#581 ⟨a, b | aaa=b, bbb=a⟩

Properties

Presentation has sum-of-sides 8
Isomorphic to ℕ_{(9 = 1)}
Finite commutative monoid with 9 elements
Commutative Gröbner basis: ⟨a, b | b⁹=b, a=b³⟩
Not cancellative, because multiplication by b⁴ is not injective:
- b⁸ ≠ 1
- b⁴ ⋅ b⁸ = b⁴
- b⁴ ⋅ 1 = b⁴
- Cancellative quotient is isomorphic to ℤ₈
Grothendieck group is isomorphic to ℤ₈
Group of units is isomorphic to ℤ₁
2 Archimedian components:
- 1, b

Element profile

2 idempotent elements:
- 1, a⁸

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

a⁹ ⇒ a
b ⇒ a³

# ab:aaa=b,bbb=a a/b
aaaaaaaaa=a
b=aaa

Staircase diagram

Cayley table

Idempotents are shown in bold.

	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸
1	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸
a	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a
a²	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a	a²
a³	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a	a²	a³
a⁴	a⁴	a⁵	a⁶	a⁷	a⁸	a	a²	a³	a⁴
a⁵	a⁵	a⁶	a⁷	a⁸	a	a²	a³	a⁴	a⁵
a⁶	a⁶	a⁷	a⁸	a	a²	a³	a⁴	a⁵	a⁶
a⁷	a⁷	a⁸	a	a²	a³	a⁴	a⁵	a⁶	a⁷
a⁸	a⁸	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

26 unique, 880 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	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	1608	⟨a, b \| aaa=bb, bbb=a⟩	Isomorphic to ℕ_{(9 = 2)}	70 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.

62 total

Σ	#	Presentation	Mapping
9	2082	⟨a, b \| aab=b, bbbb=a⟩	φ(a) = bbbb, φ(b) = b
9	2122	⟨a, b \| aba=b, bbbb=a⟩	φ(a) = bbbb, φ(b) = b
9	2922	⟨a, b \| aa=b, abbbb=a⟩	φ(a) = b, φ(b) = bb
9	2930	⟨a, b \| aa=b, babbb=a⟩	φ(a) = b, φ(b) = bb
9	2932	⟨a, b \| aa=b, bbabb=a⟩	φ(a) = b, φ(b) = bb
10	6284	⟨a, b \| aaa=b, aaabb=a⟩	φ(a) = bbb, φ(b) = b
10	6288	⟨a, b \| aaa=b, aabab=a⟩	φ(a) = bbb, φ(b) = b
10	6290	⟨a, b \| aaa=b, aabba=a⟩	φ(a) = bbb, φ(b) = b
10	6294	⟨a, b \| aaa=b, abaab=a⟩	φ(a) = bbb, φ(b) = b
10	6296	⟨a, b \| aaa=b, ababa=a⟩	φ(a) = bbb, φ(b) = b
10	6306	⟨a, b \| aaa=b, baaab=a⟩	φ(a) = bbb, φ(b) = b
10	8654	⟨a, b \| aa=b, aaabbb=a⟩	φ(a) = b, φ(b) = bb
10	8660	⟨a, b \| aa=b, aababb=a⟩	φ(a) = b, φ(b) = bb
10	8664	⟨a, b \| aa=b, aabbab=a⟩	φ(a) = b, φ(b) = bb
10	8666	⟨a, b \| aa=b, aabbba=a⟩	φ(a) = b, φ(b) = bb
10	8674	⟨a, b \| aa=b, abaabb=a⟩	φ(a) = b, φ(b) = bb
10	8676	⟨a, b \| aa=b, ababab=a⟩	φ(a) = b, φ(b) = bb
10	8678	⟨a, b \| aa=b, ababba=a⟩	φ(a) = b, φ(b) = bb
10	8682	⟨a, b \| aa=b, abbaab=a⟩	φ(a) = b, φ(b) = bb
10	8694	⟨a, b \| aa=b, baaabb=a⟩	φ(a) = b, φ(b) = bb
10	8696	⟨a, b \| aa=b, baabab=a⟩	φ(a) = b, φ(b) = bb
11	14668	⟨a, b \| aabb=a, aaaab=b⟩	φ(a) = bbbbbb, φ(b) = b
11	14670	⟨a, b \| aabb=a, aaaba=b⟩	φ(a) = bbbbbb, φ(b) = b
11	14674	⟨a, b \| aabb=a, aabaa=b⟩	φ(a) = bbbbbb, φ(b) = b
11	14682	⟨a, b \| aabb=a, abaaa=b⟩	φ(a) = bbbbbb, φ(b) = b
11	14732	⟨a, b \| abab=a, aaaab=b⟩	φ(a) = bbbbbb, φ(b) = b
11	14734	⟨a, b \| abab=a, aaaba=b⟩	φ(a) = bbbbbb, φ(b) = b
11	14738	⟨a, b \| abab=a, aabaa=b⟩	φ(a) = bbbbbb, φ(b) = b
11	14746	⟨a, b \| abab=a, abaaa=b⟩	φ(a) = bbbbbb, φ(b) = b
11	14798	⟨a, b \| abba=a, aaaba=b⟩	φ(a) = bbbbbb, φ(b) = b
11	14802	⟨a, b \| abba=a, aabaa=b⟩	φ(a) = bbbbbb, φ(b) = b
11	14859	⟨a, b \| abba=b, abbbb=a⟩	φ(a) = b, φ(b) = bbbbbb
11	14867	⟨a, b \| abba=b, babbb=a⟩	φ(a) = b, φ(b) = bbbbbb
11	18990	⟨a, b \| aab=b, aabbbb=a⟩	φ(a) = bbbb, φ(b) = b
11	19006	⟨a, b \| aab=b, ababbb=a⟩	φ(a) = bbbb, φ(b) = b
11	19014	⟨a, b \| aab=b, abbabb=a⟩	φ(a) = bbbb, φ(b) = b
11	19018	⟨a, b \| aab=b, abbbab=a⟩	φ(a) = bbbb, φ(b) = b
11	19038	⟨a, b \| aab=b, baabbb=a⟩	φ(a) = bbbb, φ(b) = b
11	19046	⟨a, b \| aab=b, bababb=a⟩	φ(a) = bbbb, φ(b) = b
11	19050	⟨a, b \| aab=b, babbab=a⟩	φ(a) = bbbb, φ(b) = b
11	19062	⟨a, b \| aab=b, bbaabb=a⟩	φ(a) = bbbb, φ(b) = b
11	19066	⟨a, b \| aab=b, bbabab=a⟩	φ(a) = bbbb, φ(b) = b
11	19074	⟨a, b \| aab=b, bbbaab=a⟩	φ(a) = bbbb, φ(b) = b
11	19188	⟨a, b \| aba=b, aabbbb=a⟩	φ(a) = bbbb, φ(b) = b
11	19200	⟨a, b \| aba=b, ababbb=a⟩	φ(a) = bbbb, φ(b) = b
11	19204	⟨a, b \| aba=b, abbabb=a⟩	φ(a) = bbbb, φ(b) = b
11	19206	⟨a, b \| aba=b, abbbab=a⟩	φ(a) = bbbb, φ(b) = b
11	19208	⟨a, b \| aba=b, abbbba=a⟩	φ(a) = bbbb, φ(b) = b
11	19220	⟨a, b \| aba=b, bababb=a⟩	φ(a) = bbbb, φ(b) = b
11	19226	⟨a, b \| aba=b, bbaabb=a⟩	φ(a) = bbbb, φ(b) = b
11	24194	⟨a, b \| aa=b, aaaaabb=a⟩	φ(a) = b, φ(b) = bb
11	24198	⟨a, b \| aa=b, aaaabab=a⟩	φ(a) = b, φ(b) = bb
11	24200	⟨a, b \| aa=b, aaaabba=a⟩	φ(a) = b, φ(b) = bb
11	24206	⟨a, b \| aa=b, aaabaab=a⟩	φ(a) = b, φ(b) = bb
11	24208	⟨a, b \| aa=b, aaababa=a⟩	φ(a) = b, φ(b) = bb
11	24212	⟨a, b \| aa=b, aaabbaa=a⟩	φ(a) = b, φ(b) = bb
11	24220	⟨a, b \| aa=b, aabaaab=a⟩	φ(a) = b, φ(b) = bb
11	24222	⟨a, b \| aa=b, aabaaba=a⟩	φ(a) = b, φ(b) = bb
11	24226	⟨a, b \| aa=b, aababaa=a⟩	φ(a) = b, φ(b) = bb
11	24248	⟨a, b \| aa=b, abaaaab=a⟩	φ(a) = b, φ(b) = bb
11	24250	⟨a, b \| aa=b, abaaaba=a⟩	φ(a) = b, φ(b) = bb
11	24292	⟨a, b \| aa=b, baaaaab=a⟩	φ(a) = b, φ(b) = bb

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