#8910 ⟨a, b | aa=a, bbbbb=ab⟩

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

Properties

Sum of relation sides is 10
Finite non-commutative monoid with 18 elements
Not cancellative, because multiplication by a is not injective:
- a ⋅ a = a and a ⋅ 1 = a, however a ≠ 1

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:
- a, ab⁴, ab⁴a
Order of generators:
- a: index 1, period 1
- b: index 5, period 4

Histogram:

index 1, period 1	3 elements	a, ab⁴, ab⁴a
index 1, period 2	2 elements	ab², ab²a
index 1, period 4	4 elements	ab, aba, ab³, ab³a
index 2, period 1	2 elements	b⁴, b⁴a
index 2, period 2	1 element	b²a
index 2, period 4	3 elements	ba, b³, b³a
index 3, period 2	1 element	b²
index 5, period 4	1 element	b

Cayley table

Idempotents are shown in bold.

	1	a	b	ab	ba	b²	aba	ab²	b²a	b³	ab²a	ab³	b³a	b⁴	ab³a	ab⁴	b⁴a	ab⁴a
1	1	a	b	ab	ba	b²	aba	ab²	b²a	b³	ab²a	ab³	b³a	b⁴	ab³a	ab⁴	b⁴a	ab⁴a
a	a	a	ab	ab	aba	ab²	aba	ab²	ab²a	ab³	ab²a	ab³	ab³a	ab⁴	ab³a	ab⁴	ab⁴a	ab⁴a
b	b	ba	b²	ab²	b²a	b³	ab²a	ab³	b³a	b⁴	ab³a	ab⁴	b⁴a	ab	ab⁴a	ab	aba	aba
ab	ab	aba	ab²	ab²	ab²a	ab³	ab²a	ab³	ab³a	ab⁴	ab³a	ab⁴	ab⁴a	ab	ab⁴a	ab	aba	aba
ba	ba	ba	ab²	ab²	ab²a	ab³	ab²a	ab³	ab³a	ab⁴	ab³a	ab⁴	ab⁴a	ab	ab⁴a	ab	aba	aba
b²	b²	b²a	b³	ab³	b³a	b⁴	ab³a	ab⁴	b⁴a	ab	ab⁴a	ab	aba	ab²	aba	ab²	ab²a	ab²a
aba	aba	aba	ab²	ab²	ab²a	ab³	ab²a	ab³	ab³a	ab⁴	ab³a	ab⁴	ab⁴a	ab	ab⁴a	ab	aba	aba
ab²	ab²	ab²a	ab³	ab³	ab³a	ab⁴	ab³a	ab⁴	ab⁴a	ab	ab⁴a	ab	aba	ab²	aba	ab²	ab²a	ab²a
b²a	b²a	b²a	ab³	ab³	ab³a	ab⁴	ab³a	ab⁴	ab⁴a	ab	ab⁴a	ab	aba	ab²	aba	ab²	ab²a	ab²a
b³	b³	b³a	b⁴	ab⁴	b⁴a	ab	ab⁴a	ab	aba	ab²	aba	ab²	ab²a	ab³	ab²a	ab³	ab³a	ab³a
ab²a	ab²a	ab²a	ab³	ab³	ab³a	ab⁴	ab³a	ab⁴	ab⁴a	ab	ab⁴a	ab	aba	ab²	aba	ab²	ab²a	ab²a
ab³	ab³	ab³a	ab⁴	ab⁴	ab⁴a	ab	ab⁴a	ab	aba	ab²	aba	ab²	ab²a	ab³	ab²a	ab³	ab³a	ab³a
b³a	b³a	b³a	ab⁴	ab⁴	ab⁴a	ab	ab⁴a	ab	aba	ab²	aba	ab²	ab²a	ab³	ab²a	ab³	ab³a	ab³a
b⁴	b⁴	b⁴a	ab	ab	aba	ab²	aba	ab²	ab²a	ab³	ab²a	ab³	ab³a	ab⁴	ab³a	ab⁴	ab⁴a	ab⁴a
ab³a	ab³a	ab³a	ab⁴	ab⁴	ab⁴a	ab	ab⁴a	ab	aba	ab²	aba	ab²	ab²a	ab³	ab²a	ab³	ab³a	ab³a
ab⁴	ab⁴	ab⁴a	ab	ab	aba	ab²	aba	ab²	ab²a	ab³	ab²a	ab³	ab³a	ab⁴	ab³a	ab⁴	ab⁴a	ab⁴a
b⁴a	b⁴a	b⁴a	ab	ab	aba	ab²	aba	ab²	ab²a	ab³	ab²a	ab³	ab³a	ab⁴	ab³a	ab⁴	ab⁴a	ab⁴a
ab⁴a	ab⁴a	ab⁴a	ab	ab	aba	ab²	aba	ab²	ab²a	ab³	ab²a	ab³	ab³a	ab⁴	ab³a	ab⁴	ab⁴a	ab⁴a

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 shortlex with a < b
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	a² ⇒ a	[1]
2.	bab ⇒ ab²	[3]
3.	b⁵ ⇒ ab	[2]

# ab:aa=a,bbbbb=ab ab - -
aa=a
bab=abb
bbbbb=ab

Same cardinality

17 unique, 103 total

Σ	#	Presentation	Properties	Description	φ
8	1121	⟨a, b \| aa=1, abbba=b⟩	Noc 18		17
9	3387	⟨a, b \| aa=1, abbbbb=b⟩	Noc 18		31
10	5521	⟨a, b \| aaab=1, bbbbbb=1⟩	Grp Com 18	ℤ₁₈	33
10	6732	⟨a, b \| aba=a, aaab=bb⟩	Noc 18
10	6788	⟨a, b \| aba=b, baab=aa⟩	Noc 18
10	6795	⟨a, b \| aba=b, bbbb=aa⟩	Noc 18
10	7039	⟨a, b \| bb=aa, aaaba=b⟩	Noc 18		2
11	12187	⟨a, b \| aaaa=ab, babb=b⟩	Noc 18		2
11	15797	⟨a, b \| aab=bb, bbbba=a⟩	Noc 18
11	16042	⟨a, b \| aaa=ab, bbbb=ab⟩	Noc 18
11	16043	⟨a, b \| aaa=ab, bbbb=ba⟩	Noc 18
11	16313	⟨a, b \| aab=bb, baba=aa⟩	Noc 18
11	18758	⟨a, b \| aaa=a, bbbbbb=a⟩	Noc Com 18	ℕ_{(18 = 6)}
11	18830	⟨a, b \| aaa=b, bbbbbb=a⟩	Noc Com 18	ℕ_{(18 = 1)}
11	18831	⟨a, b \| aaa=b, bbbbbb=b⟩	Noc Com 18	ℕ_{(18 = 3)}
11	19624	⟨a, b \| aab=b, bbbba=aa⟩	Noc 18
11	20914	⟨a, b \| bb=aa, ababa=aa⟩	Noc 18		1

Isomorphic instances

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

1 total

Σ	#	Presentation	Mapping
11	25255	⟨a, b \| aa=a, bbbbb=aab⟩	φ(a) = a, φ(b) = b

Up:	Monoid enumeration
Prev:	#8908 ⟨a, b \| aa=a, bbabb=bb⟩
Next:	#8911 ⟨a, b \| aa=a, bbbbb=bb⟩

#8910 ⟨a, b | aa=a, bbbbb=ab⟩

Contents