⟨a, b | aa=b, ab=a⟩

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Presentation has sum-of-sides 6

Isomorphic to ℕ_{(3 = 1)}

Finite commutative monoid with 3 elements

Not cancellative, because multiplication by a is not injective:

a ⋅ a² = a and a ⋅ 1 = a, however a² ≠ 1

Commutative Gröbner basis: ⟨a, b | a³=a, b=a²⟩

Cancellative quotient is isomorphic to ℤ₂

Enveloping group is isomorphic to ℤ₂

Group of units is isomorphic to ℤ₁

2 Archimedian components:

1, b

	1	a	a²
1	1	a	a²
a	a	a²	a
a²	a²	a	a²

a²

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.	a³ ⇒ a	[2]
2.	b ⇒ a²	[1]

# ab:aa=b,ab=a a/b aaa=a b=aa

Same cardinality

Σ	#	Presentation	Description	Related
5	6	⟨a, b \| aa=b, ab=1⟩	Isomorphic to ℤ₃	2029 iso
6	60	⟨a, b \| aa=b, ab=b⟩	Isomorphic to ℕ_{(3 = 2)}	23 iso
6	63	⟨a, b \| ab=a, ba=b⟩	Finite non-commutative monoid with 3 elements	61 iso, 17 anti-iso
8	891	⟨a, b \| aa=a, abba=b⟩	Finite commutative monoid with 3 elements	12 iso

Presentation

Description

⟨a, b | aa=b, ab=1⟩

Isomorphic to ℤ₃

2029 iso

⟨a, b | aa=b, ab=b⟩

Isomorphic to ℕ_{(3 = 2)}

23 iso

⟨a, b | ab=a, ba=b⟩

Finite non-commutative monoid with 3 elements

61 iso, 17 anti-iso

891

⟨a, b | aa=a, abba=b⟩

Finite commutative monoid with 3 elements

12 iso

Isomorphic instances

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

Σ	#	Presentation	Mapping
7	247	⟨a, b \| aa=b, aaa=a⟩	φ(a) = a, φ(b) = aa
7	262	⟨a, b \| ab=a, aab=b⟩	φ(a) = a, φ(b) = aa
7	264	⟨a, b \| ab=a, aba=b⟩	φ(a) = a, φ(b) = aa
8	583	⟨a, b \| aab=a, aba=b⟩	φ(a) = a, φ(b) = a
8	927	⟨a, b \| ab=a, aabb=b⟩	φ(a) = a, φ(b) = aa
8	931	⟨a, b \| ab=a, abab=b⟩	φ(a) = a, φ(b) = aa
8	933	⟨a, b \| ab=a, abba=b⟩	φ(a) = a, φ(b) = aa
9	1981	⟨a, b \| aaa=a, aaaa=b⟩	φ(a) = a, φ(b) = aa
9	2054	⟨a, b \| aab=b, aaab=a⟩	φ(a) = a, φ(b) = aa
9	2106	⟨a, b \| aba=b, aaab=a⟩	φ(a) = a, φ(b) = aa
9	2108	⟨a, b \| aba=b, aaba=a⟩	φ(a) = a, φ(b) = aa
9	2951	⟨a, b \| ab=a, aabbb=b⟩	φ(a) = a, φ(b) = aa
9	2959	⟨a, b \| ab=a, ababb=b⟩	φ(a) = a, φ(b) = aa
9	2963	⟨a, b \| ab=a, abbab=b⟩	φ(a) = a, φ(b) = aa
9	2965	⟨a, b \| ab=a, abbba=b⟩	φ(a) = a, φ(b) = aa
10	6239	⟨a, b \| aaa=a, aaaaa=b⟩	φ(a) = a, φ(b) = a
10	6325	⟨a, b \| aab=a, aaabb=b⟩	φ(a) = a, φ(b) = a
10	6329	⟨a, b \| aab=a, aabab=b⟩	φ(a) = a, φ(b) = a
10	6331	⟨a, b \| aab=a, aabba=b⟩	φ(a) = a, φ(b) = a
10	6337	⟨a, b \| aab=a, abaab=b⟩	φ(a) = a, φ(b) = a
10	6339	⟨a, b \| aab=a, ababa=b⟩	φ(a) = a, φ(b) = a
10	6343	⟨a, b \| aab=a, abbaa=b⟩	φ(a) = a, φ(b) = a
10	6349	⟨a, b \| aab=a, abbbb=b⟩	φ(a) = a, φ(b) = a
10	6384	⟨a, b \| aab=b, aaaab=a⟩	φ(a) = a, φ(b) = a
10	6459	⟨a, b \| aba=a, aabba=b⟩	φ(a) = a, φ(b) = a
10	6465	⟨a, b \| aba=a, ababa=b⟩	φ(a) = a, φ(b) = a
10	6488	⟨a, b \| aba=b, aaaab=a⟩	φ(a) = a, φ(b) = a
10	6490	⟨a, b \| aba=b, aaaba=a⟩	φ(a) = a, φ(b) = a
10	6494	⟨a, b \| aba=b, aabaa=a⟩	φ(a) = a, φ(b) = a
10	8743	⟨a, b \| ab=a, aabbbb=b⟩	φ(a) = a, φ(b) = aa
10	8759	⟨a, b \| ab=a, ababbb=b⟩	φ(a) = a, φ(b) = aa
10	8767	⟨a, b \| ab=a, abbabb=b⟩	φ(a) = a, φ(b) = aa
10	8771	⟨a, b \| ab=a, abbbab=b⟩	φ(a) = a, φ(b) = aa
10	8773	⟨a, b \| ab=a, abbbba=b⟩	φ(a) = a, φ(b) = aa
11	14424	⟨a, b \| aaab=a, aabbb=b⟩	φ(a) = a, φ(b) = aa
11	14432	⟨a, b \| aaab=a, ababb=b⟩	φ(a) = a, φ(b) = aa
11	14436	⟨a, b \| aaab=a, abbab=b⟩	φ(a) = a, φ(b) = aa
11	14438	⟨a, b \| aaab=a, abbba=b⟩	φ(a) = a, φ(b) = aa
11	14566	⟨a, b \| aaba=a, abbba=b⟩	φ(a) = a, φ(b) = aa
11	14680	⟨a, b \| aabb=a, aabbb=b⟩	φ(a) = aa, φ(b) = a
11	14692	⟨a, b \| aabb=a, abbab=b⟩	φ(a) = aa, φ(b) = a
11	14694	⟨a, b \| aabb=a, abbba=b⟩	φ(a) = aa, φ(b) = a
11	14744	⟨a, b \| abab=a, aabbb=b⟩	φ(a) = aa, φ(b) = a
11	14752	⟨a, b \| abab=a, ababb=b⟩	φ(a) = aa, φ(b) = a
11	14756	⟨a, b \| abab=a, abbab=b⟩	φ(a) = aa, φ(b) = a
11	14758	⟨a, b \| abab=a, abbba=b⟩	φ(a) = aa, φ(b) = a
11	14818	⟨a, b \| abba=a, abbba=b⟩	φ(a) = aa, φ(b) = a
11	14839	⟨a, b \| abba=b, aaabb=a⟩	φ(a) = a, φ(b) = aa
11	14845	⟨a, b \| abba=b, aabba=a⟩	φ(a) = a, φ(b) = aa
11	14849	⟨a, b \| abba=b, abaab=a⟩	φ(a) = a, φ(b) = aa
11	18689	⟨a, b \| aaa=a, aaaaaa=b⟩	φ(a) = a, φ(b) = aa
11	18962	⟨a, b \| aab=b, aaaaab=a⟩	φ(a) = a, φ(b) = aa
11	19162	⟨a, b \| aba=b, aaaaab=a⟩	φ(a) = a, φ(b) = aa
11	19164	⟨a, b \| aba=b, aaaaba=a⟩	φ(a) = a, φ(b) = aa
11	19168	⟨a, b \| aba=b, aaabaa=a⟩	φ(a) = a, φ(b) = aa
11	24395	⟨a, b \| ab=a, aabbbbb=b⟩	φ(a) = a, φ(b) = aa
11	24427	⟨a, b \| ab=a, ababbbb=b⟩	φ(a) = a, φ(b) = aa
11	24443	⟨a, b \| ab=a, abbabbb=b⟩	φ(a) = a, φ(b) = aa
11	24451	⟨a, b \| ab=a, abbbabb=b⟩	φ(a) = a, φ(b) = aa
11	24455	⟨a, b \| ab=a, abbbbab=b⟩	φ(a) = a, φ(b) = aa
11	24457	⟨a, b \| ab=a, abbbbba=b⟩	φ(a) = a, φ(b) = aa

Presentation

Mapping

247

⟨a, b | aa=b, aaa=a⟩

φ(a) = a, φ(b) = aa

262

⟨a, b | ab=a, aab=b⟩

φ(a) = a, φ(b) = aa

264

⟨a, b | ab=a, aba=b⟩

φ(a) = a, φ(b) = aa

583

⟨a, b | aab=a, aba=b⟩

φ(a) = a, φ(b) = a

927

⟨a, b | ab=a, aabb=b⟩

φ(a) = a, φ(b) = aa

931

⟨a, b | ab=a, abab=b⟩

φ(a) = a, φ(b) = aa

933

⟨a, b | ab=a, abba=b⟩

φ(a) = a, φ(b) = aa

1981

⟨a, b | aaa=a, aaaa=b⟩

φ(a) = a, φ(b) = aa

2054

⟨a, b | aab=b, aaab=a⟩

φ(a) = a, φ(b) = aa

2106

⟨a, b | aba=b, aaab=a⟩

φ(a) = a, φ(b) = aa

2108

⟨a, b | aba=b, aaba=a⟩

φ(a) = a, φ(b) = aa

2951

⟨a, b | ab=a, aabbb=b⟩

φ(a) = a, φ(b) = aa

2959

⟨a, b | ab=a, ababb=b⟩

φ(a) = a, φ(b) = aa

2963

⟨a, b | ab=a, abbab=b⟩

φ(a) = a, φ(b) = aa

2965

⟨a, b | ab=a, abbba=b⟩

φ(a) = a, φ(b) = aa

6239

⟨a, b | aaa=a, aaaaa=b⟩

φ(a) = a, φ(b) = a

6325

⟨a, b | aab=a, aaabb=b⟩

φ(a) = a, φ(b) = a

6329

⟨a, b | aab=a, aabab=b⟩

φ(a) = a, φ(b) = a

6331

⟨a, b | aab=a, aabba=b⟩

φ(a) = a, φ(b) = a

6337

⟨a, b | aab=a, abaab=b⟩

φ(a) = a, φ(b) = a

6339

⟨a, b | aab=a, ababa=b⟩

φ(a) = a, φ(b) = a

6343

⟨a, b | aab=a, abbaa=b⟩

φ(a) = a, φ(b) = a

6349

⟨a, b | aab=a, abbbb=b⟩

φ(a) = a, φ(b) = a

6384

⟨a, b | aab=b, aaaab=a⟩

φ(a) = a, φ(b) = a

6459

⟨a, b | aba=a, aabba=b⟩

φ(a) = a, φ(b) = a

6465

⟨a, b | aba=a, ababa=b⟩

φ(a) = a, φ(b) = a

6488

⟨a, b | aba=b, aaaab=a⟩

φ(a) = a, φ(b) = a

6490

⟨a, b | aba=b, aaaba=a⟩

φ(a) = a, φ(b) = a

6494

⟨a, b | aba=b, aabaa=a⟩

φ(a) = a, φ(b) = a

8743

⟨a, b | ab=a, aabbbb=b⟩

φ(a) = a, φ(b) = aa

8759

⟨a, b | ab=a, ababbb=b⟩

φ(a) = a, φ(b) = aa

8767

⟨a, b | ab=a, abbabb=b⟩

φ(a) = a, φ(b) = aa

8771

⟨a, b | ab=a, abbbab=b⟩

φ(a) = a, φ(b) = aa

8773

⟨a, b | ab=a, abbbba=b⟩

φ(a) = a, φ(b) = aa

14424

⟨a, b | aaab=a, aabbb=b⟩

φ(a) = a, φ(b) = aa

14432

⟨a, b | aaab=a, ababb=b⟩

φ(a) = a, φ(b) = aa

14436

⟨a, b | aaab=a, abbab=b⟩

φ(a) = a, φ(b) = aa

14438

⟨a, b | aaab=a, abbba=b⟩

φ(a) = a, φ(b) = aa

14566

⟨a, b | aaba=a, abbba=b⟩

φ(a) = a, φ(b) = aa

14680

⟨a, b | aabb=a, aabbb=b⟩

φ(a) = aa, φ(b) = a

14692

⟨a, b | aabb=a, abbab=b⟩

φ(a) = aa, φ(b) = a

14694

⟨a, b | aabb=a, abbba=b⟩

φ(a) = aa, φ(b) = a

14744

⟨a, b | abab=a, aabbb=b⟩

φ(a) = aa, φ(b) = a

14752

⟨a, b | abab=a, ababb=b⟩

φ(a) = aa, φ(b) = a

14756

⟨a, b | abab=a, abbab=b⟩

φ(a) = aa, φ(b) = a

14758

⟨a, b | abab=a, abbba=b⟩

φ(a) = aa, φ(b) = a

14818

⟨a, b | abba=a, abbba=b⟩

φ(a) = aa, φ(b) = a

14839

⟨a, b | abba=b, aaabb=a⟩

φ(a) = a, φ(b) = aa

14845

⟨a, b | abba=b, aabba=a⟩

φ(a) = a, φ(b) = aa

14849

⟨a, b | abba=b, abaab=a⟩

φ(a) = a, φ(b) = aa

18689

⟨a, b | aaa=a, aaaaaa=b⟩

φ(a) = a, φ(b) = aa

18962

⟨a, b | aab=b, aaaaab=a⟩

φ(a) = a, φ(b) = aa

19162

⟨a, b | aba=b, aaaaab=a⟩

φ(a) = a, φ(b) = aa

19164

⟨a, b | aba=b, aaaaba=a⟩

φ(a) = a, φ(b) = aa

19168

⟨a, b | aba=b, aaabaa=a⟩

φ(a) = a, φ(b) = aa

24395

⟨a, b | ab=a, aabbbbb=b⟩

φ(a) = a, φ(b) = aa

24427

⟨a, b | ab=a, ababbbb=b⟩

φ(a) = a, φ(b) = aa

24443

⟨a, b | ab=a, abbabbb=b⟩

φ(a) = a, φ(b) = aa

24451

⟨a, b | ab=a, abbbabb=b⟩

φ(a) = a, φ(b) = aa

24455

⟨a, b | ab=a, abbbbab=b⟩

φ(a) = a, φ(b) = aa

24457

⟨a, b | ab=a, abbbbba=b⟩

φ(a) = a, φ(b) = aa

Up:	Monoids with two generators and two relations
Prev:	#58 ⟨a, b \| aa=a, bb=b⟩
Next:	#60 ⟨a, b \| aa=b, ab=b⟩

index 1, period 1	1 element	a²
index 1, period 2	1 element	a

#59 ⟨a, b | aa=b, ab=a⟩

Quick links

Properties

Elements

Staircase diagram

Cayley table

Right Cayley graph

Rewriting system

Same cardinality

Isomorphic instances