#10301 ⟨a, b | aa=1, abbbb=bab⟩

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Properties
Elements
Right Cayley graph
Left Cayley graph
Rewriting system
Same cardinality
Isomorphic instances

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 50 elements
Not cancellative, because multiplication by b⁵ is not injective:
- b⁵ ⋅ b⁸ = b⁵ and b⁵ ⋅ 1 = b⁵, however b⁸ ≠ 1

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
2 non-trivial idempotents:
- b⁸, ab⁸a
Order of generators:
- a: order 2
- b: index 5, period 8

Histogram:

order 2	1 element	a
index 1, period 1	2 elements	b⁸, ab⁸a
index 1, period 2	10 elements	ab⁶, b⁶a, ab⁸, b⁸a, ab¹⁰, ...
index 1, period 4	12 elements	ab⁵, b⁵a, b⁶, ab⁶a, ab⁷, ...
index 1, period 8	8 elements	b⁵, ab⁵a, b⁷, ab⁷a, b⁹, ...
index 2, period 2	6 elements	ab², b²a, b⁴, ab⁴, b⁴a, ...
index 2, period 4	2 elements	ab³, b³a
index 2, period 8	2 elements	b³, ab³a
index 3, period 4	4 elements	ab, ba, b², ab²a
index 5, period 8	2 elements	b, aba

Right Cayley graph

Left Cayley graph

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
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	b¹³ ⇒ b⁵	[3]
2.	bab ⇒ ab⁴	[2]
3.	a² ⇒ 1	[1]

# ab:aa=1,abbbb=bab b/a
bbbbbbbbbbbbb=bbbbb
bab=abbbb
aa=1

Same cardinality

2 unique, 9 total

Σ	#	Presentation	Description	Related
10	9747	⟨a, b \| aa=1, abbabba=b⟩	Finite non-commutative monoid with 50 elements	5 iso
10	9757	⟨a, b \| aa=1, abbbbba=b⟩	Finite non-commutative monoid with 50 elements	2 iso

Isomorphic instances

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

1 total

Σ	#	Presentation	Mapping
10	10533	⟨a, b \| aa=1, bbbb=abab⟩	φ(a) = a, φ(b) = b

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