#2013 ⟨a, b | aaa=b, abbb=b⟩

Properties

Presentation has sum-of-sides 9
Isomorphic to ℕ_{(10 = 3)}
Finite commutative monoid with 10 elements
Commutative Gröbner basis: ⟨a, b | a¹⁰=a³, b=a³⟩
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

a¹⁰ ⇒ a³
b ⇒ a³

# ab:aaa=b,abbb=b a/b
aaaaaaaaaa=aaa
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	a³
a²	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a³	a⁴
a³	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a³	a⁴	a⁵
a⁴	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a³	a⁴	a⁵	a⁶
a⁵	a⁵	a⁶	a⁷	a⁸	a⁹	a³	a⁴	a⁵	a⁶	a⁷
a⁶	a⁶	a⁷	a⁸	a⁹	a³	a⁴	a⁵	a⁶	a⁷	a⁸
a⁷	a⁷	a⁸	a⁹	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹
a⁸	a⁸	a⁹	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a³
a⁹	a⁹	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a³	a⁴

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

31 unique, 884 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	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.

35 total

Σ	#	Presentation	Mapping
9	2017	⟨a, b \| aaa=b, babb=b⟩	φ(a) = a, φ(b) = aaa
10	5081	⟨a, b \| aaa=bb, aabb=b⟩	φ(a) = a, φ(b) = aaaaa
10	5083	⟨a, b \| aaa=bb, abab=b⟩	φ(a) = a, φ(b) = aaaaa
10	5085	⟨a, b \| aaa=bb, abba=b⟩	φ(a) = a, φ(b) = aaaaa
10	8982	⟨a, b \| aa=b, bbbbb=ab⟩	φ(a) = a, φ(b) = aa
11	15472	⟨a, b \| aaa=ab, abbbb=b⟩	φ(a) = a, φ(b) = aaaaaaaaa
11	15506	⟨a, b \| aaa=bb, aaaaa=b⟩	φ(a) = a, φ(b) = aaaaa
11	18767	⟨a, b \| aaa=b, aaaabb=b⟩	φ(a) = a, φ(b) = aaa
11	18771	⟨a, b \| aaa=b, aaabab=b⟩	φ(a) = a, φ(b) = aaa
11	18773	⟨a, b \| aaa=b, aaabba=b⟩	φ(a) = a, φ(b) = aaa
11	18777	⟨a, b \| aaa=b, aabaab=b⟩	φ(a) = a, φ(b) = aaa
11	18779	⟨a, b \| aaa=b, aababa=b⟩	φ(a) = a, φ(b) = aaa
11	18783	⟨a, b \| aaa=b, aabbaa=b⟩	φ(a) = a, φ(b) = aaa
11	18791	⟨a, b \| aaa=b, abaaab=b⟩	φ(a) = a, φ(b) = aaa
11	18793	⟨a, b \| aaa=b, abaaba=b⟩	φ(a) = a, φ(b) = aaa
11	18813	⟨a, b \| aaa=b, baaaab=b⟩	φ(a) = a, φ(b) = aaa
11	19892	⟨a, b \| aaa=b, abbb=aaa⟩	φ(a) = a, φ(b) = aaa
11	19906	⟨a, b \| aaa=b, babb=aaa⟩	φ(a) = a, φ(b) = aaa
11	19927	⟨a, b \| aab=a, aaaa=bbb⟩	φ(a) = aaaaaa, φ(b) = a
11	24779	⟨a, b \| aa=b, aabbbb=ab⟩	φ(a) = a, φ(b) = aa
11	24780	⟨a, b \| aa=b, aabbbb=ba⟩	φ(a) = a, φ(b) = aa
11	24802	⟨a, b \| aa=b, ababbb=ab⟩	φ(a) = a, φ(b) = aa
11	24803	⟨a, b \| aa=b, ababbb=ba⟩	φ(a) = a, φ(b) = aa
11	24810	⟨a, b \| aa=b, abbabb=ab⟩	φ(a) = a, φ(b) = aa
11	24811	⟨a, b \| aa=b, abbabb=ba⟩	φ(a) = a, φ(b) = aa
11	24814	⟨a, b \| aa=b, abbbab=ab⟩	φ(a) = a, φ(b) = aa
11	24815	⟨a, b \| aa=b, abbbab=ba⟩	φ(a) = a, φ(b) = aa
11	24818	⟨a, b \| aa=b, abbbba=ab⟩	φ(a) = a, φ(b) = aa
11	24836	⟨a, b \| aa=b, baabbb=ab⟩	φ(a) = a, φ(b) = aa
11	24837	⟨a, b \| aa=b, baabbb=ba⟩	φ(a) = a, φ(b) = aa
11	24840	⟨a, b \| aa=b, bababb=ab⟩	φ(a) = a, φ(b) = aa
11	24841	⟨a, b \| aa=b, bababb=ba⟩	φ(a) = a, φ(b) = aa
11	24844	⟨a, b \| aa=b, babbab=ab⟩	φ(a) = a, φ(b) = aa
11	24851	⟨a, b \| aa=b, bbaabb=ab⟩	φ(a) = a, φ(b) = aa
11	25398	⟨a, b \| aa=b, bbbbb=aaa⟩	φ(a) = a, φ(b) = aa

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