#1608 ⟨a, b | aaa=bb, bbb=a⟩

Properties

Presentation has sum-of-sides 9
Isomorphic to ℕ_{(9 = 2)}
Finite commutative monoid with 9 elements
Commutative Gröbner basis: ⟨a, b | b⁹=b², a=b³⟩
Not cancellative, because multiplication by b⁴ is not injective:
- b⁷ ≠ 1
- b⁴ ⋅ b⁷ = b⁴
- b⁴ ⋅ 1 = b⁴
- 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, b⁷

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

b⁹ ⇒ b²
a ⇒ b³

# ab:aaa=bb,bbb=a b/a
bbbbbbbbb=bb
a=bbb

Staircase diagram

Cayley table

Idempotents are shown in bold.

	1	b	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸
1	1	b	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸
b	b	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b²
b²	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b²	b³
b³	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b²	b³	b⁴
b⁴	b⁴	b⁵	b⁶	b⁷	b⁸	b²	b³	b⁴	b⁵
b⁵	b⁵	b⁶	b⁷	b⁸	b²	b³	b⁴	b⁵	b⁶
b⁶	b⁶	b⁷	b⁸	b²	b³	b⁴	b⁵	b⁶	b⁷
b⁷	b⁷	b⁸	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸
b⁸	b⁸	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b²

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

26 unique, 872 total

Σ	#	Presentation	Description	Related
7	118	⟨a, b \| aaa=b, bbb=1⟩	Isomorphic to ℤ₉	562 iso
8	568	⟨a, b \| aaa=a, abb=b⟩	Finite non-commutative monoid with 9 elements	5 iso
8	571	⟨a, b \| aaa=a, bbb=a⟩	Isomorphic to ℕ_{(9 = 3)}	48 iso
8	581	⟨a, b \| aaa=b, bbb=a⟩	Isomorphic to ℕ_{(9 = 1)}	62 iso
8	601	⟨a, b \| aba=b, bab=a⟩	Finite non-commutative monoid with 9 elements	3 iso
9	1584	⟨a, b \| aaa=aa, bbb=a⟩	Isomorphic to ℕ_{(9 = 6)}	16 iso
9	1595	⟨a, b \| aaa=ab, bab=b⟩	Finite non-commutative monoid with 9 elements	4 iso, 5 anti-iso
9	1598	⟨a, b \| aaa=ab, bbb=a⟩	Isomorphic to ℕ_{(9 = 4)}	40 iso
9	1991	⟨a, b \| aaa=a, abba=b⟩	Finite non-commutative monoid with 9 elements	3 iso
9	2064	⟨a, b \| aab=b, abba=a⟩	Finite non-commutative monoid with 9 elements	14 iso, 24 anti-iso
9	2255	⟨a, b \| ab=aa, bba=bb⟩	Finite non-commutative monoid with 9 elements
10	4125	⟨a, b \| aab=aaa, baa=b⟩	Finite non-commutative monoid with 9 elements	1 iso, 1 anti-iso
10	4130	⟨a, b \| aab=aaa, bbb=a⟩	Isomorphic to ℕ_{(9 = 7)}	1 iso
10	4156	⟨a, b \| abb=aaa, bbb=a⟩	Isomorphic to ℕ_{(9 = 5)}	12 iso
10	5035	⟨a, b \| aaa=aa, abbb=b⟩	Finite non-commutative monoid with 9 elements	3 iso
10	5079	⟨a, b \| aaa=bb, aaba=b⟩	Finite commutative monoid with 9 elements	8 iso
10	5124	⟨a, b \| aab=aa, bbbb=a⟩	Isomorphic to ℕ_{(9 = 8)}	5 iso
10	5339	⟨a, b \| aaa=bb, aab=ab⟩	Finite non-commutative monoid with 9 elements	1 iso
10	6824	⟨a, b \| aab=b, bbb=aaa⟩	Finite commutative monoid with 9 elements	5 iso
10	7139	⟨a, b \| bb=aa, aaaa=ab⟩	Finite non-commutative monoid with 9 elements	3 iso
10	9327	⟨a, b \| ab=a, baaa=bbb⟩	Finite non-commutative monoid with 9 elements	3 iso
10	9359	⟨a, b \| ab=a, bbaa=bbb⟩	Finite non-commutative monoid with 9 elements	2 iso
11	19506	⟨a, b \| aab=b, aaaaa=ba⟩	Finite non-commutative monoid with 9 elements
11	19507	⟨a, b \| aab=b, aaaaa=bb⟩	Finite commutative monoid with 9 elements
11	20723	⟨a, b \| ab=aa, aaaaa=bb⟩	Finite non-commutative monoid with 9 elements	15 iso
11	25656	⟨a, b \| ab=a, bbbbb=baa⟩	Finite non-commutative monoid with 9 elements

Other isomorphic instances

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

70 total

Σ	#	Presentation	Mapping
9	2923	⟨a, b \| aa=b, abbbb=b⟩	φ(a) = b, φ(b) = bb
9	2931	⟨a, b \| aa=b, babbb=b⟩	φ(a) = b, φ(b) = bb
9	2933	⟨a, b \| aa=b, bbabb=b⟩	φ(a) = b, φ(b) = bb
10	5076	⟨a, b \| aaa=bb, aaab=a⟩	φ(a) = bbb, φ(b) = b
10	5078	⟨a, b \| aaa=bb, aaba=a⟩	φ(a) = bbb, φ(b) = b
10	5220	⟨a, b \| aab=bb, bbbb=a⟩	φ(a) = bbbb, φ(b) = b
10	5292	⟨a, b \| aba=bb, bbbb=a⟩	φ(a) = bbbb, φ(b) = b
10	8655	⟨a, b \| aa=b, aaabbb=b⟩	φ(a) = b, φ(b) = bb
10	8661	⟨a, b \| aa=b, aababb=b⟩	φ(a) = b, φ(b) = bb
10	8665	⟨a, b \| aa=b, aabbab=b⟩	φ(a) = b, φ(b) = bb
10	8667	⟨a, b \| aa=b, aabbba=b⟩	φ(a) = b, φ(b) = bb
10	8675	⟨a, b \| aa=b, abaabb=b⟩	φ(a) = b, φ(b) = bb
10	8677	⟨a, b \| aa=b, ababab=b⟩	φ(a) = b, φ(b) = bb
10	8679	⟨a, b \| aa=b, ababba=b⟩	φ(a) = b, φ(b) = bb
10	8683	⟨a, b \| aa=b, abbaab=b⟩	φ(a) = b, φ(b) = bb
10	8695	⟨a, b \| aa=b, baaabb=b⟩	φ(a) = b, φ(b) = bb
10	8697	⟨a, b \| aa=b, baabab=b⟩	φ(a) = b, φ(b) = bb
10	8960	⟨a, b \| aa=b, abbbb=aa⟩	φ(a) = b, φ(b) = bb
10	8974	⟨a, b \| aa=b, babbb=aa⟩	φ(a) = b, φ(b) = bb
10	8978	⟨a, b \| aa=b, bbabb=aa⟩	φ(a) = b, φ(b) = bb
11	12300	⟨a, b \| aaab=bb, aabb=a⟩	φ(a) = bbbbb, φ(b) = b
11	12304	⟨a, b \| aaab=bb, abab=a⟩	φ(a) = bbbbb, φ(b) = b
11	12312	⟨a, b \| aaab=bb, baab=a⟩	φ(a) = bbbbb, φ(b) = b
11	12404	⟨a, b \| aaba=bb, aabb=a⟩	φ(a) = bbbbb, φ(b) = b
11	12408	⟨a, b \| aaba=bb, abab=a⟩	φ(a) = bbbbb, φ(b) = b
11	12410	⟨a, b \| aaba=bb, abba=a⟩	φ(a) = bbbbb, φ(b) = b
11	12416	⟨a, b \| aaba=bb, baab=a⟩	φ(a) = bbbbb, φ(b) = b
11	12418	⟨a, b \| aaba=bb, baba=a⟩	φ(a) = bbbbb, φ(b) = b
11	12422	⟨a, b \| aaba=bb, bbaa=a⟩	φ(a) = bbbbb, φ(b) = b
11	15751	⟨a, b \| aab=bb, aabbb=a⟩	φ(a) = bbbb, φ(b) = b
11	15759	⟨a, b \| aab=bb, ababb=a⟩	φ(a) = bbbb, φ(b) = b
11	15763	⟨a, b \| aab=bb, abbab=a⟩	φ(a) = bbbb, φ(b) = b
11	15775	⟨a, b \| aab=bb, baabb=a⟩	φ(a) = bbbb, φ(b) = b
11	15779	⟨a, b \| aab=bb, babab=a⟩	φ(a) = bbbb, φ(b) = b
11	15787	⟨a, b \| aab=bb, bbaab=a⟩	φ(a) = bbbb, φ(b) = b
11	15919	⟨a, b \| aba=bb, aabbb=a⟩	φ(a) = bbbb, φ(b) = b
11	15925	⟨a, b \| aba=bb, ababb=a⟩	φ(a) = bbbb, φ(b) = b
11	15927	⟨a, b \| aba=bb, abbab=a⟩	φ(a) = bbbb, φ(b) = b
11	15929	⟨a, b \| aba=bb, abbba=a⟩	φ(a) = bbbb, φ(b) = b
11	15935	⟨a, b \| aba=bb, baabb=a⟩	φ(a) = bbbb, φ(b) = b
11	15937	⟨a, b \| aba=bb, babab=a⟩	φ(a) = bbbb, φ(b) = b
11	19315	⟨a, b \| aaa=b, aaabb=aa⟩	φ(a) = b, φ(b) = bbb
11	19322	⟨a, b \| aaa=b, aabab=aa⟩	φ(a) = b, φ(b) = bbb
11	19326	⟨a, b \| aaa=b, aabba=aa⟩	φ(a) = b, φ(b) = bbb
11	19334	⟨a, b \| aaa=b, abaab=aa⟩	φ(a) = b, φ(b) = bbb
11	19338	⟨a, b \| aaa=b, ababa=aa⟩	φ(a) = b, φ(b) = bbb
11	19356	⟨a, b \| aaa=b, baaab=aa⟩	φ(a) = b, φ(b) = bbb
11	19379	⟨a, b \| aab=a, aaaaa=bb⟩	φ(a) = bbbbbb, φ(b) = b
11	24195	⟨a, b \| aa=b, aaaaabb=b⟩	φ(a) = b, φ(b) = bb
11	24199	⟨a, b \| aa=b, aaaabab=b⟩	φ(a) = b, φ(b) = bb
11	24201	⟨a, b \| aa=b, aaaabba=b⟩	φ(a) = b, φ(b) = bb
11	24207	⟨a, b \| aa=b, aaabaab=b⟩	φ(a) = b, φ(b) = bb
11	24209	⟨a, b \| aa=b, aaababa=b⟩	φ(a) = b, φ(b) = bb
11	24213	⟨a, b \| aa=b, aaabbaa=b⟩	φ(a) = b, φ(b) = bb
11	24221	⟨a, b \| aa=b, aabaaab=b⟩	φ(a) = b, φ(b) = bb
11	24223	⟨a, b \| aa=b, aabaaba=b⟩	φ(a) = b, φ(b) = bb
11	24227	⟨a, b \| aa=b, aababaa=b⟩	φ(a) = b, φ(b) = bb
11	24249	⟨a, b \| aa=b, abaaaab=b⟩	φ(a) = b, φ(b) = bb
11	24251	⟨a, b \| aa=b, abaaaba=b⟩	φ(a) = b, φ(b) = bb
11	24293	⟨a, b \| aa=b, baaaaab=b⟩	φ(a) = b, φ(b) = bb
11	24751	⟨a, b \| aa=b, aaabbb=aa⟩	φ(a) = b, φ(b) = bb
11	24763	⟨a, b \| aa=b, aababb=aa⟩	φ(a) = b, φ(b) = bb
11	24770	⟨a, b \| aa=b, aabbab=aa⟩	φ(a) = b, φ(b) = bb
11	24774	⟨a, b \| aa=b, aabbba=aa⟩	φ(a) = b, φ(b) = bb
11	24789	⟨a, b \| aa=b, abaabb=aa⟩	φ(a) = b, φ(b) = bb
11	24793	⟨a, b \| aa=b, ababab=aa⟩	φ(a) = b, φ(b) = bb
11	24797	⟨a, b \| aa=b, ababba=aa⟩	φ(a) = b, φ(b) = bb
11	24805	⟨a, b \| aa=b, abbaab=aa⟩	φ(a) = b, φ(b) = bb
11	24827	⟨a, b \| aa=b, baaabb=aa⟩	φ(a) = b, φ(b) = bb
11	24831	⟨a, b \| aa=b, baabab=aa⟩	φ(a) = b, φ(b) = bb

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