#5073 ⟨a, b | aaa=ab, bbbb=b⟩

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Properties
Elements
Right Cayley graph
Left Cayley graph
Rewriting system
Same cardinality
Isomorphic instances

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 36 elements
Not cancellative, because multiplication by a³ is not injective:
- a³ ⋅ a⁶ = a³ and a³ ⋅ 1 = a³, however a⁶ ≠ 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
5 non-trivial idempotents:
- b³, ba⁴, a⁶, b³a⁶, b²a⁸
Order of generators:
- a: index 3, period 6
- b: index 1, period 3

Histogram:

index 1, period 1	5 elements	b³, ba⁴, a⁶, b³a⁶, b²a⁸
index 1, period 2	4 elements	a³, b³a³, b²a⁵, ba⁷
index 1, period 3	10 elements	b, b², a⁴, b²a⁴, ba⁶, ...
index 1, period 6	8 elements	ba³, a⁵, b²a³, ba⁵, a⁷, ...
index 2, period 1	1 element	b²a²
index 2, period 2	1 element	ba
index 2, period 3	3 elements	a², ba², b³a²
index 2, period 6	2 elements	b²a, b³a
index 3, period 6	1 element	a

Right Cayley graph

Left Cayley graph

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.	a⁹ ⇒ a³	[3]
2.	ab ⇒ a³	[1]
3.	b⁴ ⇒ b	[2]

# ab:aaa=ab,bbbb=b a/b
aaaaaaaaa=aaa
ab=aaa
bbbb=b

Same cardinality

4 unique, 40 total

Σ	#	Presentation	Description	Related
9	1345	⟨a, b \| aaab=a, bbbb=1⟩	Finite non-commutative monoid with 36 elements	5 iso, 8 anti-iso
9	2300	⟨a, b \| aaa=1, ababbb=1⟩	Finite non-Abelian group with 36 elements	23 iso
11	16044	⟨a, b \| aaa=ab, bbbb=bb⟩	Finite non-commutative monoid with 36 elements
11	18959	⟨a, b \| aab=a, bbbbbb=b⟩	Finite non-commutative monoid with 36 elements

Isomorphic instances

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

1 total

Σ	#	Presentation	Mapping
11	12995	⟨a, b \| abb=aaa, bbbb=b⟩	φ(a) = a, φ(b) = bb

Up:	Monoids with two generators and two relations
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