#4623 ⟨a, b | aaaa=a, abba=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.

4 element center:
- 1, a³, b³, (ab)³
2 non-trivial idempotents:
- a³, (ab)³
Order of generators:
- a: index 1, period 3
- b: index 1, period 6

Histogram:

index 1, period 1	2 elements	a³, (ab)³
index 1, period 2	1 element	b³
index 1, period 3	10 elements	a, a², ab, ba, b², ...
index 1, period 4	6 elements	a²b, aba, ab², bab, a²(ba)², ...
index 1, period 6	8 elements	b, a²ba, a²b², ab³, a(ab)², ...

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 shortlex with a < b

ba² ⇒ ab²
b²a ⇒ a²b
a⁴ ⇒ a
a³b ⇒ b
bab² ⇒ a²ba
b⁵ ⇒ a(ba)²
(ba)³ ⇒ (ab)³

# ab:aaaa=a,abba=b ab
baa=abb
bba=aab
aaaa=a
aaab=b
babb=aaba
bbbbb=ababa
bababa=ababab

Right Cayley graph

Left Cayley graph

Others with same cardinality

2 unique, 7 total

Σ	#	Presentation	Description	Related
10	4625	⟨a, b \| aaaa=a, abbb=b⟩	Finite non-commutative monoid with 28 elements	2 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.

1 total

Σ	#	Presentation	Mapping
11	14348	⟨a, b \| aaaa=a, ababa=b⟩	φ(a) = a, φ(b) = aab

Up:	Monoids with two generators and two relations
Prev:	#4622 ⟨a, b \| aaaa=a, abba=a⟩
Next:	#4624 ⟨a, b \| aaaa=a, abbb=a⟩