#6292 ⟨a, b | aaa=b, aabbb=a⟩

Properties

Presentation has sum-of-sides 10
Isomorphic to ℕ_{(11 = 1)}
Finite commutative monoid with 11 elements
Commutative Gröbner basis: ⟨a, b | b¹¹=b, a=b⁷⟩
Not cancellative, because multiplication by b⁸ is not injective:
- b¹⁰ ≠ 1
- b⁸ ⋅ b¹⁰ = b⁸
- b⁸ ⋅ 1 = b⁸
- Cancellative quotient is isomorphic to ℤ₁₀
Grothendieck group is isomorphic to ℤ₁₀
Group of units is isomorphic to ℤ₁
2 Archimedian components:
- 1, b

Element profile

2 idempotent elements:
- 1, a¹⁰

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 ⇒ a
b ⇒ a³

# ab:aaa=b,aabbb=a a/b
aaaaaaaaaaa=a
b=aaa

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

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

29 unique, 445 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	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	12179	⟨a, b \| aaaa=ab, abbb=b⟩	Isomorphic to ℕ_{(11 = 4)}	20 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.

35 total

Σ	#	Presentation	Mapping
10	6298	⟨a, b \| aaa=b, ababb=a⟩	φ(a) = bbbbbbb, φ(b) = b
10	6300	⟨a, b \| aaa=b, abbab=a⟩	φ(a) = bbbbbbb, φ(b) = b
10	6302	⟨a, b \| aaa=b, abbba=a⟩	φ(a) = bbbbbbb, φ(b) = b
10	6308	⟨a, b \| aaa=b, baabb=a⟩	φ(a) = bbbbbbb, φ(b) = b
10	6310	⟨a, b \| aaa=b, babab=a⟩	φ(a) = bbbbbbb, φ(b) = b
10	6444	⟨a, b \| aab=b, bbbbb=a⟩	φ(a) = bbbbb, φ(b) = b
10	6524	⟨a, b \| aba=b, bbbbb=a⟩	φ(a) = bbbbb, φ(b) = b
10	8690	⟨a, b \| aa=b, abbbbb=a⟩	φ(a) = b, φ(b) = bb
10	8704	⟨a, b \| aa=b, babbbb=a⟩	φ(a) = b, φ(b) = bb
10	8708	⟨a, b \| aa=b, bbabbb=a⟩	φ(a) = b, φ(b) = bb
11	14375	⟨a, b \| aaaa=b, aaabb=a⟩	φ(a) = b, φ(b) = bbbb
11	14379	⟨a, b \| aaaa=b, aabab=a⟩	φ(a) = b, φ(b) = bbbb
11	14381	⟨a, b \| aaaa=b, aabba=a⟩	φ(a) = b, φ(b) = bbbb
11	14385	⟨a, b \| aaaa=b, abaab=a⟩	φ(a) = b, φ(b) = bbbb
11	14387	⟨a, b \| aaaa=b, ababa=a⟩	φ(a) = b, φ(b) = bbbb
11	14397	⟨a, b \| aaaa=b, baaab=a⟩	φ(a) = b, φ(b) = bbbb
11	24218	⟨a, b \| aa=b, aaabbbb=a⟩	φ(a) = b, φ(b) = bb
11	24232	⟨a, b \| aa=b, aababbb=a⟩	φ(a) = b, φ(b) = bb
11	24238	⟨a, b \| aa=b, aabbabb=a⟩	φ(a) = b, φ(b) = bb
11	24242	⟨a, b \| aa=b, aabbbab=a⟩	φ(a) = b, φ(b) = bb
11	24244	⟨a, b \| aa=b, aabbbba=a⟩	φ(a) = b, φ(b) = bb
11	24258	⟨a, b \| aa=b, abaabbb=a⟩	φ(a) = b, φ(b) = bb
11	24264	⟨a, b \| aa=b, abababb=a⟩	φ(a) = b, φ(b) = bb
11	24266	⟨a, b \| aa=b, ababbab=a⟩	φ(a) = b, φ(b) = bb
11	24268	⟨a, b \| aa=b, ababbba=a⟩	φ(a) = b, φ(b) = bb
11	24274	⟨a, b \| aa=b, abbaabb=a⟩	φ(a) = b, φ(b) = bb
11	24276	⟨a, b \| aa=b, abbabab=a⟩	φ(a) = b, φ(b) = bb
11	24278	⟨a, b \| aa=b, abbabba=a⟩	φ(a) = b, φ(b) = bb
11	24282	⟨a, b \| aa=b, abbbaab=a⟩	φ(a) = b, φ(b) = bb
11	24298	⟨a, b \| aa=b, baaabbb=a⟩	φ(a) = b, φ(b) = bb
11	24302	⟨a, b \| aa=b, baababb=a⟩	φ(a) = b, φ(b) = bb
11	24304	⟨a, b \| aa=b, baabbab=a⟩	φ(a) = b, φ(b) = bb
11	24308	⟨a, b \| aa=b, babaabb=a⟩	φ(a) = b, φ(b) = bb
11	24310	⟨a, b \| aa=b, bababab=a⟩	φ(a) = b, φ(b) = bb
11	24320	⟨a, b \| aa=b, bbaaabb=a⟩	φ(a) = b, φ(b) = bb

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