#6265 ⟨a, b | aaa=a, abbbb=b⟩

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 21 elements
Not cancellative, because multiplication by a is not injective:
- a ⋅ a² = a and a ⋅ 1 = a, however a² ≠ 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.

1 element center:
- 1
4 non-trivial idempotents:
- a², b³a, b⁶, b⁶a²
Order of generators:
- a: index 1, period 2
- b: index 1, period 6

Histogram:

index 1, period 1	4 elements	a², b³a, b⁶, b⁶a²
index 1, period 2	4 elements	a, b³, b³a², b⁶a
index 1, period 3	6 elements	ba, b², b²a², b⁴, b⁴a², ...
index 1, period 6	6 elements	b, ba², b²a, b⁴a, b⁵, ...

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ba	b²	ba²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b⁶a²
1	1	a	b	a²	ba	b²	ba²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b⁶a²
a	a	a²	b⁴	a	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b³	b²a²	b³a	b³a²
b	b	ba	b²	ba²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	ba²
a²	a²	a	b	a²	ba	b²	ba²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b⁶a²
ba	ba	ba²	b⁵	ba	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁴a²
b²	b²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b²a²
ba²	ba²	ba	b²	ba²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	ba²
b²a	b²a	b²a²	b⁶	b²a	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁵a²
b³	b³	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b³	b²a²	b³a	b³a²
b²a²	b²a²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b²a²
b³a	b³a	b³a²	b	b³a	ba	b²	ba²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b⁶a²
b⁴	b⁴	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁴a²
b³a²	b³a²	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b³	b²a²	b³a	b³a²
b⁴a	b⁴a	b⁴a²	b²	b⁴a	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	ba²
b⁵	b⁵	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁵a²
b⁴a²	b⁴a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁴a²
b⁵a	b⁵a	b⁵a²	b³	b⁵a	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b²a²
b⁶	b⁶	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b⁶a²
b⁵a²	b⁵a²	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁵a²
b⁶a	b⁶a	b⁶a²	b⁴	b⁶a	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b³	b²a²	b³a	b³a²
b⁶a²	b⁶a²	b⁶a	b	b⁶a²	ba	b²	ba²	b²a	b³	b²a²	b³a	b⁴	b³a²	b⁴a	b⁵	b⁴a²	b⁵a	b⁶	b⁵a²	b⁶a	b⁶a²

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(b) = 0; deg(a) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	b⁷ ⇒ b	[5]
2.	ab ⇒ b⁴	[4]
3.	a³ ⇒ a	[1]

# ab:aaa=a,abbbb=b b/a
bbbbbbb=b
ab=bbbb
aaa=a

Same cardinality

8 unique, 62 total

Σ	#	Presentation	Description	Related
8	402	⟨a, b \| aaa=ab, bbb=1⟩	Finite non-commutative monoid with 21 elements	12 iso, 27 anti-iso
9	1599	⟨a, b \| aaa=ab, bbb=b⟩	Finite non-commutative monoid with 21 elements	1 anti-iso
10	5309	⟨a, b \| aaa=aa, bbb=ab⟩	Finite non-commutative monoid with 21 elements
11	11127	⟨a, b \| aaaaa=b, abbbb=1⟩	Isomorphic to ℤ₂₁	13 iso
11	13091	⟨a, b \| bab=aaa, abba=b⟩	Finite non-commutative monoid with 21 elements
11	13248	⟨a, b \| bab=aaa, bbb=aa⟩	Finite non-commutative monoid with 21 elements
11	15158	⟨a, b \| aab=ba, ababab=1⟩	Finite non-Abelian group with 21 elements	1 iso
11	19212	⟨a, b \| aba=b, baaaab=a⟩	Finite non-commutative monoid with 21 elements

Isomorphic instances

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

2 total

Σ	#	Presentation	Mapping
11	18729	⟨a, b \| aaa=a, ababbb=b⟩	φ(a) = a, φ(b) = bb
11	18733	⟨a, b \| aaa=a, abbabb=b⟩	φ(a) = a, φ(b) = bb

Up:	Monoids with two generators and two relations
Prev:	#6264 ⟨a, b \| aaa=a, abbbb=a⟩
Next:	#6268 ⟨a, b \| aaa=a, baabb=a⟩

#6265 ⟨a, b | aaa=a, abbbb=b⟩

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