#10767 ⟨a, b | aaab=bba, bbbb=1⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 148 elements
Not cancellative, because multiplication by a is not injective:
- a ⋅ a⁸ = a and a ⋅ 1 = a, however a⁸ ≠ 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:
- (ba²)², a⁸
Order of generators:
- a: index 1, period 8
- b: order 4

Histogram:

order 2	1 element	b²
order 4	2 elements	b, b³
index 1, period 1	2 elements	(ba²)², a⁸
index 1, period 2	18 elements	a²b, aba, ba², a⁴, ba⁸b, ...
index 1, period 3	16 elements	ba⁴b, ba³ba, baba³, a⁶b, a⁵ba, ...
index 1, period 4	36 elements	a², ba²b, (ba)², a⁴b, a³ba, ...
index 1, period 8	72 elements	a, ab, ba, a³, bab, ...

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(a) = 0; deg(b) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	a⁹ ⇒ a	[23]
2.	aba⁸ ⇒ ab	[21]
3.	b²a ⇒ a³b	[1]
4.	ab² ⇒ a⁴ba⁷	[24]
5.	(ab)² ⇒ a⁶ba⁶	[13]
6.	aba²b ⇒ a⁷	[22]
7.	aba³b ⇒ a³ba³	[7]
8.	aba⁴b ⇒ a²ba⁵	[28]
9.	aba⁵b ⇒ a⁷ba	[25]
10.	aba⁶b ⇒ a⁵ba⁴	[8]
11.	aba⁷b ⇒ a⁸ba²	[27]
12.	b⁴ ⇒ 1	[2]

# ab:aaab=bba,bbbb=1 a/b
aaaaaaaaa=a
abaaaaaaaa=ab
bba=aaab
abb=aaaabaaaaaaa
abab=aaaaaabaaaaaa
abaab=aaaaaaa
abaaab=aaabaaa
abaaaab=aabaaaaa
abaaaaab=aaaaaaaba
abaaaaaab=aaaaabaaaa
abaaaaaaab=aaaaaaaabaa
bbbb=1

Same cardinality

1 unique, 1 total

Σ	#	Presentation	Description	Related
11	14354	⟨a, b \| aaaa=a, abbba=b⟩	Finite non-commutative monoid with 148 elements

Isomorphic instances

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

2 total

Σ	#	Presentation	Mapping
11	17115	⟨a, b \| aaaa=1, aabbba=b⟩	φ(a) = b, φ(b) = a
11	17644	⟨a, b \| aaaa=1, aabbb=ba⟩	φ(a) = bbb, φ(b) = a

Anti-isomorphic instances

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

1 total

Σ	#	Presentation	Mapping
11	11088	⟨a, b \| babb=aaa, bbbb=1⟩	φ(a) = ab, φ(b) = b

Up:	Monoids with two generators and two relations
Prev:	#10757 ⟨a, b \| aaab=bba, abab=1⟩
Next:	#10794 ⟨a, b \| aaba=aaa, bbbb=1⟩

#10767 ⟨a, b | aaab=bba, bbbb=1⟩

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