#4625 ⟨a, b | aaaa=a, abbb=b⟩

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 28 elements

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:
- a³, b²a, b⁴a², b⁶, b⁶a³
Order of generators:
- a: index 1, period 3
- b: index 1, period 6

Histogram:

index 1, period 1	5 elements	a³, b²a, b⁴a², b⁶, b⁶a³
index 1, period 2	4 elements	ba², b³, b³a³, b⁵a
index 1, period 3	10 elements	a, a², b², b²a², b⁴, ...
index 1, period 6	8 elements	b, ba, ba³, b³a, b³a², ...

Complete 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

b⁷ ⇒ b
ab ⇒ b⁵
a⁴ ⇒ a

# ab:aaaa=a,abbb=b b/a
bbbbbbb=b
ab=bbbbb
aaaa=a

Right Cayley graph

Left Cayley graph

Others with same cardinality

2 unique, 6 total

Σ	#	Presentation	Description	Related
10	4623	⟨a, b \| aaaa=a, abba=b⟩	Finite non-commutative monoid with 28 elements	1 iso
10	4691	⟨a, b \| aaab=b, babb=a⟩	Finite non-commutative monoid with 28 elements	3 iso

Other isomorphic instances

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

2 total

Σ	#	Presentation	Mapping
11	14344	⟨a, b \| aaaa=a, aabbb=b⟩	φ(a) = aa, φ(b) = b
11	14352	⟨a, b \| aaaa=a, abbab=b⟩	φ(a) = aa, φ(b) = b

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