#4413 ⟨a, b | aaab=a, bbbbb=1⟩

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 55 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
5 non-trivial idempotents:
- ba², b²a⁴, b³a⁶, a¹⁰, b⁴a⁸
Order of generators:
- a: index 1, period 10
- b: order 5

Histogram:

order 5	4 elements	b, b², b³, b⁴
index 1, period 1	5 elements	ba², b²a⁴, b³a⁶, a¹⁰, b⁴a⁸
index 1, period 2	5 elements	b³a, a⁵, b⁴a³, ba⁷, b²a⁹
index 1, period 5	20 elements	a², a⁴, b²a², ba⁴, b³a², ...
index 1, period 10	20 elements	a, ba, a³, b²a, ba³, ...

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

a¹¹ ⇒ a
ab ⇒ a⁹
b⁵ ⇒ 1

# ab:aaab=a,bbbbb=1 a/b
aaaaaaaaaaa=a
ab=aaaaaaaaa
bbbbb=1

Right Cayley graph

Left Cayley graph

Others with same cardinality

1 unique, 1 total

Σ	#	Presentation	Description	Related
11	15504	⟨a, b \| aaa=ab, bbbbb=b⟩	Finite non-commutative monoid with 55 elements

Other isomorphic instances

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

5 total

Σ	#	Presentation	Mapping
10	4801	⟨a, b \| aaa=ab, bbbbb=1⟩	φ(a) = ba, φ(b) = b
11	11279	⟨a, b \| aaabb=a, bbbbb=1⟩	φ(a) = bba, φ(b) = b
11	11475	⟨a, b \| abaab=a, bbbbb=1⟩	φ(a) = bba, φ(b) = b
11	11883	⟨a, b \| aaba=ab, bbbbb=1⟩	φ(a) = bbba, φ(b) = b
11	12755	⟨a, b \| abb=aaa, bbbbb=1⟩	φ(a) = aaa, φ(b) = b

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