#14395 ⟨a, b | aaaa=b, abbbb=a⟩

Contents

Properties
Commutative structure
Finite structure
Staircase diagram
Cayley table
Right Cayley graph
Rewriting system
Same cardinality
Isomorphic instances

Properties

Sum of relation sides is 11
Isomorphic to ℕ_{(17 = 1)}
Finite commutative monoid with 17 elements
Not cancellative, because multiplication by a is not injective:
- a ⋅ a¹⁶ = a and a ⋅ 1 = a, however a¹⁶ ≠ 1

Commutative structure

Gröbner basis: ⟨a, b | a¹⁷=a, b=a⁴⟩
Cancellative quotient is isomorphic to ℤ₁₆
Enveloping group is isomorphic to ℤ₁₆
Group of units is isomorphic to ℤ₁
2 Archimedian components:
- 1, b

Finite structure

An idempotent element x satisfies x² = x.
The index and period of x is the least m (index) and n (period) such that x^(m+n) = x^m.

1 non-trivial idempotent:
- a¹⁶
Order of generators:
- a: index 1, period 16
- b: index 1, period 4
Histogram:

index 1, period 1 1 element a¹⁶

index 1, period 2 1 element a⁸

index 1, period 4 2 elements a⁴, a¹²

index 1, period 8 4 elements a², a⁶, a¹⁰, a¹⁴

index 1, period 16 8 elements a, a³, a⁵, a⁷, a⁹, ...

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	a¹⁶
1	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶
a	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a
a²	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a	a²
a³	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a	a²	a³
a⁴	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a	a²	a³	a⁴
a⁵	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a	a²	a³	a⁴	a⁵
a⁶	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a	a²	a³	a⁴	a⁵	a⁶
a⁷	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a	a²	a³	a⁴	a⁵	a⁶	a⁷
a⁸	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸
a⁹	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹
a¹⁰	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰
a¹¹	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹
a¹²	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²
a¹³	a¹³	a¹⁴	a¹⁵	a¹⁶	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³
a¹⁴	a¹⁴	a¹⁵	a¹⁶	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴
a¹⁵	a¹⁵	a¹⁶	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵
a¹⁶	a¹⁶	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶

Right Cayley graph

Idempotents are shown in bold.

Rewriting system

Format:	Pretty Plain
Word:
	Enter a word above to compute its normal form. Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Strategy:	Leftmost Rightmost
Result:	1 1

Reduction order:
- Left-to-right recursive path with deg(a) = 0; deg(b) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	a¹⁷ ⇒ a	[2]
2.	b ⇒ a⁴	[1]

# ab:aaaa=b,abbbb=a a/b - -
aaaaaaaaaaaaaaaaa=a
b=aaaa

Same cardinality

16 unique, 69 total

Σ	#	Presentation	Properties	Description	φ
10	4375	⟨a, b \| aaaa=b, abbbb=1⟩	Grp Com 17	ℤ₁₇	39
10	6428	⟨a, b \| aab=b, babbb=a⟩	Noc 17		2
10	6514	⟨a, b \| aba=b, baaab=a⟩	Noc 17
10	7152	⟨a, b \| bb=aa, abab=aa⟩	Noc 17
10	7153	⟨a, b \| bb=aa, abab=ab⟩	Noc 17
10	7154	⟨a, b \| bb=aa, abab=ba⟩	Noc 17
11	13141	⟨a, b \| aab=aaa, bbb=ab⟩	Noc 17
11	13142	⟨a, b \| aab=aaa, bbb=ba⟩	Noc 17
11	13151	⟨a, b \| aba=aaa, baa=bb⟩	Noc 17
11	13156	⟨a, b \| aba=aaa, bbb=ab⟩	Noc 17
11	13218	⟨a, b \| abb=aba, bbb=aa⟩	Noc 17
11	14396	⟨a, b \| aaaa=b, abbbb=b⟩	Noc Com 17	ℕ_{(17 = 4)}	2
11	14698	⟨a, b \| aabb=a, baaaa=b⟩	Noc 17		7
11	15488	⟨a, b \| aaa=ab, babbb=b⟩	Noc 17		3
11	16143	⟨a, b \| aab=aa, bbbb=ba⟩	Noc 17
11	16521	⟨a, b \| aab=bb, bab=aaa⟩	Noc 17

Isomorphic instances

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

2 total

Σ	#	Presentation	Mapping
11	14403	⟨a, b \| aaaa=b, babbb=a⟩	φ(a) = a, φ(b) = aaaa
11	14405	⟨a, b \| aaaa=b, bbabb=a⟩	φ(a) = a, φ(b) = aaaa