#16053 ⟨a, b | aaa=bb, aaba=ab⟩

Properties
Finite structure
Cayley table
Right Cayley graph
Left Cayley graph
Rewriting system
Same cardinality

Properties

Sum of relation sides is 11
Finite non-commutative monoid with 19 elements
Not cancellative, because multiplication by a is not injective:
- a ⋅ a⁹ = a⁴ and a ⋅ a³ = a⁴, however a⁹ ≠ a³

Finite structure

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, a³, a⁶, a⁹
1 non-trivial idempotent:
- a⁶
Order of generators:
- a: index 4, period 6
- b: index 3, period 4

Histogram:

index 1, period 1	1 element	a⁶
index 1, period 2	1 element	a⁹
index 1, period 3	2 elements	a⁴, a⁸
index 1, period 4	6 elements	ba³, ba⁴, ba⁵, ba⁶, ba⁷, ...
index 1, period 6	2 elements	a⁵, a⁷
index 2, period 2	1 element	a³
index 2, period 3	1 element	a²
index 2, period 4	2 elements	ba, ba²
index 3, period 4	1 element	b
index 4, period 6	1 element	a

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸
1	1	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸
a	a	a²	ba⁵	a³	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a⁴	ba⁷
b	b	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a⁴	ba³	a⁵
a²	a²	a³	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba³	a⁹	ba⁴	a⁴	ba⁵	a⁵	ba⁶
ba	ba	ba²	a⁸	ba³	a⁹	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba³	a⁹	ba⁴	a⁴
a³	a³	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵
ba²	ba²	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹
a⁴	a⁴	a⁵	ba⁸	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba³	a⁷	ba⁴
ba³	ba³	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸
a⁵	a⁵	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba³	a⁹	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba³
ba⁴	ba⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba³	a⁹	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷
a⁶	a⁶	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸
ba⁵	ba⁵	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶
a⁷	a⁷	a⁸	ba⁵	a⁹	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a⁴	ba⁷
ba⁶	ba⁶	ba⁷	a⁹	ba⁸	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a⁴	ba³	a⁵
a⁸	a⁸	a⁹	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba³	a⁹	ba⁴	a⁴	ba⁵	a⁵	ba⁶
ba⁷	ba⁷	ba⁸	a⁸	ba³	a⁹	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba³	a⁹	ba⁴	a⁴
a⁹	a⁹	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵
ba⁸	ba⁸	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹

Right Cayley graph

Idempotents are shown in bold.

Left 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.	a¹⁰ ⇒ a⁴	[6]
2.	ba⁹ ⇒ ba³	[7]
3.	ab ⇒ ba⁵	[5]
4.	b² ⇒ a³	[1]

# ab:aaa=bb,aaba=ab a/b - -
aaaaaaaaaa=aaaa
baaaaaaaaa=baaa
ab=baaaaa
bb=aaa

Same cardinality

5 unique, 29 total

Σ	#	Presentation	Properties	Description	φ
10	6263	⟨a, b \| aaa=a, abbba=b⟩	Noc 19
10	6838	⟨a, b \| aba=b, bab=aaa⟩	Noc 19
11	13340	⟨a, b \| aaaab=1, abbbbb=1⟩	Grp Com 19	ℤ₁₉	20
11	14501	⟨a, b \| aaab=b, abbba=a⟩	Noc 19		3
11	16036	⟨a, b \| aaa=ab, bbab=bb⟩	Noc 19		1

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#16053 ⟨a, b | aaa=bb, aaba=ab⟩

Contents

Properties

Finite structure

Cayley table

Right Cayley graph

Left Cayley graph

Rewriting system

Same cardinality