#7154 ⟨a, b | bb=aa, abab=ba⟩

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Properties
Elements
Cayley table
Right Cayley graph
Left Cayley graph
Rewriting system
Same cardinality

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 17 elements
Not cancellative, because multiplication by a³ is not injective:
- a³ ⋅ a⁴ = a³ and a³ ⋅ 1 = a³, however a⁴ ≠ 1

Elements

Elements in the center commute with all other elements.
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.

4 element center:
- 1, a², a⁴, a⁶
1 non-trivial idempotent:
- a⁴
Order of generators:
- a: index 3, period 4
- b: index 3, period 4

Histogram:

index 1, period 1	1 element	a⁴
index 1, period 2	1 element	a⁶
index 1, period 3	2 elements	ba, aba⁴
index 1, period 4	6 elements	a³, aba, ba², a⁵, aba³, ...
index 1, period 6	2 elements	aba², ba³
index 2, period 2	1 element	a²
index 2, period 3	1 element	ab
index 3, period 4	2 elements	a, b

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	ba	a³	aba	ba²	a⁴	aba²	ba³	a⁵	aba³	ba⁴	a⁶	aba⁴
1	1	a	b	a²	ab	ba	a³	aba	ba²	a⁴	aba²	ba³	a⁵	aba³	ba⁴	a⁶	aba⁴
a	a	a²	ab	a³	ba²	aba	a⁴	ba³	aba²	a⁵	ba⁴	aba³	a⁶	ba	aba⁴	a³	ba²
b	b	ba	a²	ba²	aba³	a³	ba³	aba⁴	a⁴	ba⁴	aba	a⁵	ba	aba²	a⁶	ba²	aba³
a²	a²	a³	ba²	a⁴	aba²	ba³	a⁵	aba³	ba⁴	a⁶	aba⁴	ba	a³	aba	ba²	a⁴	aba²
ab	ab	aba	a³	aba²	ba	a⁴	aba³	ba²	a⁵	aba⁴	ba³	a⁶	aba	ba⁴	a³	aba²	ba
ba	ba	ba²	aba³	ba³	a⁴	aba⁴	ba⁴	a⁵	aba	ba	a⁶	aba²	ba²	a³	aba³	ba³	a⁴
a³	a³	a⁴	aba²	a⁵	ba⁴	aba³	a⁶	ba	aba⁴	a³	ba²	aba	a⁴	ba³	aba²	a⁵	ba⁴
aba	aba	aba²	ba	aba³	a⁵	ba²	aba⁴	a⁶	ba³	aba	a³	ba⁴	aba²	a⁴	ba	aba³	a⁵
ba²	ba²	ba³	a⁴	ba⁴	aba	a⁵	ba	aba²	a⁶	ba²	aba³	a³	ba³	aba⁴	a⁴	ba⁴	aba
a⁴	a⁴	a⁵	ba⁴	a⁶	aba⁴	ba	a³	aba	ba²	a⁴	aba²	ba³	a⁵	aba³	ba⁴	a⁶	aba⁴
aba²	aba²	aba³	a⁵	aba⁴	ba³	a⁶	aba	ba⁴	a³	aba²	ba	a⁴	aba³	ba²	a⁵	aba⁴	ba³
ba³	ba³	ba⁴	aba	ba	a⁶	aba²	ba²	a³	aba³	ba³	a⁴	aba⁴	ba⁴	a⁵	aba	ba	a⁶
a⁵	a⁵	a⁶	aba⁴	a³	ba²	aba	a⁴	ba³	aba²	a⁵	ba⁴	aba³	a⁶	ba	aba⁴	a³	ba²
aba³	aba³	aba⁴	ba³	aba	a³	ba⁴	aba²	a⁴	ba	aba³	a⁵	ba²	aba⁴	a⁶	ba³	aba	a³
ba⁴	ba⁴	ba	a⁶	ba²	aba³	a³	ba³	aba⁴	a⁴	ba⁴	aba	a⁵	ba	aba²	a⁶	ba²	aba³
a⁶	a⁶	a³	ba²	a⁴	aba²	ba³	a⁵	aba³	ba⁴	a⁶	aba⁴	ba	a³	aba	ba²	a⁴	aba²
aba⁴	aba⁴	aba	a³	aba²	ba	a⁴	aba³	ba²	a⁵	aba⁴	ba³	a⁶	aba	ba⁴	a³	aba²	ba

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

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
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	a⁷ ⇒ a³	[6]
2.	ba⁵ ⇒ ba	[7]
3.	a²b ⇒ ba²	[3]
4.	b² ⇒ a²	[1]
5.	bab ⇒ aba³	[4]

# ab:bb=aa,abab=ba a/b
aaaaaaa=aaa
baaaaa=ba
aab=baa
bb=aa
bab=abaaa

Same cardinality

16 unique, 71 total

Σ	#	Presentation	Description	Related
10	4375	⟨a, b \| aaaa=b, abbbb=1⟩	Isomorphic to ℤ₁₇	39 iso
10	6428	⟨a, b \| aab=b, babbb=a⟩	Finite non-commutative monoid with 17 elements	2 iso
10	6514	⟨a, b \| aba=b, baaab=a⟩	Finite non-commutative monoid with 17 elements
10	7152	⟨a, b \| bb=aa, abab=aa⟩	Finite non-commutative monoid with 17 elements
10	7153	⟨a, b \| bb=aa, abab=ab⟩	Finite non-commutative monoid with 17 elements
11	13141	⟨a, b \| aab=aaa, bbb=ab⟩	Finite non-commutative monoid with 17 elements
11	13142	⟨a, b \| aab=aaa, bbb=ba⟩	Finite non-commutative monoid with 17 elements
11	13151	⟨a, b \| aba=aaa, baa=bb⟩	Finite non-commutative monoid with 17 elements
11	13156	⟨a, b \| aba=aaa, bbb=ab⟩	Finite non-commutative monoid with 17 elements
11	13218	⟨a, b \| abb=aba, bbb=aa⟩	Finite non-commutative monoid with 17 elements
11	14395	⟨a, b \| aaaa=b, abbbb=a⟩	Isomorphic to ℕ_{(17 = 1)}	2 iso
11	14396	⟨a, b \| aaaa=b, abbbb=b⟩	Isomorphic to ℕ_{(17 = 4)}	2 iso
11	14698	⟨a, b \| aabb=a, baaaa=b⟩	Finite non-commutative monoid with 17 elements	1 iso, 6 anti-iso
11	15488	⟨a, b \| aaa=ab, babbb=b⟩	Finite non-commutative monoid with 17 elements	3 iso
11	16143	⟨a, b \| aab=aa, bbbb=ba⟩	Finite non-commutative monoid with 17 elements
11	16521	⟨a, b \| aab=bb, bab=aaa⟩	Finite non-commutative monoid with 17 elements

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