#1632 ⟨a, b | aab=ab, bbb=a⟩

Properties

Presentation has sum-of-sides 9
Isomorphic to ℕ_{(7 = 4)}
Finite commutative monoid with 7 elements
Commutative Gröbner basis: ⟨a, b | b⁷=b⁴, a=b³⟩
Not cancellative, because multiplication by b⁴ is not injective:
- b³ ≠ 1
- b⁴ ⋅ b³ = b⁴
- b⁴ ⋅ 1 = b⁴
- Cancellative quotient is isomorphic to ℤ₃
Grothendieck group is isomorphic to ℤ₃
Group of units is isomorphic to ℤ₁
2 Archimedian components:
- 1, b

Element profile

2 idempotent elements:
- 1, b⁶

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(b) = 0; deg(a) = 1

b⁷ ⇒ b⁴
a ⇒ b³

# ab:aab=ab,bbb=a b/a
bbbbbbb=bbbb
a=bbb

Staircase diagram

Cayley table

Idempotents are shown in bold.

	1	b	b²	b³	b⁴	b⁵	b⁶
1	1	b	b²	b³	b⁴	b⁵	b⁶
b	b	b²	b³	b⁴	b⁵	b⁶	b⁴
b²	b²	b³	b⁴	b⁵	b⁶	b⁴	b⁵
b³	b³	b⁴	b⁵	b⁶	b⁴	b⁵	b⁶
b⁴	b⁴	b⁵	b⁶	b⁴	b⁵	b⁶	b⁴
b⁵	b⁵	b⁶	b⁴	b⁵	b⁶	b⁴	b⁵
b⁶	b⁶	b⁴	b⁵	b⁶	b⁴	b⁵	b⁶

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

22 unique, 1173 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	913	⟨a, b \| aa=b, abbb=b⟩	Isomorphic to ℕ_{(7 = 2)}	82 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	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.

50 total

Σ	#	Presentation	Mapping
9	1644	⟨a, b \| aab=ba, bbb=a⟩	φ(a) = bbb, φ(b) = b
9	1674	⟨a, b \| aba=ab, bbb=a⟩	φ(a) = bbb, φ(b) = b
9	3061	⟨a, b \| aa=b, abbb=bb⟩	φ(a) = b, φ(b) = bb
9	3068	⟨a, b \| aa=b, babb=bb⟩	φ(a) = b, φ(b) = bb
10	4633	⟨a, b \| aaaa=b, aaab=b⟩	φ(a) = b, φ(b) = bbbb
10	4635	⟨a, b \| aaaa=b, aaba=b⟩	φ(a) = b, φ(b) = bbbb
10	8926	⟨a, b \| aa=b, aaabb=bb⟩	φ(a) = b, φ(b) = bb
10	8933	⟨a, b \| aa=b, aabab=bb⟩	φ(a) = b, φ(b) = bb
10	8937	⟨a, b \| aa=b, aabba=bb⟩	φ(a) = b, φ(b) = bb
10	8945	⟨a, b \| aa=b, abaab=bb⟩	φ(a) = b, φ(b) = bb
10	8948	⟨a, b \| aa=b, ababa=bb⟩	φ(a) = b, φ(b) = bb
10	8966	⟨a, b \| aa=b, baaab=bb⟩	φ(a) = b, φ(b) = bb
10	9229	⟨a, b \| aa=b, abbb=aab⟩	φ(a) = b, φ(b) = bb
10	9230	⟨a, b \| aa=b, abbb=aba⟩	φ(a) = b, φ(b) = bb
10	9232	⟨a, b \| aa=b, abbb=baa⟩	φ(a) = b, φ(b) = bb
10	9243	⟨a, b \| aa=b, babb=aab⟩	φ(a) = b, φ(b) = bb
10	9244	⟨a, b \| aa=b, babb=aba⟩	φ(a) = b, φ(b) = bb
10	9246	⟨a, b \| aa=b, babb=baa⟩	φ(a) = b, φ(b) = bb
11	12167	⟨a, b \| aaaa=ab, aaab=b⟩	φ(a) = b, φ(b) = bbbbbb
11	12169	⟨a, b \| aaaa=ab, aaba=b⟩	φ(a) = b, φ(b) = bbbbbb
11	12173	⟨a, b \| aaaa=ab, abaa=b⟩	φ(a) = b, φ(b) = bbbbbb
11	19308	⟨a, b \| aaa=b, aaaab=ab⟩	φ(a) = b, φ(b) = bbb
11	19309	⟨a, b \| aaa=b, aaaab=ba⟩	φ(a) = b, φ(b) = bbb
11	19312	⟨a, b \| aaa=b, aaaba=ab⟩	φ(a) = b, φ(b) = bbb
11	19313	⟨a, b \| aaa=b, aaaba=ba⟩	φ(a) = b, φ(b) = bbb
11	19320	⟨a, b \| aaa=b, aabaa=ab⟩	φ(a) = b, φ(b) = bbb
11	19851	⟨a, b \| aaa=b, aaaa=abb⟩	φ(a) = b, φ(b) = bbb
11	19852	⟨a, b \| aaa=b, aaaa=bab⟩	φ(a) = b, φ(b) = bbb
11	20043	⟨a, b \| aab=a, bbbb=abb⟩	φ(a) = bbbbb, φ(b) = b
11	24730	⟨a, b \| aa=b, aaaaab=bb⟩	φ(a) = b, φ(b) = bb
11	24734	⟨a, b \| aa=b, aaaaba=bb⟩	φ(a) = b, φ(b) = bb
11	24742	⟨a, b \| aa=b, aaabaa=bb⟩	φ(a) = b, φ(b) = bb
11	25283	⟨a, b \| aa=b, aaabb=aab⟩	φ(a) = b, φ(b) = bb
11	25284	⟨a, b \| aa=b, aaabb=aba⟩	φ(a) = b, φ(b) = bb
11	25286	⟨a, b \| aa=b, aaabb=baa⟩	φ(a) = b, φ(b) = bb
11	25297	⟨a, b \| aa=b, aabab=aab⟩	φ(a) = b, φ(b) = bb
11	25298	⟨a, b \| aa=b, aabab=aba⟩	φ(a) = b, φ(b) = bb
11	25300	⟨a, b \| aa=b, aabab=baa⟩	φ(a) = b, φ(b) = bb
11	25305	⟨a, b \| aa=b, aabba=aab⟩	φ(a) = b, φ(b) = bb
11	25306	⟨a, b \| aa=b, aabba=aba⟩	φ(a) = b, φ(b) = bb
11	25308	⟨a, b \| aa=b, aabba=baa⟩	φ(a) = b, φ(b) = bb
11	25321	⟨a, b \| aa=b, abaab=aab⟩	φ(a) = b, φ(b) = bb
11	25322	⟨a, b \| aa=b, abaab=aba⟩	φ(a) = b, φ(b) = bb
11	25324	⟨a, b \| aa=b, abaab=baa⟩	φ(a) = b, φ(b) = bb
11	25329	⟨a, b \| aa=b, ababa=aab⟩	φ(a) = b, φ(b) = bb
11	25330	⟨a, b \| aa=b, ababa=aba⟩	φ(a) = b, φ(b) = bb
11	25365	⟨a, b \| aa=b, baaab=aab⟩	φ(a) = b, φ(b) = bb
11	25366	⟨a, b \| aa=b, baaab=aba⟩	φ(a) = b, φ(b) = bb
11	25745	⟨a, b \| aa=b, abbb=aaaa⟩	φ(a) = b, φ(b) = bb
11	25766	⟨a, b \| aa=b, babb=aaaa⟩	φ(a) = b, φ(b) = bb

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