#16036 ⟨a, b | aaa=ab, bbab=bb⟩

Properties
Finite structure
Cayley table
Right Cayley graph
Left Cayley graph
Rewriting system
Same cardinality
Isomorphic instances

Properties

Sum of relation sides is 11
Finite non-commutative monoid with 19 elements
Not cancellative, because multiplication by a is not injective:
- a ⋅ b = a³ and a ⋅ a² = a³, however b ≠ a²

Finite structure

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.

1 element center:
- 1
3 non-trivial idempotents:
- b²a², a⁶, ba⁷
Order of generators:
- a: index 5, period 3
- b: index 2, period 3

Histogram:

index 1, period 1	3 elements	b²a², a⁶, ba⁷
index 1, period 3	6 elements	b², b²a, a⁵, ba⁵, a⁷, ...
index 2, period 1	2 elements	a³, ba⁴
index 2, period 3	4 elements	b, ba², a⁴, ba³
index 3, period 1	1 element	ba
index 3, period 3	1 element	a²
index 5, period 3	1 element	a

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ba	b²	a³	ba²	b²a	a⁴	ba³	b²a²	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	ba⁷
1	1	a	b	a²	ba	b²	a³	ba²	b²a	a⁴	ba³	b²a²	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	ba⁷
a	a	a²	a³	a³	a⁴	a⁵	a⁴	a⁵	a⁶	a⁵	a⁶	a⁷	a⁶	a⁷	a⁷	a⁵	a⁵	a⁶	a⁷
b	b	ba	b²	ba²	b²a	b²a²	ba³	b²a²	b²	ba⁴	b²	b²a	ba⁵	b²a	ba⁶	b²a²	ba⁷	b²	b²a
a²	a²	a³	a⁴	a⁴	a⁵	a⁶	a⁵	a⁶	a⁷	a⁶	a⁷	a⁵	a⁷	a⁵	a⁵	a⁶	a⁶	a⁷	a⁵
ba	ba	ba²	ba³	ba³	ba⁴	ba⁵	ba⁴	ba⁵	ba⁶	ba⁵	ba⁶	ba⁷	ba⁶	ba⁷	ba⁷	ba⁵	ba⁵	ba⁶	ba⁷
b²	b²	b²a	b²a²	b²a²	b²	b²a	b²	b²a	b²a²	b²a	b²a²	b²	b²a²	b²	b²	b²a	b²a	b²a²	b²
a³	a³	a⁴	a⁵	a⁵	a⁶	a⁷	a⁶	a⁷	a⁵	a⁷	a⁵	a⁶	a⁵	a⁶	a⁶	a⁷	a⁷	a⁵	a⁶
ba²	ba²	ba³	ba⁴	ba⁴	ba⁵	ba⁶	ba⁵	ba⁶	ba⁷	ba⁶	ba⁷	ba⁵	ba⁷	ba⁵	ba⁵	ba⁶	ba⁶	ba⁷	ba⁵
b²a	b²a	b²a²	b²	b²	b²a	b²a²	b²a	b²a²	b²	b²a²	b²	b²a	b²	b²a	b²a	b²a²	b²a²	b²	b²a
a⁴	a⁴	a⁵	a⁶	a⁶	a⁷	a⁵	a⁷	a⁵	a⁶	a⁵	a⁶	a⁷	a⁶	a⁷	a⁷	a⁵	a⁵	a⁶	a⁷
ba³	ba³	ba⁴	ba⁵	ba⁵	ba⁶	ba⁷	ba⁶	ba⁷	ba⁵	ba⁷	ba⁵	ba⁶	ba⁵	ba⁶	ba⁶	ba⁷	ba⁷	ba⁵	ba⁶
b²a²	b²a²	b²	b²a	b²a	b²a²	b²	b²a²	b²	b²a	b²	b²a	b²a²	b²a	b²a²	b²a²	b²	b²	b²a	b²a²
a⁵	a⁵	a⁶	a⁷	a⁷	a⁵	a⁶	a⁵	a⁶	a⁷	a⁶	a⁷	a⁵	a⁷	a⁵	a⁵	a⁶	a⁶	a⁷	a⁵
ba⁴	ba⁴	ba⁵	ba⁶	ba⁶	ba⁷	ba⁵	ba⁷	ba⁵	ba⁶	ba⁵	ba⁶	ba⁷	ba⁶	ba⁷	ba⁷	ba⁵	ba⁵	ba⁶	ba⁷
a⁶	a⁶	a⁷	a⁵	a⁵	a⁶	a⁷	a⁶	a⁷	a⁵	a⁷	a⁵	a⁶	a⁵	a⁶	a⁶	a⁷	a⁷	a⁵	a⁶
ba⁵	ba⁵	ba⁶	ba⁷	ba⁷	ba⁵	ba⁶	ba⁵	ba⁶	ba⁷	ba⁶	ba⁷	ba⁵	ba⁷	ba⁵	ba⁵	ba⁶	ba⁶	ba⁷	ba⁵
a⁷	a⁷	a⁵	a⁶	a⁶	a⁷	a⁵	a⁷	a⁵	a⁶	a⁵	a⁶	a⁷	a⁶	a⁷	a⁷	a⁵	a⁵	a⁶	a⁷
ba⁶	ba⁶	ba⁷	ba⁵	ba⁵	ba⁶	ba⁷	ba⁶	ba⁷	ba⁵	ba⁷	ba⁵	ba⁶	ba⁵	ba⁶	ba⁶	ba⁷	ba⁷	ba⁵	ba⁶
ba⁷	ba⁷	ba⁵	ba⁶	ba⁶	ba⁷	ba⁵	ba⁷	ba⁵	ba⁶	ba⁵	ba⁶	ba⁷	ba⁶	ba⁷	ba⁷	ba⁵	ba⁵	ba⁶	ba⁷

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Rewriting system

Format:	Pretty Plain
Word:
	Enter a word above to compute its normal form. Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Strategy:	Leftmost Rightmost
Result:	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.	a⁸ ⇒ a⁵	[3]
2.	ab ⇒ a³	[1]
3.	b²a³ ⇒ b²	[2]
4.	b³ ⇒ b²a²	[4]

# ab:aaa=ab,bbab=bb a/b - -
aaaaaaaa=aaaaa
ab=aaa
bbaaa=bb
bbb=bbaa

Same cardinality

5 unique, 28 total

Σ	#	Presentation	Properties	Description	φ
10	6263	⟨a, b \| aaa=a, abbba=b⟩	Noc 19
10	6838	⟨a, b \| aba=b, bab=aaa⟩	Noc 19
11	13340	⟨a, b \| aaaab=1, abbbbb=1⟩	Grp Com 19	ℤ₁₉	20
11	14501	⟨a, b \| aaab=b, abbba=a⟩	Noc 19		3
11	16053	⟨a, b \| aaa=bb, aaba=ab⟩	Noc 19

Isomorphic instances

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

1 total

Σ	#	Presentation	Mapping
11	16040	⟨a, b \| aaa=ab, bbba=bb⟩	φ(a) = a, φ(b) = b

Up:	Monoid enumeration
Prev:	#16035 ⟨a, b \| aaa=ab, bbab=ba⟩
Next:	#16037 ⟨a, b \| aaa=ab, bbba=aa⟩

#16036 ⟨a, b | aaa=ab, bbab=bb⟩

Contents