#4648 ⟨a, b | aaaa=b, bbbb=a⟩

Properties

Presentation has sum-of-sides 10
Isomorphic to ℕ_{(16 = 1)}
Finite commutative monoid with 16 elements
Commutative Gröbner basis: ⟨a, b | b¹⁶=b, a=b⁴⟩
Not cancellative, because multiplication by b⁵ is not injective:
- b¹⁵ ≠ 1
- b⁵ ⋅ b¹⁵ = b⁵
- b⁵ ⋅ 1 = b⁵
- Cancellative quotient is isomorphic to ℤ₁₅
Grothendieck group is isomorphic to ℤ₁₅
Group of units is isomorphic to ℤ₁
2 Archimedian components:
- 1, b

Element profile

2 idempotent elements:
- 1, a¹⁵

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

a¹⁶ ⇒ a
b ⇒ a⁴

# ab:aaaa=b,bbbb=a a/b
aaaaaaaaaaaaaaaa=a
b=aaaa

Staircase diagram

Cayley table

Idempotents are shown in bold.

	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵
1	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵
a	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a
a²	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a	a²
a³	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a	a²	a³
a⁴	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a	a²	a³	a⁴
a⁵	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a	a²	a³	a⁴	a⁵
a⁶	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a	a²	a³	a⁴	a⁵	a⁶
a⁷	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a	a²	a³	a⁴	a⁵	a⁶	a⁷
a⁸	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸
a⁹	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹
a¹⁰	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰
a¹¹	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹
a¹²	a¹²	a¹³	a¹⁴	a¹⁵	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²
a¹³	a¹³	a¹⁴	a¹⁵	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³
a¹⁴	a¹⁴	a¹⁵	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴
a¹⁵	a¹⁵	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

20 unique, 170 total

Σ	#	Presentation	Description	Related
8	628	⟨a, b \| bb=aa, abab=1⟩	Finite non-Abelian group with 16 elements	58 iso
9	1331	⟨a, b \| aaaa=b, bbbb=1⟩	Isomorphic to ℤ₁₆	67 iso
9	2051	⟨a, b \| aab=a, bbbb=b⟩	Finite non-commutative monoid with 16 elements	4 anti-iso
10	3808	⟨a, b \| aaab=ba, abab=1⟩	Finite non-Abelian group with 16 elements	7 iso
10	4630	⟨a, b \| aaaa=a, bbbb=a⟩	Isomorphic to ℕ_{(16 = 4)}	1 iso
10	6205	⟨a, b \| aba=b, aaaabb=1⟩	Finite non-Abelian group with 16 elements	3 iso
11	12164	⟨a, b \| aaaa=aa, bbbb=a⟩	Isomorphic to ℕ_{(16 = 8)}
11	12194	⟨a, b \| aaaa=ab, bbbb=a⟩	Isomorphic to ℕ_{(16 = 5)}	2 iso
11	12212	⟨a, b \| aaaa=bb, bbbb=a⟩	Isomorphic to ℕ_{(16 = 2)}
11	12306	⟨a, b \| aaab=bb, abba=a⟩	Finite non-commutative monoid with 16 elements	6 iso
11	13251	⟨a, b \| bab=aab, bbb=aa⟩	Finite non-commutative monoid with 16 elements
11	13259	⟨a, b \| bab=aba, bbb=aa⟩	Finite non-commutative monoid with 16 elements
11	16028	⟨a, b \| aaa=ab, babb=bb⟩	Finite non-commutative monoid with 16 elements
11	16032	⟨a, b \| aaa=ab, bbaa=bb⟩	Finite non-commutative monoid with 16 elements
11	16060	⟨a, b \| aaa=bb, abab=aa⟩	Finite non-commutative monoid with 16 elements
11	16371	⟨a, b \| aba=aa, bbbb=ab⟩	Finite non-commutative monoid with 16 elements
11	18811	⟨a, b \| aaa=b, abbbbb=b⟩	Isomorphic to ℕ_{(16 = 3)}	2 iso
11	20251	⟨a, b \| aba=b, aaaa=abb⟩	Finite non-commutative monoid with 16 elements
11	21039	⟨a, b \| ab=aa, bbba=bbb⟩	Finite non-commutative monoid with 16 elements
11	24660	⟨a, b \| aa=a, ababab=bb⟩	Finite non-commutative monoid with 16 elements

Other isomorphic instances

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

5 total

Σ	#	Presentation	Mapping
11	14535	⟨a, b \| aaab=b, bbbbb=a⟩	φ(a) = bbbbb, φ(b) = b
11	14663	⟨a, b \| aaba=b, bbbbb=a⟩	φ(a) = bbbbb, φ(b) = b
11	18810	⟨a, b \| aaa=b, abbbbb=a⟩	φ(a) = b, φ(b) = bbb
11	18824	⟨a, b \| aaa=b, babbbb=a⟩	φ(a) = b, φ(b) = bbb
11	18828	⟨a, b \| aaa=b, bbabbb=a⟩	φ(a) = b, φ(b) = bbb

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