⟨a, b | aa=b, bb=a⟩

Up:

Presentation has sum-of-sides 6

Isomorphic to ℕ_{(4 = 1)}

Finite commutative monoid with 4 elements

Not cancellative, because multiplication by a is not injective:

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

Commutative Gröbner basis: ⟨a, b | b⁴=b, a=b²⟩

Cancellative quotient is isomorphic to ℤ₃

Enveloping group is isomorphic to ℤ₃

Group of units is isomorphic to ℤ₁

2 Archimedian components:

1, b

	1	a	a²	a³
1	1	a	a²	a³
a	a	a²	a³	a
a²	a²	a³	a	a²
a³	a³	a	a²	a³

a²

a³

a²

a³

a²

a³

a²

a³

a²

a³

a²

a³

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.	a⁴ ⇒ a	[2]
2.	b ⇒ a²	[1]

# ab:aa=b,bb=a a/b aaaa=a b=aa

Same cardinality

Σ	#	Presentation	Description	Related
5	7	⟨a, b \| aa=b, bb=1⟩	Isomorphic to ℤ₄	1419 iso
6	57	⟨a, b \| aa=a, bb=a⟩	Isomorphic to ℕ_{(4 = 2)}	37 iso
7	158	⟨a, b \| ab=aa, ba=b⟩	Finite non-commutative monoid with 4 elements	8 iso, 6 anti-iso
7	159	⟨a, b \| ab=aa, bb=a⟩	Isomorphic to ℕ_{(4 = 3)}	16 iso
7	242	⟨a, b \| aa=a, abb=b⟩	Finite non-commutative monoid with 4 elements	14 iso
7	280	⟨a, b \| ab=a, bb=aa⟩	Finite commutative monoid with 4 elements	17 iso
9	2881	⟨a, b \| aa=a, abbba=b⟩	Finite commutative monoid with 4 elements	8 iso
10	5033	⟨a, b \| aaa=aa, abba=b⟩	Finite commutative monoid with 4 elements	2 iso

Presentation

Description

⟨a, b | aa=b, bb=1⟩

Isomorphic to ℤ₄

1419 iso

⟨a, b | aa=a, bb=a⟩

Isomorphic to ℕ_{(4 = 2)}

37 iso

158

⟨a, b | ab=aa, ba=b⟩

Finite non-commutative monoid with 4 elements

8 iso, 6 anti-iso

159

⟨a, b | ab=aa, bb=a⟩

Isomorphic to ℕ_{(4 = 3)}

16 iso

242

⟨a, b | aa=a, abb=b⟩

Finite non-commutative monoid with 4 elements

14 iso

280

⟨a, b | ab=a, bb=aa⟩

Finite commutative monoid with 4 elements

17 iso

2881

⟨a, b | aa=a, abbba=b⟩

Finite commutative monoid with 4 elements

8 iso

5033

⟨a, b | aaa=aa, abba=b⟩

Finite commutative monoid with 4 elements

2 iso

Isomorphic instances

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

Σ	#	Presentation	Mapping
7	249	⟨a, b \| aa=b, aab=a⟩	φ(a) = bb, φ(b) = b
7	251	⟨a, b \| aa=b, aba=a⟩	φ(a) = bb, φ(b) = b
7	260	⟨a, b \| ab=a, aaa=b⟩	φ(a) = b, φ(b) = bbb
8	900	⟨a, b \| aa=b, aaaa=a⟩	φ(a) = bb, φ(b) = b
8	923	⟨a, b \| ab=a, aaab=b⟩	φ(a) = b, φ(b) = bbb
8	925	⟨a, b \| ab=a, aaba=b⟩	φ(a) = b, φ(b) = bbb
8	929	⟨a, b \| ab=a, abaa=b⟩	φ(a) = b, φ(b) = bbb
9	2000	⟨a, b \| aaa=b, aaaa=a⟩	φ(a) = b, φ(b) = bbb
9	2023	⟨a, b \| aab=a, aaab=b⟩	φ(a) = bb, φ(b) = b
9	2025	⟨a, b \| aab=a, aaba=b⟩	φ(a) = bb, φ(b) = b
9	2029	⟨a, b \| aab=a, abaa=b⟩	φ(a) = bb, φ(b) = b
9	2089	⟨a, b \| aba=a, aaba=b⟩	φ(a) = bb, φ(b) = b
9	2943	⟨a, b \| ab=a, aaabb=b⟩	φ(a) = b, φ(b) = bbb
9	2947	⟨a, b \| ab=a, aabab=b⟩	φ(a) = b, φ(b) = bbb
9	2949	⟨a, b \| ab=a, aabba=b⟩	φ(a) = b, φ(b) = bbb
9	2955	⟨a, b \| ab=a, abaab=b⟩	φ(a) = b, φ(b) = bbb
9	2957	⟨a, b \| ab=a, ababa=b⟩	φ(a) = b, φ(b) = bbb
9	2961	⟨a, b \| ab=a, abbaa=b⟩	φ(a) = b, φ(b) = bbb
10	4650	⟨a, b \| aaab=a, aaba=b⟩	φ(a) = b, φ(b) = b
10	4654	⟨a, b \| aaab=a, abaa=b⟩	φ(a) = b, φ(b) = b
10	4660	⟨a, b \| aaab=a, abbb=b⟩	φ(a) = b, φ(b) = b
10	4701	⟨a, b \| aaba=a, abaa=b⟩	φ(a) = b, φ(b) = b
10	4732	⟨a, b \| aabb=a, abab=b⟩	φ(a) = b, φ(b) = b
10	4734	⟨a, b \| aabb=a, abba=b⟩	φ(a) = b, φ(b) = b
10	4742	⟨a, b \| abab=a, abba=b⟩	φ(a) = b, φ(b) = b
10	8727	⟨a, b \| ab=a, aaabbb=b⟩	φ(a) = b, φ(b) = bbb
10	8735	⟨a, b \| ab=a, aababb=b⟩	φ(a) = b, φ(b) = bbb
10	8739	⟨a, b \| ab=a, aabbab=b⟩	φ(a) = b, φ(b) = bbb
10	8741	⟨a, b \| ab=a, aabbba=b⟩	φ(a) = b, φ(b) = bbb
10	8751	⟨a, b \| ab=a, abaabb=b⟩	φ(a) = b, φ(b) = bbb
10	8755	⟨a, b \| ab=a, ababab=b⟩	φ(a) = b, φ(b) = bbb
10	8757	⟨a, b \| ab=a, ababba=b⟩	φ(a) = b, φ(b) = bbb
10	8763	⟨a, b \| ab=a, abbaab=b⟩	φ(a) = b, φ(b) = bbb
10	8765	⟨a, b \| ab=a, abbaba=b⟩	φ(a) = b, φ(b) = bbb
10	8769	⟨a, b \| ab=a, abbbaa=b⟩	φ(a) = b, φ(b) = bbb
11	14330	⟨a, b \| aaaa=a, aaaaa=b⟩	φ(a) = bb, φ(b) = b
11	14475	⟨a, b \| aaab=b, aaaab=a⟩	φ(a) = b, φ(b) = bbb
11	14603	⟨a, b \| aaba=b, aaaab=a⟩	φ(a) = b, φ(b) = bbb
11	14605	⟨a, b \| aaba=b, aaaba=a⟩	φ(a) = b, φ(b) = bbb
11	14609	⟨a, b \| aaba=b, aabaa=a⟩	φ(a) = b, φ(b) = bbb
11	14617	⟨a, b \| aaba=b, abaaa=a⟩	φ(a) = b, φ(b) = bbb
11	14633	⟨a, b \| aaba=b, baaaa=a⟩	φ(a) = b, φ(b) = bbb
11	18839	⟨a, b \| aab=a, aaaabb=b⟩	φ(a) = bb, φ(b) = b
11	18843	⟨a, b \| aab=a, aaabab=b⟩	φ(a) = bb, φ(b) = b
11	18845	⟨a, b \| aab=a, aaabba=b⟩	φ(a) = bb, φ(b) = b
11	18851	⟨a, b \| aab=a, aabaab=b⟩	φ(a) = bb, φ(b) = b
11	18853	⟨a, b \| aab=a, aababa=b⟩	φ(a) = bb, φ(b) = b
11	18857	⟨a, b \| aab=a, aabbaa=b⟩	φ(a) = bb, φ(b) = b
11	18867	⟨a, b \| aab=a, abaaab=b⟩	φ(a) = bb, φ(b) = b
11	18869	⟨a, b \| aab=a, abaaba=b⟩	φ(a) = bb, φ(b) = b
11	18873	⟨a, b \| aab=a, ababaa=b⟩	φ(a) = bb, φ(b) = b
11	18881	⟨a, b \| aab=a, abbaaa=b⟩	φ(a) = bb, φ(b) = b
11	18895	⟨a, b \| aab=a, abbbbb=b⟩	φ(a) = bb, φ(b) = b
11	19101	⟨a, b \| aba=a, aaabba=b⟩	φ(a) = bb, φ(b) = b
11	19107	⟨a, b \| aba=a, aababa=b⟩	φ(a) = bb, φ(b) = b
11	19111	⟨a, b \| aba=a, aabbaa=b⟩	φ(a) = bb, φ(b) = b
11	19121	⟨a, b \| aba=a, abaaba=b⟩	φ(a) = bb, φ(b) = b
11	24363	⟨a, b \| ab=a, aaabbbb=b⟩	φ(a) = b, φ(b) = bbb
11	24379	⟨a, b \| ab=a, aababbb=b⟩	φ(a) = b, φ(b) = bbb
11	24387	⟨a, b \| ab=a, aabbabb=b⟩	φ(a) = b, φ(b) = bbb
11	24391	⟨a, b \| ab=a, aabbbab=b⟩	φ(a) = b, φ(b) = bbb
11	24393	⟨a, b \| ab=a, aabbbba=b⟩	φ(a) = b, φ(b) = bbb
11	24411	⟨a, b \| ab=a, abaabbb=b⟩	φ(a) = b, φ(b) = bbb
11	24419	⟨a, b \| ab=a, abababb=b⟩	φ(a) = b, φ(b) = bbb
11	24423	⟨a, b \| ab=a, ababbab=b⟩	φ(a) = b, φ(b) = bbb
11	24425	⟨a, b \| ab=a, ababbba=b⟩	φ(a) = b, φ(b) = bbb
11	24435	⟨a, b \| ab=a, abbaabb=b⟩	φ(a) = b, φ(b) = bbb
11	24439	⟨a, b \| ab=a, abbabab=b⟩	φ(a) = b, φ(b) = bbb
11	24441	⟨a, b \| ab=a, abbabba=b⟩	φ(a) = b, φ(b) = bbb
11	24447	⟨a, b \| ab=a, abbbaab=b⟩	φ(a) = b, φ(b) = bbb
11	24449	⟨a, b \| ab=a, abbbaba=b⟩	φ(a) = b, φ(b) = bbb
11	24453	⟨a, b \| ab=a, abbbbaa=b⟩	φ(a) = b, φ(b) = bbb

Presentation

Mapping

249

⟨a, b | aa=b, aab=a⟩

φ(a) = bb, φ(b) = b

251

⟨a, b | aa=b, aba=a⟩

φ(a) = bb, φ(b) = b

260

⟨a, b | ab=a, aaa=b⟩

φ(a) = b, φ(b) = bbb

900

⟨a, b | aa=b, aaaa=a⟩

φ(a) = bb, φ(b) = b

923

⟨a, b | ab=a, aaab=b⟩

φ(a) = b, φ(b) = bbb

925

⟨a, b | ab=a, aaba=b⟩

φ(a) = b, φ(b) = bbb

929

⟨a, b | ab=a, abaa=b⟩

φ(a) = b, φ(b) = bbb

2000

⟨a, b | aaa=b, aaaa=a⟩

φ(a) = b, φ(b) = bbb

2023

⟨a, b | aab=a, aaab=b⟩

φ(a) = bb, φ(b) = b

2025

⟨a, b | aab=a, aaba=b⟩

φ(a) = bb, φ(b) = b

2029

⟨a, b | aab=a, abaa=b⟩

φ(a) = bb, φ(b) = b

2089

⟨a, b | aba=a, aaba=b⟩

φ(a) = bb, φ(b) = b

2943

⟨a, b | ab=a, aaabb=b⟩

φ(a) = b, φ(b) = bbb

2947

⟨a, b | ab=a, aabab=b⟩

φ(a) = b, φ(b) = bbb

2949

⟨a, b | ab=a, aabba=b⟩

φ(a) = b, φ(b) = bbb

2955

⟨a, b | ab=a, abaab=b⟩

φ(a) = b, φ(b) = bbb

2957

⟨a, b | ab=a, ababa=b⟩

φ(a) = b, φ(b) = bbb

2961

⟨a, b | ab=a, abbaa=b⟩

φ(a) = b, φ(b) = bbb

4650

⟨a, b | aaab=a, aaba=b⟩

φ(a) = b, φ(b) = b

4654

⟨a, b | aaab=a, abaa=b⟩

φ(a) = b, φ(b) = b

4660

⟨a, b | aaab=a, abbb=b⟩

φ(a) = b, φ(b) = b

4701

⟨a, b | aaba=a, abaa=b⟩

φ(a) = b, φ(b) = b

4732

⟨a, b | aabb=a, abab=b⟩

φ(a) = b, φ(b) = b

4734

⟨a, b | aabb=a, abba=b⟩

φ(a) = b, φ(b) = b

4742

⟨a, b | abab=a, abba=b⟩

φ(a) = b, φ(b) = b

8727

⟨a, b | ab=a, aaabbb=b⟩

φ(a) = b, φ(b) = bbb

8735

⟨a, b | ab=a, aababb=b⟩

φ(a) = b, φ(b) = bbb

8739

⟨a, b | ab=a, aabbab=b⟩

φ(a) = b, φ(b) = bbb

8741

⟨a, b | ab=a, aabbba=b⟩

φ(a) = b, φ(b) = bbb

8751

⟨a, b | ab=a, abaabb=b⟩

φ(a) = b, φ(b) = bbb

8755

⟨a, b | ab=a, ababab=b⟩

φ(a) = b, φ(b) = bbb

8757

⟨a, b | ab=a, ababba=b⟩

φ(a) = b, φ(b) = bbb

8763

⟨a, b | ab=a, abbaab=b⟩

φ(a) = b, φ(b) = bbb

8765

⟨a, b | ab=a, abbaba=b⟩

φ(a) = b, φ(b) = bbb

8769

⟨a, b | ab=a, abbbaa=b⟩

φ(a) = b, φ(b) = bbb

14330

⟨a, b | aaaa=a, aaaaa=b⟩

φ(a) = bb, φ(b) = b

14475

⟨a, b | aaab=b, aaaab=a⟩

φ(a) = b, φ(b) = bbb

14603

⟨a, b | aaba=b, aaaab=a⟩

φ(a) = b, φ(b) = bbb

14605

⟨a, b | aaba=b, aaaba=a⟩

φ(a) = b, φ(b) = bbb

14609

⟨a, b | aaba=b, aabaa=a⟩

φ(a) = b, φ(b) = bbb

14617

⟨a, b | aaba=b, abaaa=a⟩

φ(a) = b, φ(b) = bbb

14633

⟨a, b | aaba=b, baaaa=a⟩

φ(a) = b, φ(b) = bbb

18839

⟨a, b | aab=a, aaaabb=b⟩

φ(a) = bb, φ(b) = b

18843

⟨a, b | aab=a, aaabab=b⟩

φ(a) = bb, φ(b) = b

18845

⟨a, b | aab=a, aaabba=b⟩

φ(a) = bb, φ(b) = b

18851

⟨a, b | aab=a, aabaab=b⟩

φ(a) = bb, φ(b) = b

18853

⟨a, b | aab=a, aababa=b⟩

φ(a) = bb, φ(b) = b

18857

⟨a, b | aab=a, aabbaa=b⟩

φ(a) = bb, φ(b) = b

18867

⟨a, b | aab=a, abaaab=b⟩

φ(a) = bb, φ(b) = b

18869

⟨a, b | aab=a, abaaba=b⟩

φ(a) = bb, φ(b) = b

18873

⟨a, b | aab=a, ababaa=b⟩

φ(a) = bb, φ(b) = b

18881

⟨a, b | aab=a, abbaaa=b⟩

φ(a) = bb, φ(b) = b

18895

⟨a, b | aab=a, abbbbb=b⟩

φ(a) = bb, φ(b) = b

19101

⟨a, b | aba=a, aaabba=b⟩

φ(a) = bb, φ(b) = b

19107

⟨a, b | aba=a, aababa=b⟩

φ(a) = bb, φ(b) = b

19111

⟨a, b | aba=a, aabbaa=b⟩

φ(a) = bb, φ(b) = b

19121

⟨a, b | aba=a, abaaba=b⟩

φ(a) = bb, φ(b) = b

24363

⟨a, b | ab=a, aaabbbb=b⟩

φ(a) = b, φ(b) = bbb

24379

⟨a, b | ab=a, aababbb=b⟩

φ(a) = b, φ(b) = bbb

24387

⟨a, b | ab=a, aabbabb=b⟩

φ(a) = b, φ(b) = bbb

24391

⟨a, b | ab=a, aabbbab=b⟩

φ(a) = b, φ(b) = bbb

24393

⟨a, b | ab=a, aabbbba=b⟩

φ(a) = b, φ(b) = bbb

24411

⟨a, b | ab=a, abaabbb=b⟩

φ(a) = b, φ(b) = bbb

24419

⟨a, b | ab=a, abababb=b⟩

φ(a) = b, φ(b) = bbb

24423

⟨a, b | ab=a, ababbab=b⟩

φ(a) = b, φ(b) = bbb

24425

⟨a, b | ab=a, ababbba=b⟩

φ(a) = b, φ(b) = bbb

24435

⟨a, b | ab=a, abbaabb=b⟩

φ(a) = b, φ(b) = bbb

24439

⟨a, b | ab=a, abbabab=b⟩

φ(a) = b, φ(b) = bbb

24441

⟨a, b | ab=a, abbabba=b⟩

φ(a) = b, φ(b) = bbb

24447

⟨a, b | ab=a, abbbaab=b⟩

φ(a) = b, φ(b) = bbb

24449

⟨a, b | ab=a, abbbaba=b⟩

φ(a) = b, φ(b) = bbb

24453

⟨a, b | ab=a, abbbbaa=b⟩

φ(a) = b, φ(b) = bbb

Up:	Monoids with two generators and two relations
Prev:	#60 ⟨a, b \| aa=b, ab=b⟩
Next:	#62 ⟨a, b \| ab=a, ba=a⟩

index 1, period 1	1 element	a³
index 1, period 3	2 elements	a, a²

#61 ⟨a, b | aa=b, bb=a⟩

Quick links

Properties

Elements

Staircase diagram

Cayley table

Right Cayley graph

Rewriting system

Same cardinality

Isomorphic instances