=a⟩

φ(a) = b, φ(b) = bb

8666

⟨a, b | aa=b, aabbba=a⟩

φ(a) = b, φ(b) = bb

8674

⟨a, b | aa=b, abaabb=a⟩

φ(a) = b, φ(b) = bb

8676

⟨a, b | aa=b, ababab=a⟩

φ(a) = b, φ(b) = bb

8678

⟨a, b | aa=b, ababba=a⟩

φ(a) = b, φ(b) = bb

8682

⟨a, b | aa=b, abbaab=a⟩

φ(a) = b, φ(b) = bb

8694

⟨a, b | aa=b, baaabb=a⟩

φ(a) = b, φ(b) = bb

8696

⟨a, b | aa=b, baabab=a⟩

φ(a) = b, φ(b) = bb

14668

⟨a, b | aabb=a, aaaab=b⟩

φ(a) = bbbbbb, φ(b) = b

14670

⟨a, b | aabb=a, aaaba=b⟩

φ(a) = bbbbbb, φ(b) = b

14674

⟨a, b | aabb=a, aabaa=b⟩

φ(a) = bbbbbb, φ(b) = b

14682

⟨a, b | aabb=a, abaaa=b⟩

φ(a) = bbbbbb, φ(b) = b

14732

⟨a, b | abab=a, aaaab=b⟩

φ(a) = bbbbbb, φ(b) = b

14734

⟨a, b | abab=a, aaaba=b⟩

φ(a) = bbbbbb, φ(b) = b

14738

⟨a, b | abab=a, aabaa=b⟩

φ(a) = bbbbbb, φ(b) = b

14746

⟨a, b | abab=a, abaaa=b⟩

φ(a) = bbbbbb, φ(b) = b

14798

⟨a, b | abba=a, aaaba=b⟩

φ(a) = bbbbbb, φ(b) = b

14802

⟨a, b | abba=a, aabaa=b⟩

φ(a) = bbbbbb, φ(b) = b

14859

⟨a, b | abba=b, abbbb=a⟩

φ(a) = b, φ(b) = bbbbbb

14867

⟨a, b | abba=b, babbb=a⟩

φ(a) = b, φ(b) = bbbbbb

18990

⟨a, b | aab=b, aabbbb=a⟩

φ(a) = bbbb, φ(b) = b

19006

⟨a, b | aab=b, ababbb=a⟩

φ(a) = bbbb, φ(b) = b

19014

⟨a, b | aab=b, abbabb=a⟩

φ(a) = bbbb, φ(b) = b

19018

⟨a, b | aab=b, abbbab=a⟩

φ(a) = bbbb, φ(b) = b

19038

⟨a, b | aab=b, baabbb=a⟩

φ(a) = bbbb, φ(b) = b

19046

⟨a, b | aab=b, bababb=a⟩

φ(a) = bbbb, φ(b) = b

19050

⟨a, b | aab=b, babbab=a⟩

φ(a) = bbbb, φ(b) = b

19062

⟨a, b | aab=b, bbaabb=a⟩

φ(a) = bbbb, φ(b) = b

19066

⟨a, b | aab=b, bbabab=a⟩

φ(a) = bbbb, φ(b) = b

19074

⟨a, b | aab=b, bbbaab=a⟩

φ(a) = bbbb, φ(b) = b

19188

⟨a, b | aba=b, aabbbb=a⟩

φ(a) = bbbb, φ(b) = b

19200

⟨a, b | aba=b, ababbb=a⟩

φ(a) = bbbb, φ(b) = b

19204

⟨a, b | aba=b, abbabb=a⟩

φ(a) = bbbb, φ(b) = b

19206

⟨a, b | aba=b, abbbab=a⟩

φ(a) = bbbb, φ(b) = b

19208

⟨a, b | aba=b, abbbba=a⟩

φ(a) = bbbb, φ(b) = b

19220

⟨a, b | aba=b, bababb=a⟩

φ(a) = bbbb, φ(b) = b

19226

⟨a, b | aba=b, bbaabb=a⟩

φ(a) = bbbb, φ(b) = b

24194

⟨a, b | aa=b, aaaaabb=a⟩

φ(a) = b, φ(b) = bb

24198

⟨a, b | aa=b, aaaabab=a⟩

φ(a) = b, φ(b) = bb

24200

⟨a, b | aa=b, aaaabba=a⟩

φ(a) = b, φ(b) = bb

24206

⟨a, b | aa=b, aaabaab=a⟩

φ(a) = b, φ(b) = bb

24208

⟨a, b | aa=b, aaababa=a⟩

φ(a) = b, φ(b) = bb

24212

⟨a, b | aa=b, aaabbaa=a⟩

φ(a) = b, φ(b) = bb

24220

⟨a, b | aa=b, aabaaab=a⟩

φ(a) = b, φ(b) = bb

24222

⟨a, b | aa=b, aabaaba=a⟩

φ(a) = b, φ(b) = bb

24226

⟨a, b | aa=b, aababaa=a⟩

φ(a) = b, φ(b) = bb

24248

⟨a, b | aa=b, abaaaab=a⟩

φ(a) = b, φ(b) = bb

24250

⟨a, b | aa=b, abaaaba=a⟩

φ(a) = b, φ(b) = bb

24292

⟨a, b | aa=b, baaaaab=a⟩

φ(a) = b, φ(b) = bb

index 1, period 1	1 element	a⁸
index 1, period 2	1 element	a⁴
index 1, period 4	2 elements	a², a⁶
index 1, period 8	4 elements	a, a³, a⁵, a⁷

#581 ⟨a, b | aaa=b, bbb=a⟩

Quick links

Properties

Elements

Staircase diagram

Cayley table

Right Cayley graph

Rewriting system

Same cardinality

Isomorphic instances

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