#2012 ⟨a, b | aaa=b, abbb=a⟩

Properties

Presentation has sum-of-sides 9
Isomorphic to ℕ_{(10 = 1)}
Finite commutative monoid with 10 elements
Commutative Gröbner basis: ⟨a, b | a¹⁰=a, b=a³⟩
Not cancellative, because multiplication by a⁴ is not injective:
- a⁹ ≠ 1
- a⁴ ⋅ a⁹ = a⁴
- a⁴ ⋅ 1 = a⁴
- Cancellative quotient is isomorphic to ℤ₉
Grothendieck group is isomorphic to ℤ₉
Group of units is isomorphic to ℤ₁
2 Archimedian components:
- 1, b

Element profile

2 idempotent elements:
- 1, a⁹

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

a¹⁰ ⇒ a
b ⇒ a³

# ab:aaa=b,abbb=a a/b
aaaaaaaaaa=a
b=aaa

Staircase diagram

Cayley table

Idempotents are shown in bold.

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

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

31 unique, 860 total

Σ	#	Presentation	Description	Related
7	332	⟨a, b \| aa=1, abbb=b⟩	Finite non-commutative monoid with 10 elements	51 iso, 34 anti-iso
8	507	⟨a, b \| aaa=b, abbb=1⟩	Isomorphic to ℤ₁₀	536 iso
8	656	⟨a, b \| bb=aa, aba=a⟩	Finite non-commutative monoid with 10 elements	1 iso
8	657	⟨a, b \| bb=aa, aba=b⟩	Finite non-commutative monoid with 10 elements	1 iso
8	970	⟨a, b \| aa=a, bbb=ab⟩	Finite non-commutative monoid with 10 elements	3 iso
9	1654	⟨a, b \| aab=bb, bba=a⟩	Finite non-commutative monoid with 10 elements	6 iso
9	2013	⟨a, b \| aaa=b, abbb=b⟩	Isomorphic to ℕ_{(10 = 3)}	35 iso
9	2277	⟨a, b \| bb=aa, aba=aa⟩	Finite non-commutative monoid with 10 elements	1 iso
9	2894	⟨a, b \| aa=a, bbbbb=a⟩	Isomorphic to ℕ_{(10 = 5)}	17 iso
9	2935	⟨a, b \| aa=b, bbbbb=b⟩	Isomorphic to ℕ_{(10 = 2)}	57 iso
10	4637	⟨a, b \| aaaa=b, aabb=b⟩	Isomorphic to ℕ_{(10 = 4)}	20 iso
10	5346	⟨a, b \| aaa=bb, abb=ab⟩	Finite non-commutative monoid with 10 elements	2 iso, 1 anti-iso
10	5356	⟨a, b \| aab=aa, aba=bb⟩	Finite non-commutative monoid with 10 elements
10	6587	⟨a, b \| aaa=b, abbb=bb⟩	Isomorphic to ℕ_{(10 = 6)}	3 iso
10	6633	⟨a, b \| aab=a, baaa=bb⟩	Finite non-commutative monoid with 10 elements	20 iso, 1 anti-iso
10	7089	⟨a, b \| ab=aa, baaa=bb⟩	Finite non-commutative monoid with 10 elements	3 iso
10	8619	⟨a, b \| aa=a, abbbbb=b⟩	Finite non-commutative monoid with 10 elements	5 iso
10	9051	⟨a, b \| ab=a, baaaa=bb⟩	Finite non-commutative monoid with 10 elements	4 iso
11	10729	⟨a, b \| aaab=baa, abab=1⟩	Finite non-Abelian group with 10 elements	3 iso
11	12157	⟨a, b \| aaaa=aa, abba=b⟩	Finite non-commutative monoid with 10 elements
11	12181	⟨a, b \| aaaa=ab, baaa=b⟩	Finite non-commutative monoid with 10 elements	1 iso
11	12196	⟨a, b \| aaaa=bb, aaab=a⟩	Finite commutative monoid with 10 elements	7 iso
11	12197	⟨a, b \| aaaa=bb, aaab=b⟩	Finite commutative monoid with 10 elements	1 iso
11	12452	⟨a, b \| aabb=aa, bbbb=a⟩	Isomorphic to ℕ_{(10 = 8)}	3 iso
11	15738	⟨a, b \| aab=bb, aaaaa=b⟩	Isomorphic to ℕ_{(10 = 7)}	7 iso
11	19564	⟨a, b \| aab=b, abbbb=aa⟩	Finite commutative monoid with 10 elements	5 iso
11	19899	⟨a, b \| aaa=b, abbb=bbb⟩	Isomorphic to ℕ_{(10 = 9)}	1 iso
11	20052	⟨a, b \| aab=b, aaaa=baa⟩	Finite non-commutative monoid with 10 elements
11	20253	⟨a, b \| aba=b, aaaa=bbb⟩	Finite commutative monoid with 10 elements
11	25905	⟨a, b \| ab=a, bbbb=baaa⟩	Finite non-commutative monoid with 10 elements
11	25909	⟨a, b \| ab=a, bbbb=bbaa⟩	Finite non-commutative monoid with 10 elements

Other isomorphic instances

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

59 total

Σ	#	Presentation	Mapping
9	2016	⟨a, b \| aaa=b, babb=a⟩	φ(a) = a, φ(b) = aaa
9	2934	⟨a, b \| aa=b, bbbbb=a⟩	φ(a) = aaaaa, φ(b) = a
10	4636	⟨a, b \| aaaa=b, aabb=a⟩	φ(a) = aaaaaaa, φ(b) = a
10	4638	⟨a, b \| aaaa=b, abab=a⟩	φ(a) = aaaaaaa, φ(b) = a
10	4640	⟨a, b \| aaaa=b, abba=a⟩	φ(a) = aaaaaaa, φ(b) = a
10	4644	⟨a, b \| aaaa=b, baab=a⟩	φ(a) = aaaaaaa, φ(b) = a
10	8668	⟨a, b \| aa=b, aabbbb=a⟩	φ(a) = aaaaa, φ(b) = a
10	8680	⟨a, b \| aa=b, ababbb=a⟩	φ(a) = aaaaa, φ(b) = a
10	8684	⟨a, b \| aa=b, abbabb=a⟩	φ(a) = aaaaa, φ(b) = a
10	8686	⟨a, b \| aa=b, abbbab=a⟩	φ(a) = aaaaa, φ(b) = a
10	8688	⟨a, b \| aa=b, abbbba=a⟩	φ(a) = aaaaa, φ(b) = a
10	8698	⟨a, b \| aa=b, baabbb=a⟩	φ(a) = aaaaa, φ(b) = a
10	8700	⟨a, b \| aa=b, bababb=a⟩	φ(a) = aaaaa, φ(b) = a
10	8702	⟨a, b \| aa=b, babbab=a⟩	φ(a) = aaaaa, φ(b) = a
10	8706	⟨a, b \| aa=b, bbaabb=a⟩	φ(a) = aaaaa, φ(b) = a
11	14487	⟨a, b \| aaab=b, aabbb=a⟩	φ(a) = aaaaaa, φ(b) = a
11	14495	⟨a, b \| aaab=b, ababb=a⟩	φ(a) = aaaaaa, φ(b) = a
11	14499	⟨a, b \| aaab=b, abbab=a⟩	φ(a) = aaaaaa, φ(b) = a
11	14511	⟨a, b \| aaab=b, baabb=a⟩	φ(a) = aaaaaa, φ(b) = a
11	14515	⟨a, b \| aaab=b, babab=a⟩	φ(a) = aaaaaa, φ(b) = a
11	14523	⟨a, b \| aaab=b, bbaab=a⟩	φ(a) = aaaaaa, φ(b) = a
11	14615	⟨a, b \| aaba=b, aabbb=a⟩	φ(a) = aaaaaa, φ(b) = a
11	14623	⟨a, b \| aaba=b, ababb=a⟩	φ(a) = aaaaaa, φ(b) = a
11	14627	⟨a, b \| aaba=b, abbab=a⟩	φ(a) = aaaaaa, φ(b) = a
11	14629	⟨a, b \| aaba=b, abbba=a⟩	φ(a) = aaaaaa, φ(b) = a
11	14639	⟨a, b \| aaba=b, baabb=a⟩	φ(a) = aaaaaa, φ(b) = a
11	14643	⟨a, b \| aaba=b, babab=a⟩	φ(a) = aaaaaa, φ(b) = a
11	14645	⟨a, b \| aaba=b, babba=a⟩	φ(a) = aaaaaa, φ(b) = a
11	14651	⟨a, b \| aaba=b, bbaab=a⟩	φ(a) = aaaaaa, φ(b) = a
11	14653	⟨a, b \| aaba=b, bbaba=a⟩	φ(a) = aaaaaa, φ(b) = a
11	14657	⟨a, b \| aaba=b, bbbaa=a⟩	φ(a) = aaaaaa, φ(b) = a
11	18766	⟨a, b \| aaa=b, aaaabb=a⟩	φ(a) = a, φ(b) = aaa
11	18770	⟨a, b \| aaa=b, aaabab=a⟩	φ(a) = a, φ(b) = aaa
11	18772	⟨a, b \| aaa=b, aaabba=a⟩	φ(a) = a, φ(b) = aaa
11	18776	⟨a, b \| aaa=b, aabaab=a⟩	φ(a) = a, φ(b) = aaa
11	18778	⟨a, b \| aaa=b, aababa=a⟩	φ(a) = a, φ(b) = aaa
11	18782	⟨a, b \| aaa=b, aabbaa=a⟩	φ(a) = a, φ(b) = aaa
11	18790	⟨a, b \| aaa=b, abaaab=a⟩	φ(a) = a, φ(b) = aaa
11	18792	⟨a, b \| aaa=b, abaaba=a⟩	φ(a) = a, φ(b) = aaa
11	18812	⟨a, b \| aaa=b, baaaab=a⟩	φ(a) = a, φ(b) = aaa
11	24202	⟨a, b \| aa=b, aaaabbb=a⟩	φ(a) = aaaaa, φ(b) = a
11	24210	⟨a, b \| aa=b, aaababb=a⟩	φ(a) = aaaaa, φ(b) = a
11	24214	⟨a, b \| aa=b, aaabbab=a⟩	φ(a) = aaaaa, φ(b) = a
11	24216	⟨a, b \| aa=b, aaabbba=a⟩	φ(a) = aaaaa, φ(b) = a
11	24224	⟨a, b \| aa=b, aabaabb=a⟩	φ(a) = aaaaa, φ(b) = a
11	24228	⟨a, b \| aa=b, aababab=a⟩	φ(a) = aaaaa, φ(b) = a
11	24230	⟨a, b \| aa=b, aababba=a⟩	φ(a) = aaaaa, φ(b) = a
11	24234	⟨a, b \| aa=b, aabbaab=a⟩	φ(a) = aaaaa, φ(b) = a
11	24236	⟨a, b \| aa=b, aabbaba=a⟩	φ(a) = aaaaa, φ(b) = a
11	24240	⟨a, b \| aa=b, aabbbaa=a⟩	φ(a) = aaaaa, φ(b) = a
11	24252	⟨a, b \| aa=b, abaaabb=a⟩	φ(a) = aaaaa, φ(b) = a
11	24254	⟨a, b \| aa=b, abaabab=a⟩	φ(a) = aaaaa, φ(b) = a
11	24256	⟨a, b \| aa=b, abaabba=a⟩	φ(a) = aaaaa, φ(b) = a
11	24260	⟨a, b \| aa=b, ababaab=a⟩	φ(a) = aaaaa, φ(b) = a
11	24262	⟨a, b \| aa=b, abababa=a⟩	φ(a) = aaaaa, φ(b) = a
11	24272	⟨a, b \| aa=b, abbaaab=a⟩	φ(a) = aaaaa, φ(b) = a
11	24294	⟨a, b \| aa=b, baaaabb=a⟩	φ(a) = aaaaa, φ(b) = a
11	24296	⟨a, b \| aa=b, baaabab=a⟩	φ(a) = aaaaa, φ(b) = a
11	24300	⟨a, b \| aa=b, baabaab=a⟩	φ(a) = aaaaa, φ(b) = a

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