#918 ⟨a, b | aa=b, bbbb=a⟩

Properties

Presentation has sum-of-sides 8
Isomorphic to ℕ_{(8 = 1)}
Finite commutative monoid with 8 elements
Commutative Gröbner basis: ⟨a, b | b⁸=b, a=b⁴⟩
Not cancellative, because multiplication by b⁵ is not injective:
- b⁷ ≠ 1
- b⁵ ⋅ b⁷ = b⁵
- b⁵ ⋅ 1 = b⁵
- 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

a⁸ ⇒ a
b ⇒ a²

# ab:aa=b,bbbb=a a/b
aaaaaaaa=a
b=aa

Staircase diagram

Cayley table

Idempotents are shown in bold.

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

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

33 unique, 1177 total

Σ	#	Presentation	Description	Related
7	188	⟨a, b \| aab=1, bbbb=1⟩	Isomorphic to ℤ₈	727 iso
7	330	⟨a, b \| aa=1, abba=b⟩	Finite non-commutative monoid with 8 elements	60 iso
8	556	⟨a, b \| aba=b, aabb=1⟩	Finite non-Abelian group with 8 elements	28 iso
8	898	⟨a, b \| aa=a, bbbb=a⟩	Isomorphic to ℕ_{(8 = 4)}	48 iso
8	919	⟨a, b \| aa=b, bbbb=b⟩	Isomorphic to ℕ_{(8 = 2)}	46 iso
8	961	⟨a, b \| aa=a, aba=bb⟩	Finite non-commutative monoid with 8 elements	5 iso
9	1605	⟨a, b \| aaa=bb, abb=b⟩	Isomorphic to ℕ_{(8 = 3)}	55 iso
9	1606	⟨a, b \| aaa=bb, bab=a⟩	Finite commutative monoid with 8 elements	14 iso
9	1615	⟨a, b \| aab=aa, baa=b⟩	Finite non-commutative monoid with 8 elements	9 iso, 5 anti-iso
9	1650	⟨a, b \| aab=bb, baa=a⟩	Finite non-commutative monoid with 8 elements	4 iso, 11 anti-iso
9	2206	⟨a, b \| ab=aa, bbbb=a⟩	Isomorphic to ℕ_{(8 = 5)}	32 iso
9	2220	⟨a, b \| bb=aa, aaab=a⟩	Finite commutative monoid with 8 elements	22 iso
9	2247	⟨a, b \| ab=aa, baa=bb⟩	Finite non-commutative monoid with 8 elements	1 iso
9	2256	⟨a, b \| ab=aa, bbb=aa⟩	Finite non-commutative monoid with 8 elements	1 iso
9	2258	⟨a, b \| ab=aa, bbb=ba⟩	Finite non-commutative monoid with 8 elements
9	2883	⟨a, b \| aa=a, abbbb=b⟩	Finite non-commutative monoid with 8 elements	14 iso
9	3107	⟨a, b \| ab=a, baaa=bb⟩	Finite non-commutative monoid with 8 elements	9 iso
9	3123	⟨a, b \| ab=a, bbaa=bb⟩	Finite non-commutative monoid with 8 elements	10 iso
10	5191	⟨a, b \| aab=bb, aaaa=b⟩	Isomorphic to ℕ_{(8 = 6)}	18 iso
10	6664	⟨a, b \| aab=b, aaaa=ba⟩	Finite non-commutative monoid with 8 elements	1 iso
10	7057	⟨a, b \| ab=aa, aaaa=bb⟩	Finite non-commutative monoid with 8 elements	7 iso
10	9380	⟨a, b \| ab=a, bbbb=baa⟩	Finite non-commutative monoid with 8 elements	2 iso
11	12606	⟨a, b \| abbb=aa, bbbb=a⟩	Isomorphic to ℕ_{(8 = 7)}	14 iso
11	20047	⟨a, b \| aab=a, bbbb=bbb⟩	Finite non-commutative monoid with 8 elements
11	20051	⟨a, b \| aab=b, aaaa=abb⟩	Finite commutative monoid with 8 elements	1 iso
11	24863	⟨a, b \| ab=a, aaaaaa=bb⟩	Finite commutative monoid with 8 elements
11	25112	⟨a, b \| ab=a, bbbbbb=aa⟩	Finite commutative monoid with 8 elements
11	25114	⟨a, b \| ab=a, bbbbbb=ba⟩	Finite non-commutative monoid with 8 elements
11	25411	⟨a, b \| ab=a, aaaaa=bbb⟩	Finite commutative monoid with 8 elements
11	25652	⟨a, b \| ab=a, bbbbb=aaa⟩	Finite commutative monoid with 8 elements
11	25658	⟨a, b \| ab=a, bbbbb=bba⟩	Finite non-commutative monoid with 8 elements
11	25897	⟨a, b \| ab=a, bbbb=aaaa⟩	Finite commutative monoid with 8 elements
11	25911	⟨a, b \| ab=a, bbbb=bbba⟩	Finite non-commutative monoid with 8 elements

Other isomorphic instances

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

34 total

Σ	#	Presentation	Mapping
9	2006	⟨a, b \| aaa=b, aabb=a⟩	φ(a) = bbbbb, φ(b) = b
9	2008	⟨a, b \| aaa=b, abab=a⟩	φ(a) = bbbbb, φ(b) = b
9	2010	⟨a, b \| aaa=b, abba=a⟩	φ(a) = bbbbb, φ(b) = b
9	2014	⟨a, b \| aaa=b, baab=a⟩	φ(a) = bbbbb, φ(b) = b
9	2910	⟨a, b \| aa=b, aabbb=a⟩	φ(a) = bbbb, φ(b) = b
9	2916	⟨a, b \| aa=b, ababb=a⟩	φ(a) = bbbb, φ(b) = b
9	2918	⟨a, b \| aa=b, abbab=a⟩	φ(a) = bbbb, φ(b) = b
9	2920	⟨a, b \| aa=b, abbba=a⟩	φ(a) = bbbb, φ(b) = b
9	2926	⟨a, b \| aa=b, baabb=a⟩	φ(a) = bbbb, φ(b) = b
9	2928	⟨a, b \| aa=b, babab=a⟩	φ(a) = bbbb, φ(b) = b
10	8646	⟨a, b \| aa=b, aaaabb=a⟩	φ(a) = bbbb, φ(b) = b
10	8650	⟨a, b \| aa=b, aaabab=a⟩	φ(a) = bbbb, φ(b) = b
10	8652	⟨a, b \| aa=b, aaabba=a⟩	φ(a) = bbbb, φ(b) = b
10	8656	⟨a, b \| aa=b, aabaab=a⟩	φ(a) = bbbb, φ(b) = b
10	8658	⟨a, b \| aa=b, aababa=a⟩	φ(a) = bbbb, φ(b) = b
10	8662	⟨a, b \| aa=b, aabbaa=a⟩	φ(a) = bbbb, φ(b) = b
10	8670	⟨a, b \| aa=b, abaaab=a⟩	φ(a) = bbbb, φ(b) = b
10	8672	⟨a, b \| aa=b, abaaba=a⟩	φ(a) = bbbb, φ(b) = b
10	8692	⟨a, b \| aa=b, baaaab=a⟩	φ(a) = bbbb, φ(b) = b
11	14371	⟨a, b \| aaaa=b, aaaab=a⟩	φ(a) = bb, φ(b) = b
11	14373	⟨a, b \| aaaa=b, aaaba=a⟩	φ(a) = bb, φ(b) = b
11	14377	⟨a, b \| aaaa=b, aabaa=a⟩	φ(a) = bb, φ(b) = b
11	14410	⟨a, b \| aaab=a, aaaaa=b⟩	φ(a) = bbb, φ(b) = b
11	14538	⟨a, b \| aaba=a, aaaaa=b⟩	φ(a) = bbb, φ(b) = b
11	18762	⟨a, b \| aaa=b, aaaaab=a⟩	φ(a) = bbbbb, φ(b) = b
11	18764	⟨a, b \| aaa=b, aaaaba=a⟩	φ(a) = bbbbb, φ(b) = b
11	18768	⟨a, b \| aaa=b, aaabaa=a⟩	φ(a) = bbbbb, φ(b) = b
11	18833	⟨a, b \| aab=a, aaaaaa=b⟩	φ(a) = bbbbbb, φ(b) = b
11	19089	⟨a, b \| aba=a, aaaaaa=b⟩	φ(a) = bbbbbb, φ(b) = b
11	24190	⟨a, b \| aa=b, aaaaaab=a⟩	φ(a) = bbbb, φ(b) = b
11	24192	⟨a, b \| aa=b, aaaaaba=a⟩	φ(a) = bbbb, φ(b) = b
11	24196	⟨a, b \| aa=b, aaaabaa=a⟩	φ(a) = bbbb, φ(b) = b
11	24204	⟨a, b \| aa=b, aaabaaa=a⟩	φ(a) = bbbb, φ(b) = b
11	24333	⟨a, b \| ab=a, aaaaaaa=b⟩	φ(a) = b, φ(b) = bbbbbbb

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