#258 ⟨a, b | aa=b, bbb=b⟩

Properties

Presentation has sum-of-sides 7
Isomorphic to ℕ_{(6 = 2)}
Finite commutative monoid with 6 elements
Commutative Gröbner basis: ⟨a, b | a⁶=a², b=a²⟩
Not cancellative, because multiplication by a³ is not injective:
- a⁴ ≠ 1
- a³ ⋅ a⁴ = a³
- a³ ⋅ 1 = a³
- Cancellative quotient is isomorphic to ℤ₄
Grothendieck group is isomorphic to ℤ₄
Group of units is isomorphic to ℤ₁
2 Archimedian components:
- 1, b

Element profile

2 idempotent elements:
- 1, a⁴

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

a⁶ ⇒ a²
b ⇒ a²

# ab:aa=b,bbb=b a/b
aaaaaa=aa
b=aa

Staircase diagram

Cayley table

Idempotents are shown in bold.

	1	a	a²	a³	a⁴	a⁵
1	1	a	a²	a³	a⁴	a⁵
a	a	a²	a³	a⁴	a⁵	a²
a²	a²	a³	a⁴	a⁵	a²	a³
a³	a³	a⁴	a⁵	a²	a³	a⁴
a⁴	a⁴	a⁵	a²	a³	a⁴	a⁵
a⁵	a⁵	a²	a³	a⁴	a⁵	a²

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

19 unique, 1870 total

Σ	#	Presentation	Description	Related
6	22	⟨a, b \| ab=aa, bb=1⟩	Finite non-commutative monoid with 6 elements	44 iso, 63 anti-iso
6	26	⟨a, b \| aaa=1, abb=1⟩	Isomorphic to ℤ₆	1373 iso
7	160	⟨a, b \| ab=aa, bb=b⟩	Finite non-commutative monoid with 6 elements	4 anti-iso
7	245	⟨a, b \| aa=a, bbb=a⟩	Isomorphic to ℕ_{(6 = 3)}	49 iso
7	257	⟨a, b \| aa=b, bbb=a⟩	Isomorphic to ℕ_{(6 = 1)}	61 iso
8	639	⟨a, b \| ab=aa, baa=b⟩	Finite non-commutative monoid with 6 elements	5 iso, 8 anti-iso
8	644	⟨a, b \| ab=aa, bbb=a⟩	Isomorphic to ℕ_{(6 = 4)}	46 iso
8	893	⟨a, b \| aa=a, abbb=b⟩	Finite non-commutative monoid with 6 elements	19 iso
8	1011	⟨a, b \| ab=a, baa=bb⟩	Finite non-commutative monoid with 6 elements	12 iso, 1 anti-iso
9	1427	⟨a, b \| abba=b, baba=1⟩	Finite non-Abelian group with 6 elements	66 iso
9	1581	⟨a, b \| aaa=aa, abb=b⟩	Finite non-commutative monoid with 6 elements	5 iso
9	1648	⟨a, b \| aab=bb, abb=a⟩	Finite commutative monoid with 6 elements	13 iso
9	1686	⟨a, b \| abb=aa, bbb=a⟩	Isomorphic to ℕ_{(6 = 5)}	49 iso
9	3075	⟨a, b \| ab=a, aaaa=bb⟩	Finite commutative monoid with 6 elements	14 iso
9	3132	⟨a, b \| ab=a, bbbb=aa⟩	Finite commutative monoid with 6 elements	5 iso
9	3134	⟨a, b \| ab=a, bbbb=ba⟩	Finite non-commutative monoid with 6 elements	2 iso
9	3193	⟨a, b \| ab=a, bbb=aaa⟩	Finite commutative monoid with 6 elements	9 iso
9	3199	⟨a, b \| ab=a, bbb=bba⟩	Finite non-commutative monoid with 6 elements	3 iso
11	24145	⟨a, b \| aa=a, abbbbba=b⟩	Finite commutative monoid with 6 elements

Other isomorphic instances

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

55 total

Σ	#	Presentation	Mapping
8	653	⟨a, b \| bb=aa, aaa=b⟩	φ(a) = a, φ(b) = aaa
8	654	⟨a, b \| bb=aa, aab=a⟩	φ(a) = aaa, φ(b) = a
8	907	⟨a, b \| aa=b, aabb=b⟩	φ(a) = a, φ(b) = aa
8	909	⟨a, b \| aa=b, abab=b⟩	φ(a) = a, φ(b) = aa
8	911	⟨a, b \| aa=b, abba=b⟩	φ(a) = a, φ(b) = aa
8	915	⟨a, b \| aa=b, baab=b⟩	φ(a) = a, φ(b) = aa
8	989	⟨a, b \| aa=b, bbb=aa⟩	φ(a) = a, φ(b) = aa
9	2899	⟨a, b \| aa=b, aaaab=b⟩	φ(a) = a, φ(b) = aa
9	2901	⟨a, b \| aa=b, aaaba=b⟩	φ(a) = a, φ(b) = aa
9	2905	⟨a, b \| aa=b, aabaa=b⟩	φ(a) = a, φ(b) = aa
9	3047	⟨a, b \| aa=b, aabb=aa⟩	φ(a) = a, φ(b) = aa
9	3051	⟨a, b \| aa=b, abab=aa⟩	φ(a) = a, φ(b) = aa
9	3055	⟨a, b \| aa=b, abba=aa⟩	φ(a) = a, φ(b) = aa
9	3062	⟨a, b \| aa=b, baab=aa⟩	φ(a) = a, φ(b) = aa
10	5095	⟨a, b \| aab=aa, aaaa=b⟩	φ(a) = a, φ(b) = aaaa
10	5223	⟨a, b \| aba=aa, aaaa=b⟩	φ(a) = a, φ(b) = aaaa
10	6565	⟨a, b \| aaa=b, aaab=aa⟩	φ(a) = a, φ(b) = aaa
10	6569	⟨a, b \| aaa=b, aaba=aa⟩	φ(a) = a, φ(b) = aaa
10	6605	⟨a, b \| aab=a, aaab=bb⟩	φ(a) = aaa, φ(b) = a
10	6609	⟨a, b \| aab=a, aaba=bb⟩	φ(a) = aaa, φ(b) = a
10	6617	⟨a, b \| aab=a, abaa=bb⟩	φ(a) = aaa, φ(b) = a
10	6951	⟨a, b \| ab=aa, aaaaa=b⟩	φ(a) = a, φ(b) = aaaaa
10	6953	⟨a, b \| ab=aa, aaaab=b⟩	φ(a) = a, φ(b) = aaaaa
10	6955	⟨a, b \| ab=aa, aaaba=b⟩	φ(a) = a, φ(b) = aaaaa
10	6957	⟨a, b \| ab=aa, aaabb=b⟩	φ(a) = a, φ(b) = aaaaa
10	6959	⟨a, b \| ab=aa, aabaa=b⟩	φ(a) = a, φ(b) = aaaaa
10	6961	⟨a, b \| ab=aa, aabab=b⟩	φ(a) = a, φ(b) = aaaaa
10	6963	⟨a, b \| ab=aa, aabba=b⟩	φ(a) = a, φ(b) = aaaaa
10	6965	⟨a, b \| ab=aa, aabbb=b⟩	φ(a) = a, φ(b) = aaaaa
10	6967	⟨a, b \| ab=aa, abaaa=b⟩	φ(a) = a, φ(b) = aaaaa
10	6969	⟨a, b \| ab=aa, abaab=b⟩	φ(a) = a, φ(b) = aaaaa
10	6971	⟨a, b \| ab=aa, ababa=b⟩	φ(a) = a, φ(b) = aaaaa
10	6973	⟨a, b \| ab=aa, ababb=b⟩	φ(a) = a, φ(b) = aaaaa
10	6975	⟨a, b \| ab=aa, abbaa=b⟩	φ(a) = a, φ(b) = aaaaa
10	6977	⟨a, b \| ab=aa, abbab=b⟩	φ(a) = a, φ(b) = aaaaa
10	6979	⟨a, b \| ab=aa, abbba=b⟩	φ(a) = a, φ(b) = aaaaa
10	6981	⟨a, b \| ab=aa, abbbb=b⟩	φ(a) = a, φ(b) = aaaaa
10	8641	⟨a, b \| aa=b, aaaaaa=b⟩	φ(a) = a, φ(b) = aa
10	8915	⟨a, b \| aa=b, aaaab=aa⟩	φ(a) = a, φ(b) = aa
10	8919	⟨a, b \| aa=b, aaaba=aa⟩	φ(a) = a, φ(b) = aa
10	8927	⟨a, b \| aa=b, aabaa=aa⟩	φ(a) = a, φ(b) = aa
11	12433	⟨a, b \| aabb=aa, abab=b⟩	φ(a) = a, φ(b) = aa
11	12435	⟨a, b \| aabb=aa, abba=b⟩	φ(a) = a, φ(b) = aa
11	12596	⟨a, b \| abba=bb, baab=a⟩	φ(a) = aa, φ(b) = a
11	12598	⟨a, b \| abba=bb, baba=a⟩	φ(a) = aa, φ(b) = a
11	12602	⟨a, b \| abba=bb, bbaa=a⟩	φ(a) = aa, φ(b) = a
11	12629	⟨a, b \| baba=aa, bbaa=b⟩	φ(a) = a, φ(b) = aa
11	15548	⟨a, b \| aab=aa, aaaab=b⟩	φ(a) = a, φ(b) = aaaa
11	15550	⟨a, b \| aab=aa, aaaba=b⟩	φ(a) = a, φ(b) = aaaa
11	15554	⟨a, b \| aab=aa, aabaa=b⟩	φ(a) = a, φ(b) = aaaa
11	15562	⟨a, b \| aab=aa, abaaa=b⟩	φ(a) = a, φ(b) = aaaa
11	15806	⟨a, b \| aba=aa, aaaba=b⟩	φ(a) = a, φ(b) = aaaa
11	15810	⟨a, b \| aba=aa, aabaa=b⟩	φ(a) = a, φ(b) = aaaa
11	19234	⟨a, b \| aaa=a, aaaaa=bb⟩	φ(a) = aa, φ(b) = a
11	24724	⟨a, b \| aa=b, aaaaaa=aa⟩	φ(a) = a, φ(b) = aa

Up:	Monoids with two generators and two relations
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Next:	#259 ⟨a, b \| ab=a, aaa=a⟩