#2935 ⟨a, b | aa=b, bbbbb=b⟩

Properties

Presentation has sum-of-sides 9
Isomorphic to ℕ_{(10 = 2)}
Finite commutative monoid with 10 elements
Commutative Gröbner basis: ⟨a, b | a¹⁰=a², b=a²⟩
Not cancellative, because multiplication by a³ is not injective:
- a⁸ ≠ 1
- a³ ⋅ a⁸ = a³
- a³ ⋅ 1 = a³
- Cancellative quotient is isomorphic to ℤ₈
Grothendieck group is isomorphic to ℤ₈
Group of units is isomorphic to ℤ₁
2 Archimedian components:
- 1, b

Element profile

2 idempotent elements:
- 1, a⁸

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 ⇒ a²
b ⇒ a²

# ab:aa=b,bbbbb=b a/b
aaaaaaaaaa=aa
b=aa

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

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

31 unique, 862 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	2012	⟨a, b \| aaa=b, abbb=a⟩	Isomorphic to ℕ_{(10 = 1)}	59 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
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.

57 total

Σ	#	Presentation	Mapping
10	5080	⟨a, b \| aaa=bb, aabb=a⟩	φ(a) = aaaaaa, φ(b) = a
10	5082	⟨a, b \| aaa=bb, abab=a⟩	φ(a) = aaaaaa, φ(b) = a
10	5084	⟨a, b \| aaa=bb, abba=a⟩	φ(a) = aaaaaa, φ(b) = a
10	6584	⟨a, b \| aaa=b, abbb=aa⟩	φ(a) = a, φ(b) = aaa
10	6591	⟨a, b \| aaa=b, babb=aa⟩	φ(a) = a, φ(b) = aaa
10	7035	⟨a, b \| bb=aa, aaaaa=b⟩	φ(a) = a, φ(b) = aaaaa
10	7036	⟨a, b \| bb=aa, aaaab=a⟩	φ(a) = aaaaa, φ(b) = a
10	7038	⟨a, b \| bb=aa, aaaba=a⟩	φ(a) = aaaaa, φ(b) = a
10	7041	⟨a, b \| bb=aa, aaabb=b⟩	φ(a) = a, φ(b) = aaaaa
10	7042	⟨a, b \| bb=aa, aabaa=a⟩	φ(a) = aaaaa, φ(b) = a
10	7045	⟨a, b \| bb=aa, aabab=b⟩	φ(a) = a, φ(b) = aaaaa
10	7047	⟨a, b \| bb=aa, aabba=b⟩	φ(a) = a, φ(b) = aaaaa
10	7049	⟨a, b \| bb=aa, abaab=b⟩	φ(a) = a, φ(b) = aaaaa
10	7052	⟨a, b \| bb=aa, abbba=a⟩	φ(a) = aaaaa, φ(b) = a
10	8669	⟨a, b \| aa=b, aabbbb=b⟩	φ(a) = a, φ(b) = aa
10	8681	⟨a, b \| aa=b, ababbb=b⟩	φ(a) = a, φ(b) = aa
10	8685	⟨a, b \| aa=b, abbabb=b⟩	φ(a) = a, φ(b) = aa
10	8687	⟨a, b \| aa=b, abbbab=b⟩	φ(a) = a, φ(b) = aa
10	8689	⟨a, b \| aa=b, abbbba=b⟩	φ(a) = a, φ(b) = aa
10	8699	⟨a, b \| aa=b, baabbb=b⟩	φ(a) = a, φ(b) = aa
10	8701	⟨a, b \| aa=b, bababb=b⟩	φ(a) = a, φ(b) = aa
10	8703	⟨a, b \| aa=b, babbab=b⟩	φ(a) = a, φ(b) = aa
10	8707	⟨a, b \| aa=b, bbaabb=b⟩	φ(a) = a, φ(b) = aa
10	8981	⟨a, b \| aa=b, bbbbb=aa⟩	φ(a) = a, φ(b) = aa
11	12488	⟨a, b \| aabb=bb, bbbb=a⟩	φ(a) = aaaa, φ(b) = a
11	12552	⟨a, b \| abab=bb, bbbb=a⟩	φ(a) = aaaa, φ(b) = a
11	12604	⟨a, b \| abba=bb, bbbb=a⟩	φ(a) = aaaa, φ(b) = a
11	12626	⟨a, b \| baab=bb, bbbb=a⟩	φ(a) = aaaa, φ(b) = a
11	15505	⟨a, b \| aaa=bb, aaaaa=a⟩	φ(a) = aaaaaa, φ(b) = a
11	24203	⟨a, b \| aa=b, aaaabbb=b⟩	φ(a) = a, φ(b) = aa
11	24211	⟨a, b \| aa=b, aaababb=b⟩	φ(a) = a, φ(b) = aa
11	24215	⟨a, b \| aa=b, aaabbab=b⟩	φ(a) = a, φ(b) = aa
11	24217	⟨a, b \| aa=b, aaabbba=b⟩	φ(a) = a, φ(b) = aa
11	24225	⟨a, b \| aa=b, aabaabb=b⟩	φ(a) = a, φ(b) = aa
11	24229	⟨a, b \| aa=b, aababab=b⟩	φ(a) = a, φ(b) = aa
11	24231	⟨a, b \| aa=b, aababba=b⟩	φ(a) = a, φ(b) = aa
11	24235	⟨a, b \| aa=b, aabbaab=b⟩	φ(a) = a, φ(b) = aa
11	24237	⟨a, b \| aa=b, aabbaba=b⟩	φ(a) = a, φ(b) = aa
11	24241	⟨a, b \| aa=b, aabbbaa=b⟩	φ(a) = a, φ(b) = aa
11	24253	⟨a, b \| aa=b, abaaabb=b⟩	φ(a) = a, φ(b) = aa
11	24255	⟨a, b \| aa=b, abaabab=b⟩	φ(a) = a, φ(b) = aa
11	24257	⟨a, b \| aa=b, abaabba=b⟩	φ(a) = a, φ(b) = aa
11	24261	⟨a, b \| aa=b, ababaab=b⟩	φ(a) = a, φ(b) = aa
11	24263	⟨a, b \| aa=b, abababa=b⟩	φ(a) = a, φ(b) = aa
11	24273	⟨a, b \| aa=b, abbaaab=b⟩	φ(a) = a, φ(b) = aa
11	24295	⟨a, b \| aa=b, baaaabb=b⟩	φ(a) = a, φ(b) = aa
11	24297	⟨a, b \| aa=b, baaabab=b⟩	φ(a) = a, φ(b) = aa
11	24301	⟨a, b \| aa=b, baabaab=b⟩	φ(a) = a, φ(b) = aa
11	24778	⟨a, b \| aa=b, aabbbb=aa⟩	φ(a) = a, φ(b) = aa
11	24801	⟨a, b \| aa=b, ababbb=aa⟩	φ(a) = a, φ(b) = aa
11	24809	⟨a, b \| aa=b, abbabb=aa⟩	φ(a) = a, φ(b) = aa
11	24813	⟨a, b \| aa=b, abbbab=aa⟩	φ(a) = a, φ(b) = aa
11	24817	⟨a, b \| aa=b, abbbba=aa⟩	φ(a) = a, φ(b) = aa
11	24835	⟨a, b \| aa=b, baabbb=aa⟩	φ(a) = a, φ(b) = aa
11	24839	⟨a, b \| aa=b, bababb=aa⟩	φ(a) = a, φ(b) = aa
11	24843	⟨a, b \| aa=b, babbab=aa⟩	φ(a) = a, φ(b) = aa
11	24850	⟨a, b \| aa=b, bbaabb=aa⟩	φ(a) = a, φ(b) = aa

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