#5092 ⟨a, b | aaa=bb, bbbb=a⟩

Properties

Presentation has sum-of-sides 10
Isomorphic to ℕ_{(12 = 2)}
Finite commutative monoid with 12 elements
Commutative Gröbner basis: ⟨a, b | b¹²=b², a=b⁴⟩
Not cancellative, because multiplication by b⁵ is not injective:
- b¹⁰ ≠ 1
- b⁵ ⋅ b¹⁰ = b⁵
- b⁵ ⋅ 1 = b⁵
- Cancellative quotient is isomorphic to ℤ₁₀
Grothendieck group is isomorphic to ℤ₁₀
Group of units is isomorphic to ℤ₁
2 Archimedian components:
- 1, b

Element profile

2 idempotent elements:
- 1, b¹⁰

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

b¹² ⇒ b²
a ⇒ b⁴

# ab:aaa=bb,bbbb=a b/a
bbbbbbbbbbbb=bb
a=bbbb

Staircase diagram

Cayley table

Idempotents are shown in bold.

	1	b	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹
1	1	b	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹
b	b	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b²
b²	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b²	b³
b³	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b²	b³	b⁴
b⁴	b⁴	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b²	b³	b⁴	b⁵
b⁵	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b²	b³	b⁴	b⁵	b⁶
b⁶	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b²	b³	b⁴	b⁵	b⁶	b⁷
b⁷	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸
b⁸	b⁸	b⁹	b¹⁰	b¹¹	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b⁹
b⁹	b⁹	b¹⁰	b¹¹	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰
b¹⁰	b¹⁰	b¹¹	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹
b¹¹	b¹¹	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b²

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

37 unique, 616 total

Σ	#	Presentation	Description	Related
7	124	⟨a, b \| aab=a, bbb=1⟩	Finite non-commutative monoid with 12 elements	23 iso, 38 anti-iso
8	458	⟨a, b \| aaaa=1, abbb=1⟩	Isomorphic to ℤ₁₂	325 iso
8	645	⟨a, b \| ab=aa, bbb=b⟩	Finite non-commutative monoid with 12 elements	1 anti-iso
9	1646	⟨a, b \| aab=bb, aba=a⟩	Finite non-commutative monoid with 12 elements	2 iso
9	1963	⟨a, b \| aba=b, aaabb=1⟩	Finite non-Abelian group with 12 elements	30 iso
9	1998	⟨a, b \| aaa=a, bbbb=a⟩	Isomorphic to ℕ_{(12 = 4)}	6 iso
9	2018	⟨a, b \| aaa=b, bbbb=a⟩	Isomorphic to ℕ_{(12 = 1)}	27 iso
9	2019	⟨a, b \| aaa=b, bbbb=b⟩	Isomorphic to ℕ_{(12 = 3)}	23 iso
9	2259	⟨a, b \| ab=aa, bbb=bb⟩	Finite non-commutative monoid with 12 elements
10	5040	⟨a, b \| aaa=aa, bbbb=a⟩	Isomorphic to ℕ_{(12 = 8)}
10	5072	⟨a, b \| aaa=ab, bbbb=a⟩	Isomorphic to ℕ_{(12 = 5)}	7 iso
10	5202	⟨a, b \| aab=bb, abba=a⟩	Finite non-commutative monoid with 12 elements	9 iso, 32 anti-iso
10	5330	⟨a, b \| aaa=ab, bab=bb⟩	Finite non-commutative monoid with 12 elements	1 iso
10	6597	⟨a, b \| aaa=b, bbbb=bb⟩	Isomorphic to ℕ_{(12 = 6)}	4 iso
10	6660	⟨a, b \| aab=a, bbbb=ba⟩	Finite non-commutative monoid with 12 elements	1 anti-iso
10	6661	⟨a, b \| aab=a, bbbb=bb⟩	Finite non-commutative monoid with 12 elements	1 anti-iso
10	7105	⟨a, b \| ab=aa, bbaa=bb⟩	Finite non-commutative monoid with 12 elements	2 iso
11	12897	⟨a, b \| aab=aaa, baaa=b⟩	Finite non-commutative monoid with 12 elements	5 iso
11	12910	⟨a, b \| aab=aaa, bbbb=a⟩	Isomorphic to ℕ_{(12 = 9)}	2 iso
11	15428	⟨a, b \| aaa=aa, abbbb=b⟩	Finite non-commutative monoid with 12 elements
11	15996	⟨a, b \| aaa=ab, aabb=bb⟩	Finite non-commutative monoid with 12 elements	2 iso
11	16057	⟨a, b \| aaa=bb, aabb=ab⟩	Finite non-commutative monoid with 12 elements	2 iso, 2 anti-iso
11	16104	⟨a, b \| aab=aa, abab=bb⟩	Finite non-commutative monoid with 12 elements
11	16274	⟨a, b \| aab=bb, aaaa=ab⟩	Finite non-commutative monoid with 12 elements	1 iso
11	16275	⟨a, b \| aab=bb, aaaa=ba⟩	Finite non-commutative monoid with 12 elements
11	16448	⟨a, b \| aba=bb, aabb=aa⟩	Finite non-commutative monoid with 12 elements	2 iso
11	19773	⟨a, b \| aba=b, bbbbb=aa⟩	Finite commutative monoid with 12 elements
11	19917	⟨a, b \| aaa=b, bbbb=abb⟩	Isomorphic to ℕ_{(12 = 7)}	2 iso
11	20054	⟨a, b \| aab=b, aaaa=bba⟩	Finite non-commutative monoid with 12 elements
11	20686	⟨a, b \| bb=aa, aaaaab=a⟩	Finite commutative monoid with 12 elements	15 iso
11	20787	⟨a, b \| ab=aa, baaaa=bb⟩	Finite non-commutative monoid with 12 elements	7 iso
11	21086	⟨a, b \| bb=aa, aaaa=aba⟩	Finite non-commutative monoid with 12 elements	4 iso
11	21092	⟨a, b \| bb=aa, aaab=aba⟩	Finite non-commutative monoid with 12 elements	3 iso
11	21112	⟨a, b \| bb=aa, abab=aba⟩	Finite non-commutative monoid with 12 elements
11	24147	⟨a, b \| aa=a, abbbbbb=b⟩	Finite non-commutative monoid with 12 elements
11	24991	⟨a, b \| ab=a, baaaaa=bb⟩	Finite non-commutative monoid with 12 elements
11	25603	⟨a, b \| ab=a, bbaaa=bbb⟩	Finite non-commutative monoid with 12 elements

Other isomorphic instances

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

43 total

Σ	#	Presentation	Mapping
10	6595	⟨a, b \| aaa=b, bbbb=aa⟩	φ(a) = b, φ(b) = bbb
10	8711	⟨a, b \| aa=b, bbbbbb=b⟩	φ(a) = b, φ(b) = bb
11	12200	⟨a, b \| aaaa=bb, aabb=a⟩	φ(a) = bbbbbbbb, φ(b) = b
11	12202	⟨a, b \| aaaa=bb, abab=a⟩	φ(a) = bbbbbbbb, φ(b) = b
11	12204	⟨a, b \| aaaa=bb, abba=a⟩	φ(a) = bbbbbbbb, φ(b) = b
11	12208	⟨a, b \| aaaa=bb, baab=a⟩	φ(a) = bbbbbbbb, φ(b) = b
11	15511	⟨a, b \| aaa=bb, aaabb=a⟩	φ(a) = bbbb, φ(b) = b
11	15517	⟨a, b \| aaa=bb, aabba=a⟩	φ(a) = bbbb, φ(b) = b
11	15521	⟨a, b \| aaa=bb, abaab=a⟩	φ(a) = bbbb, φ(b) = b
11	15533	⟨a, b \| aaa=bb, baaab=a⟩	φ(a) = bbbb, φ(b) = b
11	20684	⟨a, b \| bb=aa, aaaaaa=a⟩	φ(a) = bbbbbb, φ(b) = b
11	20685	⟨a, b \| bb=aa, aaaaaa=b⟩	φ(a) = b, φ(b) = bbbbbb
11	20690	⟨a, b \| bb=aa, aaaabb=a⟩	φ(a) = bbbbbb, φ(b) = b
11	20691	⟨a, b \| bb=aa, aaaabb=b⟩	φ(a) = b, φ(b) = bbbbbb
11	20694	⟨a, b \| bb=aa, aaabab=a⟩	φ(a) = bbbbbb, φ(b) = b
11	20695	⟨a, b \| bb=aa, aaabab=b⟩	φ(a) = b, φ(b) = bbbbbb
11	20696	⟨a, b \| bb=aa, aaabba=a⟩	φ(a) = bbbbbb, φ(b) = b
11	20697	⟨a, b \| bb=aa, aaabba=b⟩	φ(a) = b, φ(b) = bbbbbb
11	20699	⟨a, b \| bb=aa, aabaab=a⟩	φ(a) = bbbbbb, φ(b) = b
11	20700	⟨a, b \| bb=aa, aabaab=b⟩	φ(a) = b, φ(b) = bbbbbb
11	20701	⟨a, b \| bb=aa, aababa=a⟩	φ(a) = bbbbbb, φ(b) = b
11	20702	⟨a, b \| bb=aa, aababa=b⟩	φ(a) = b, φ(b) = bbbbbb
11	20704	⟨a, b \| bb=aa, aabbaa=a⟩	φ(a) = bbbbbb, φ(b) = b
11	20705	⟨a, b \| bb=aa, aabbaa=b⟩	φ(a) = b, φ(b) = bbbbbb
11	20710	⟨a, b \| bb=aa, abaaab=a⟩	φ(a) = bbbbbb, φ(b) = b
11	20711	⟨a, b \| bb=aa, abaaab=b⟩	φ(a) = b, φ(b) = bbbbbb
11	20712	⟨a, b \| bb=aa, abaaba=a⟩	φ(a) = bbbbbb, φ(b) = b
11	20713	⟨a, b \| bb=aa, abaaba=b⟩	φ(a) = b, φ(b) = bbbbbb
11	20718	⟨a, b \| bb=aa, abbbba=a⟩	φ(a) = bbbbbb, φ(b) = b
11	20719	⟨a, b \| bb=aa, abbbba=b⟩	φ(a) = b, φ(b) = bbbbbb
11	24247	⟨a, b \| aa=b, aabbbbb=b⟩	φ(a) = b, φ(b) = bb
11	24271	⟨a, b \| aa=b, ababbbb=b⟩	φ(a) = b, φ(b) = bb
11	24281	⟨a, b \| aa=b, abbabbb=b⟩	φ(a) = b, φ(b) = bb
11	24285	⟨a, b \| aa=b, abbbabb=b⟩	φ(a) = b, φ(b) = bb
11	24287	⟨a, b \| aa=b, abbbbab=b⟩	φ(a) = b, φ(b) = bb
11	24289	⟨a, b \| aa=b, abbbbba=b⟩	φ(a) = b, φ(b) = bb
11	24307	⟨a, b \| aa=b, baabbbb=b⟩	φ(a) = b, φ(b) = bb
11	24313	⟨a, b \| aa=b, bababbb=b⟩	φ(a) = b, φ(b) = bb
11	24315	⟨a, b \| aa=b, babbabb=b⟩	φ(a) = b, φ(b) = bb
11	24317	⟨a, b \| aa=b, babbbab=b⟩	φ(a) = b, φ(b) = bb
11	24323	⟨a, b \| aa=b, bbaabbb=b⟩	φ(a) = b, φ(b) = bb
11	24325	⟨a, b \| aa=b, bbababb=b⟩	φ(a) = b, φ(b) = bb
11	24857	⟨a, b \| aa=b, bbbbbb=aa⟩	φ(a) = b, φ(b) = bb

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