#12164 ⟨a, b | aaaa=aa, bbbb=a⟩

Properties

Presentation has sum-of-sides 11
Isomorphic to ℕ_{(16 = 8)}
Finite commutative monoid with 16 elements
Commutative Gröbner basis: ⟨a, b | b¹⁶=b⁸, a=b⁴⟩
Not cancellative, because multiplication by b¹⁰ is not injective:
- b⁸ ≠ 1
- b¹⁰ ⋅ b⁸ = b¹⁰
- b¹⁰ ⋅ 1 = b¹⁰
- 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, b⁸

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

b¹⁶ ⇒ b⁸
a ⇒ b⁴

# ab:aaaa=aa,bbbb=a b/a
bbbbbbbbbbbbbbbb=bbbbbbbb
a=bbbb

Staircase diagram

Cayley table

Idempotents are shown in bold.

	1	b	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵
1	1	b	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵
b	b	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b⁸
b²	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b⁸	b⁹
b³	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b⁸	b⁹	b¹⁰
b⁴	b⁴	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b⁸	b⁹	b¹⁰	b¹¹
b⁵	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b⁸	b⁹	b¹⁰	b¹¹	b¹²
b⁶	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³
b⁷	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴
b⁸	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵
b⁹	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b⁸
b¹⁰	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b⁸	b⁹
b¹¹	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b⁸	b⁹	b¹⁰
b¹²	b¹²	b¹³	b¹⁴	b¹⁵	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b⁸	b⁹	b¹⁰	b¹¹
b¹³	b¹³	b¹⁴	b¹⁵	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b⁸	b⁹	b¹⁰	b¹¹	b¹²
b¹⁴	b¹⁴	b¹⁵	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³
b¹⁵	b¹⁵	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

20 unique, 175 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	4648	⟨a, b \| aaaa=b, bbbb=a⟩	Isomorphic to ℕ_{(16 = 1)}	5 iso
10	6205	⟨a, b \| aba=b, aaaabb=1⟩	Finite non-Abelian group with 16 elements	3 iso
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

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