#3767 ⟨a, b | aaaa=ab, bbbb=1⟩

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 52 elements

Elements

Elements in the center commute with all other elements.
An idempotent element x satisfies x² = x.
The order of x is the least n (if it exists) such that xⁿ = 1.
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:
- b³a³, b²a⁶, ba⁹, a¹²
Order of generators:
- a: index 1, period 12
- b: order 4

Histogram:

order 2	1 element	b²
order 4	2 elements	b, b³
index 1, period 1	4 elements	b³a³, b²a⁶, ba⁹, a¹²
index 1, period 2	4 elements	ba³, a⁶, b³a⁹, b²a¹²
index 1, period 3	8 elements	ba, a⁴, b²a², ba⁵, a⁸, ...
index 1, period 4	8 elements	a³, b²a³, ba⁶, a⁹, b³a⁶, ...
index 1, period 6	8 elements	a², b³a, b²a⁴, ba⁷, b³a⁵, ...
index 1, period 12	16 elements	a, ba², b²a, a⁵, ba⁴, ...

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(a) = 0; deg(b) = 1

a¹³ ⇒ a
ab ⇒ a⁴
b⁴ ⇒ 1

# ab:aaaa=ab,bbbb=1 a/b
aaaaaaaaaaaaa=a
ab=aaaa
bbbb=1

Right Cayley graph

Left Cayley graph

Others with same cardinality

1 unique, 1 total

Σ	#	Presentation	Description	Related
11	12195	⟨a, b \| aaaa=ab, bbbb=b⟩	Finite non-commutative monoid with 52 elements

Other anti-isomorphic instances

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

3 total

Σ	#	Presentation	Mapping
10	5721	⟨a, b \| aaaa=1, abbbb=b⟩	φ(a) = bbb, φ(b) = a
11	17129	⟨a, b \| aaaa=1, ababbb=b⟩	φ(a) = b, φ(b) = aa
11	17133	⟨a, b \| aaaa=1, abbabb=b⟩	φ(a) = b, φ(b) = aa

Up:	Monoids with two generators and two relations
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Next:	#3770 ⟨a, b \| aaaa=bb, aabb=1⟩