#913 ⟨a, b | aa=b, abbb=b⟩

Properties

Presentation has sum-of-sides 8
Isomorphic to ℕ_{(7 = 2)}
Finite commutative monoid with 7 elements
Commutative Gröbner basis: ⟨a, b | a⁷=a², b=a²⟩
Not cancellative, because multiplication by a³ is not injective:
- a⁶ ≠ a
- a³ ⋅ a⁶ = a⁴
- a³ ⋅ a = a⁴
- Cancellative quotient is isomorphic to ℤ₅
Grothendieck group is isomorphic to ℤ₅
Group of units is isomorphic to ℤ₁
2 Archimedian components:
- 1, b

Element profile

2 idempotent elements:
- 1, a⁵

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

a⁷ ⇒ a²
b ⇒ a²

# ab:aa=b,abbb=b a/b
aaaaaaa=aa
b=aa

Staircase diagram

Cayley table

Idempotents are shown in bold.

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

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

22 unique, 1141 total

Σ	#	Presentation	Description	Related
7	116	⟨a, b \| aaa=b, abb=1⟩	Isomorphic to ℤ₇	763 iso
8	577	⟨a, b \| aaa=b, abb=a⟩	Isomorphic to ℕ_{(7 = 1)}	59 iso
8	578	⟨a, b \| aaa=b, abb=b⟩	Isomorphic to ℕ_{(7 = 3)}	44 iso
8	937	⟨a, b \| ab=a, baaa=b⟩	Finite non-commutative monoid with 7 elements	54 iso, 12 anti-iso
9	1593	⟨a, b \| aaa=ab, baa=b⟩	Finite non-commutative monoid with 7 elements	1 iso, 2 anti-iso
9	1601	⟨a, b \| aaa=bb, aab=b⟩	Finite commutative monoid with 7 elements	5 iso
9	1620	⟨a, b \| aab=aa, bbb=a⟩	Isomorphic to ℕ_{(7 = 6)}	35 iso
9	1632	⟨a, b \| aab=ab, bbb=a⟩	Isomorphic to ℕ_{(7 = 4)}	50 iso
9	2074	⟨a, b \| aab=b, babb=a⟩	Finite commutative monoid with 7 elements	20 iso
9	2231	⟨a, b \| ab=aa, aaa=bb⟩	Finite non-commutative monoid with 7 elements	3 iso
9	2271	⟨a, b \| bb=aa, aaa=ab⟩	Finite non-commutative monoid with 7 elements	1 iso, 2 anti-iso
9	3197	⟨a, b \| ab=a, bbb=baa⟩	Finite non-commutative monoid with 7 elements	5 iso
10	4162	⟨a, b \| abb=aab, bbb=a⟩	Isomorphic to ℕ_{(7 = 5)}	42 iso
10	6251	⟨a, b \| aaa=a, aabba=b⟩	Finite non-commutative monoid with 7 elements	1 iso
10	8987	⟨a, b \| ab=a, aaaaa=bb⟩	Finite commutative monoid with 7 elements	5 iso
10	9108	⟨a, b \| ab=a, bbbbb=aa⟩	Finite commutative monoid with 7 elements	2 iso
10	9110	⟨a, b \| ab=a, bbbbb=ba⟩	Finite non-commutative monoid with 7 elements	1 iso
10	9263	⟨a, b \| ab=a, aaaa=bbb⟩	Finite commutative monoid with 7 elements	4 iso
10	9375	⟨a, b \| ab=a, bbba=bbb⟩	Finite non-commutative monoid with 7 elements	3 iso
10	9376	⟨a, b \| ab=a, bbbb=aaa⟩	Finite commutative monoid with 7 elements	3 iso
10	9382	⟨a, b \| ab=a, bbbb=bba⟩	Finite non-commutative monoid with 7 elements	1 iso
11	14843	⟨a, b \| abba=b, aabab=a⟩	Finite non-commutative monoid with 7 elements	1 iso

Other isomorphic instances

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

82 total

Σ	#	Presentation	Mapping
8	917	⟨a, b \| aa=b, babb=b⟩	φ(a) = a, φ(b) = aa
9	1600	⟨a, b \| aaa=bb, aab=a⟩	φ(a) = aaaa, φ(b) = a
9	1656	⟨a, b \| aab=bb, bbb=a⟩	φ(a) = aaa, φ(b) = a
9	1684	⟨a, b \| aba=bb, bbb=a⟩	φ(a) = aaa, φ(b) = a
9	2903	⟨a, b \| aa=b, aaabb=b⟩	φ(a) = a, φ(b) = aa
9	2907	⟨a, b \| aa=b, aabab=b⟩	φ(a) = a, φ(b) = aa
9	2909	⟨a, b \| aa=b, aabba=b⟩	φ(a) = a, φ(b) = aa
9	2913	⟨a, b \| aa=b, abaab=b⟩	φ(a) = a, φ(b) = aa
9	2915	⟨a, b \| aa=b, ababa=b⟩	φ(a) = a, φ(b) = aa
9	2925	⟨a, b \| aa=b, baaab=b⟩	φ(a) = a, φ(b) = aa
9	3058	⟨a, b \| aa=b, abbb=aa⟩	φ(a) = a, φ(b) = aa
9	3065	⟨a, b \| aa=b, babb=aa⟩	φ(a) = a, φ(b) = aa
10	5196	⟨a, b \| aab=bb, aabb=a⟩	φ(a) = aaa, φ(b) = a
10	5200	⟨a, b \| aab=bb, abab=a⟩	φ(a) = aaa, φ(b) = a
10	5208	⟨a, b \| aab=bb, baab=a⟩	φ(a) = aaa, φ(b) = a
10	5280	⟨a, b \| aba=bb, aabb=a⟩	φ(a) = aaa, φ(b) = a
10	5282	⟨a, b \| aba=bb, abab=a⟩	φ(a) = aaa, φ(b) = a
10	5284	⟨a, b \| aba=bb, abba=a⟩	φ(a) = aaa, φ(b) = a
10	5288	⟨a, b \| aba=bb, baab=a⟩	φ(a) = aaa, φ(b) = a
10	8643	⟨a, b \| aa=b, aaaaab=b⟩	φ(a) = a, φ(b) = aa
10	8645	⟨a, b \| aa=b, aaaaba=b⟩	φ(a) = a, φ(b) = aa
10	8649	⟨a, b \| aa=b, aaabaa=b⟩	φ(a) = a, φ(b) = aa
10	8923	⟨a, b \| aa=b, aaabb=aa⟩	φ(a) = a, φ(b) = aa
10	8930	⟨a, b \| aa=b, aabab=aa⟩	φ(a) = a, φ(b) = aa
10	8934	⟨a, b \| aa=b, aabba=aa⟩	φ(a) = a, φ(b) = aa
10	8942	⟨a, b \| aa=b, abaab=aa⟩	φ(a) = a, φ(b) = aa
10	8946	⟨a, b \| aa=b, ababa=aa⟩	φ(a) = a, φ(b) = aa
10	8964	⟨a, b \| aa=b, baaab=aa⟩	φ(a) = a, φ(b) = aa
11	12406	⟨a, b \| aaba=bb, abaa=a⟩	φ(a) = aa, φ(b) = a
11	12414	⟨a, b \| aaba=bb, baaa=a⟩	φ(a) = aa, φ(b) = a
11	12496	⟨a, b \| abaa=bb, baaa=a⟩	φ(a) = aa, φ(b) = a
11	12612	⟨a, b \| abbb=bb, bbbb=a⟩	φ(a) = aaaa, φ(b) = a
11	12638	⟨a, b \| babb=bb, bbbb=a⟩	φ(a) = aaaa, φ(b) = a
11	15546	⟨a, b \| aab=aa, aaaaa=b⟩	φ(a) = a, φ(b) = aaaaa
11	15739	⟨a, b \| aab=bb, aaaab=a⟩	φ(a) = aaa, φ(b) = a
11	15802	⟨a, b \| aba=aa, aaaaa=b⟩	φ(a) = a, φ(b) = aaaaa
11	15907	⟨a, b \| aba=bb, aaaab=a⟩	φ(a) = aaa, φ(b) = a
11	15909	⟨a, b \| aba=bb, aaaba=a⟩	φ(a) = aaa, φ(b) = a
11	15913	⟨a, b \| aba=bb, aabaa=a⟩	φ(a) = aaa, φ(b) = a
11	19307	⟨a, b \| aaa=b, aaaab=aa⟩	φ(a) = a, φ(b) = aaa
11	19311	⟨a, b \| aaa=b, aaaba=aa⟩	φ(a) = a, φ(b) = aaa
11	19319	⟨a, b \| aaa=b, aabaa=aa⟩	φ(a) = a, φ(b) = aaa
11	19383	⟨a, b \| aab=a, aaaab=bb⟩	φ(a) = aaaa, φ(b) = a
11	19387	⟨a, b \| aab=a, aaaba=bb⟩	φ(a) = aaaa, φ(b) = a
11	19395	⟨a, b \| aab=a, aabaa=bb⟩	φ(a) = aaaa, φ(b) = a
11	19411	⟨a, b \| aab=a, abaaa=bb⟩	φ(a) = aaaa, φ(b) = a
11	20521	⟨a, b \| ab=aa, aaaaaa=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20523	⟨a, b \| ab=aa, aaaaab=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20525	⟨a, b \| ab=aa, aaaaba=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20527	⟨a, b \| ab=aa, aaaabb=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20529	⟨a, b \| ab=aa, aaabaa=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20531	⟨a, b \| ab=aa, aaabab=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20533	⟨a, b \| ab=aa, aaabba=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20535	⟨a, b \| ab=aa, aaabbb=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20537	⟨a, b \| ab=aa, aabaaa=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20539	⟨a, b \| ab=aa, aabaab=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20541	⟨a, b \| ab=aa, aababa=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20543	⟨a, b \| ab=aa, aababb=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20545	⟨a, b \| ab=aa, aabbaa=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20547	⟨a, b \| ab=aa, aabbab=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20549	⟨a, b \| ab=aa, aabbba=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20551	⟨a, b \| ab=aa, aabbbb=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20553	⟨a, b \| ab=aa, abaaaa=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20555	⟨a, b \| ab=aa, abaaab=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20557	⟨a, b \| ab=aa, abaaba=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20559	⟨a, b \| ab=aa, abaabb=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20561	⟨a, b \| ab=aa, ababaa=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20563	⟨a, b \| ab=aa, ababab=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20565	⟨a, b \| ab=aa, ababba=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20567	⟨a, b \| ab=aa, ababbb=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20569	⟨a, b \| ab=aa, abbaaa=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20571	⟨a, b \| ab=aa, abbaab=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20573	⟨a, b \| ab=aa, abbaba=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20575	⟨a, b \| ab=aa, abbabb=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20577	⟨a, b \| ab=aa, abbbaa=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20579	⟨a, b \| ab=aa, abbbab=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20581	⟨a, b \| ab=aa, abbbba=b⟩	φ(a) = a, φ(b) = aaaaaa
11	20583	⟨a, b \| ab=aa, abbbbb=b⟩	φ(a) = a, φ(b) = aaaaaa
11	24189	⟨a, b \| aa=b, aaaaaaa=b⟩	φ(a) = a, φ(b) = aa
11	24727	⟨a, b \| aa=b, aaaaab=aa⟩	φ(a) = a, φ(b) = aa
11	24731	⟨a, b \| aa=b, aaaaba=aa⟩	φ(a) = a, φ(b) = aa
11	24739	⟨a, b \| aa=b, aaabaa=aa⟩	φ(a) = a, φ(b) = aa

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