#16044 ⟨a, b | aaa=ab, bbbb=bb⟩

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

Properties

Sum of relation sides is 11
Finite non-commutative monoid with 36 elements
Not cancellative, because multiplication by a is not injective:
- a ⋅ b = a³ and a ⋅ a² = a³, however b ≠ a²

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.

1 element center:
- 1
5 non-trivial idempotents:
- b², ba⁶, a⁸, b³a⁶, b²a⁸
Order of generators:
- a: index 5, period 4
- b: index 2, period 2

Histogram:

index 1, period 1	5 elements	b², ba⁶, a⁸, b³a⁶, b²a⁸
index 1, period 2	5 elements	b³, a⁶, b²a⁶, ba⁸, b³a⁸
index 1, period 4	8 elements	a⁵, ba⁵, a⁷, b²a⁵, ba⁷, ...
index 2, period 1	4 elements	ba², a⁴, b³a², b²a⁴
index 2, period 2	4 elements	b, b²a², ba⁴, b³a⁴
index 2, period 4	6 elements	a³, b²a, ba³, b³a, b²a³, ...
index 3, period 2	1 element	a²
index 3, period 4	1 element	ba
index 5, period 4	1 element	a

Right Cayley graph

Left Cayley graph

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.	a⁹ ⇒ a⁵	[3]
2.	ab ⇒ a³	[1]
3.	b⁴ ⇒ b²	[2]

# ab:aaa=ab,bbbb=bb a/b - -
aaaaaaaaa=aaaaa
ab=aaa
bbbb=bb

Same cardinality

4 unique, 41 total

Σ	#	Presentation	Properties	φ
9	1345	⟨a, b \| aaab=a, bbbb=1⟩	Noc 36	13
9	2300	⟨a, b \| aaa=1, ababbb=1⟩	Grp 36	23
10	5073	⟨a, b \| aaa=ab, bbbb=b⟩	Noc 36	1
11	18959	⟨a, b \| aab=a, bbbbbb=b⟩	Noc 36

Up:	Monoid enumeration
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Next:	#16052 ⟨a, b \| aaa=bb, aaba=aa⟩