#12179 ⟨a, b | aaaa=ab, abbb=b⟩

Properties

Presentation has sum-of-sides 11
Isomorphic to ℕ_{(11 = 4)}
Finite commutative monoid with 11 elements
Commutative Gröbner basis: ⟨a, b | a¹¹=a⁴, b=a¹⁰⟩
Not cancellative, because multiplication by a⁴ is not injective:
- a⁹ ≠ a²
- a⁴ ⋅ a⁹ = a⁶
- a⁴ ⋅ a² = a⁶
- Cancellative quotient is isomorphic to ℤ₇
Grothendieck group is isomorphic to ℤ₇
Group of units is isomorphic to ℤ₁
2 Archimedian components:
- 1, b

Element profile

2 idempotent elements:
- 1, a⁷

Complete 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

a¹¹ ⇒ a⁴
b ⇒ a¹⁰

# ab:aaaa=ab,abbb=b a/b
aaaaaaaaaaa=aaaa
b=aaaaaaaaaa

Staircase diagram

Cayley table

Idempotents are shown in bold.

	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰
1	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰
a	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a⁴
a²	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a⁴	a⁵
a³	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a⁴	a⁵	a⁶
a⁴	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a⁴	a⁵	a⁶	a⁷
a⁵	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a⁴	a⁵	a⁶	a⁷	a⁸
a⁶	a⁶	a⁷	a⁸	a⁹	a¹⁰	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹
a⁷	a⁷	a⁸	a⁹	a¹⁰	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰
a⁸	a⁸	a⁹	a¹⁰	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a⁴
a⁹	a⁹	a¹⁰	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a⁴	a⁵
a¹⁰	a¹⁰	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a⁴	a⁵	a⁶

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

29 unique, 460 total

Σ	#	Presentation	Description	Related
9	1472	⟨a, b \| aaa=bb, abbb=1⟩	Isomorphic to ℤ₁₁	259 iso
10	4155	⟨a, b \| abb=aaa, bba=b⟩	Finite non-commutative monoid with 11 elements	4 iso
10	5086	⟨a, b \| aaa=bb, abbb=a⟩	Finite commutative monoid with 11 elements	11 iso
10	5087	⟨a, b \| aaa=bb, abbb=b⟩	Finite commutative monoid with 11 elements	3 iso
10	5111	⟨a, b \| aab=aa, baaa=b⟩	Finite non-commutative monoid with 11 elements	7 iso
10	5322	⟨a, b \| aaa=ab, abb=bb⟩	Finite non-commutative monoid with 11 elements	3 iso
10	5335	⟨a, b \| aaa=ab, bbb=aa⟩	Finite non-commutative monoid with 11 elements	2 iso, 1 anti-iso
10	5410	⟨a, b \| aab=bb, bab=aa⟩	Finite non-commutative monoid with 11 elements	2 iso
10	6292	⟨a, b \| aaa=b, aabbb=a⟩	Isomorphic to ℕ_{(11 = 1)}	35 iso
10	6293	⟨a, b \| aaa=b, aabbb=b⟩	Isomorphic to ℕ_{(11 = 3)}	11 iso
10	6380	⟨a, b \| aab=a, bbbbb=a⟩	Isomorphic to ℕ_{(11 = 5)}	1 iso
10	6834	⟨a, b \| aba=b, abb=aaa⟩	Finite non-commutative monoid with 11 elements
10	7114	⟨a, b \| ab=aa, bbbb=aa⟩	Finite non-commutative monoid with 11 elements	1 iso
10	7116	⟨a, b \| ab=aa, bbbb=ba⟩	Finite non-commutative monoid with 11 elements
10	8691	⟨a, b \| aa=b, abbbbb=b⟩	Isomorphic to ℕ_{(11 = 2)}	32 iso
10	8777	⟨a, b \| ab=a, baaaaa=b⟩	Finite non-commutative monoid with 11 elements	20 iso, 5 anti-iso
10	9083	⟨a, b \| ab=a, bbaaa=bb⟩	Finite non-commutative monoid with 11 elements	4 iso
11	15607	⟨a, b \| aab=aa, bbbbb=a⟩	Isomorphic to ℕ_{(11 = 10)}	1 iso
11	15671	⟨a, b \| aab=ab, bbbbb=a⟩	Isomorphic to ℕ_{(11 = 6)}	8 iso
11	16148	⟨a, b \| aab=ab, aaaa=bb⟩	Finite non-commutative monoid with 11 elements	1 iso
11	16212	⟨a, b \| aab=ba, aaaa=bb⟩	Finite non-commutative monoid with 11 elements
11	18715	⟨a, b \| aaa=a, aabbba=b⟩	Finite non-commutative monoid with 11 elements
11	19070	⟨a, b \| aab=b, bbabbb=a⟩	Finite commutative monoid with 11 elements	4 iso
11	20096	⟨a, b \| aab=b, abba=aaa⟩	Finite non-commutative monoid with 11 elements	1 iso
11	20112	⟨a, b \| aab=b, baaa=aaa⟩	Finite non-commutative monoid with 11 elements
11	20168	⟨a, b \| aab=b, bbbb=aaa⟩	Finite commutative monoid with 11 elements
11	20885	⟨a, b \| bb=aa, aaaaa=ab⟩	Finite non-commutative monoid with 11 elements	7 iso, 8 anti-iso
11	25539	⟨a, b \| ab=a, baaaa=bbb⟩	Finite non-commutative monoid with 11 elements
11	25635	⟨a, b \| ab=a, bbbaa=bbb⟩	Finite non-commutative monoid with 11 elements

Other isomorphic instances

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

20 total

Σ	#	Presentation	Mapping
11	14376	⟨a, b \| aaaa=b, aaabb=b⟩	φ(a) = a, φ(b) = aaaa
11	14380	⟨a, b \| aaaa=b, aabab=b⟩	φ(a) = a, φ(b) = aaaa
11	14382	⟨a, b \| aaaa=b, aabba=b⟩	φ(a) = a, φ(b) = aaaa
11	14386	⟨a, b \| aaaa=b, abaab=b⟩	φ(a) = a, φ(b) = aaaa
11	14388	⟨a, b \| aaaa=b, ababa=b⟩	φ(a) = a, φ(b) = aaaa
11	14398	⟨a, b \| aaaa=b, baaab=b⟩	φ(a) = a, φ(b) = aaaa
11	19331	⟨a, b \| aaa=b, aabbb=ab⟩	φ(a) = a, φ(b) = aaa
11	19332	⟨a, b \| aaa=b, aabbb=ba⟩	φ(a) = a, φ(b) = aaa
11	19342	⟨a, b \| aaa=b, ababb=ab⟩	φ(a) = a, φ(b) = aaa
11	19343	⟨a, b \| aaa=b, ababb=ba⟩	φ(a) = a, φ(b) = aaa
11	19346	⟨a, b \| aaa=b, abbab=ab⟩	φ(a) = a, φ(b) = aaa
11	19347	⟨a, b \| aaa=b, abbab=ba⟩	φ(a) = a, φ(b) = aaa
11	19350	⟨a, b \| aaa=b, abbba=ab⟩	φ(a) = a, φ(b) = aaa
11	19360	⟨a, b \| aaa=b, baabb=ab⟩	φ(a) = a, φ(b) = aaa
11	19361	⟨a, b \| aaa=b, baabb=ba⟩	φ(a) = a, φ(b) = aaa
11	19364	⟨a, b \| aaa=b, babab=ab⟩	φ(a) = a, φ(b) = aaa
11	20040	⟨a, b \| aab=a, bbbb=aaa⟩	φ(a) = aaaaaa, φ(b) = a
11	24823	⟨a, b \| aa=b, abbbbb=bb⟩	φ(a) = a, φ(b) = aa
11	24849	⟨a, b \| aa=b, babbbb=bb⟩	φ(a) = a, φ(b) = aa
11	24856	⟨a, b \| aa=b, bbabbb=bb⟩	φ(a) = a, φ(b) = aa

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