#1599 ⟨a, b | aaa=ab, bbb=b⟩

Properties
Finite structure
Cayley table
Right Cayley graph
Left Cayley graph
Rewriting system
Same cardinality
Anti-isomorphic instances

Properties

Sum of relation sides is 9
Finite non-commutative monoid with 21 elements
Not cancellative, because multiplication by b is not injective:
- b ⋅ b² = b and b ⋅ 1 = b, however b² ≠ 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
4 non-trivial idempotents:
- b², a⁴, b²a⁴, ba⁶
Order of generators:
- a: index 3, period 4
- b: index 1, period 2

Histogram:

index 1, period 1	4 elements	b², a⁴, b²a⁴, ba⁶
index 1, period 2	4 elements	b, ba⁴, a⁶, b²a⁶
index 1, period 4	6 elements	a³, ba³, a⁵, b²a³, ba⁵, ...
index 2, period 1	1 element	ba²
index 2, period 2	2 elements	a², b²a²
index 2, period 4	2 elements	ba, b²a
index 3, period 4	1 element	a

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ba	b²	a³	ba²	b²a	a⁴	ba³	b²a²	a⁵	ba⁴	b²a³	a⁶	ba⁵	b²a⁴	ba⁶	b²a⁵	b²a⁶
1	1	a	b	a²	ba	b²	a³	ba²	b²a	a⁴	ba³	b²a²	a⁵	ba⁴	b²a³	a⁶	ba⁵	b²a⁴	ba⁶	b²a⁵	b²a⁶
a	a	a²	a³	a³	a⁴	a⁵	a⁴	a⁵	a⁶	a⁵	a⁶	a³	a⁶	a³	a⁴	a³	a⁴	a⁵	a⁵	a⁶	a³
b	b	ba	b²	ba²	b²a	b	ba³	b²a²	ba	ba⁴	b²a³	ba²	ba⁵	b²a⁴	ba³	ba⁶	b²a⁵	ba⁴	b²a⁶	ba⁵	ba⁶
a²	a²	a³	a⁴	a⁴	a⁵	a⁶	a⁵	a⁶	a³	a⁶	a³	a⁴	a³	a⁴	a⁵	a⁴	a⁵	a⁶	a⁶	a³	a⁴
ba	ba	ba²	ba³	ba³	ba⁴	ba⁵	ba⁴	ba⁵	ba⁶	ba⁵	ba⁶	ba³	ba⁶	ba³	ba⁴	ba³	ba⁴	ba⁵	ba⁵	ba⁶	ba³
b²	b²	b²a	b	b²a²	ba	b²	b²a³	ba²	b²a	b²a⁴	ba³	b²a²	b²a⁵	ba⁴	b²a³	b²a⁶	ba⁵	b²a⁴	ba⁶	b²a⁵	b²a⁶
a³	a³	a⁴	a⁵	a⁵	a⁶	a³	a⁶	a³	a⁴	a³	a⁴	a⁵	a⁴	a⁵	a⁶	a⁵	a⁶	a³	a³	a⁴	a⁵
ba²	ba²	ba³	ba⁴	ba⁴	ba⁵	ba⁶	ba⁵	ba⁶	ba³	ba⁶	ba³	ba⁴	ba³	ba⁴	ba⁵	ba⁴	ba⁵	ba⁶	ba⁶	ba³	ba⁴
b²a	b²a	b²a²	b²a³	b²a³	b²a⁴	b²a⁵	b²a⁴	b²a⁵	b²a⁶	b²a⁵	b²a⁶	b²a³	b²a⁶	b²a³	b²a⁴	b²a³	b²a⁴	b²a⁵	b²a⁵	b²a⁶	b²a³
a⁴	a⁴	a⁵	a⁶	a⁶	a³	a⁴	a³	a⁴	a⁵	a⁴	a⁵	a⁶	a⁵	a⁶	a³	a⁶	a³	a⁴	a⁴	a⁵	a⁶
ba³	ba³	ba⁴	ba⁵	ba⁵	ba⁶	ba³	ba⁶	ba³	ba⁴	ba³	ba⁴	ba⁵	ba⁴	ba⁵	ba⁶	ba⁵	ba⁶	ba³	ba³	ba⁴	ba⁵
b²a²	b²a²	b²a³	b²a⁴	b²a⁴	b²a⁵	b²a⁶	b²a⁵	b²a⁶	b²a³	b²a⁶	b²a³	b²a⁴	b²a³	b²a⁴	b²a⁵	b²a⁴	b²a⁵	b²a⁶	b²a⁶	b²a³	b²a⁴
a⁵	a⁵	a⁶	a³	a³	a⁴	a⁵	a⁴	a⁵	a⁶	a⁵	a⁶	a³	a⁶	a³	a⁴	a³	a⁴	a⁵	a⁵	a⁶	a³
ba⁴	ba⁴	ba⁵	ba⁶	ba⁶	ba³	ba⁴	ba³	ba⁴	ba⁵	ba⁴	ba⁵	ba⁶	ba⁵	ba⁶	ba³	ba⁶	ba³	ba⁴	ba⁴	ba⁵	ba⁶
b²a³	b²a³	b²a⁴	b²a⁵	b²a⁵	b²a⁶	b²a³	b²a⁶	b²a³	b²a⁴	b²a³	b²a⁴	b²a⁵	b²a⁴	b²a⁵	b²a⁶	b²a⁵	b²a⁶	b²a³	b²a³	b²a⁴	b²a⁵
a⁶	a⁶	a³	a⁴	a⁴	a⁵	a⁶	a⁵	a⁶	a³	a⁶	a³	a⁴	a³	a⁴	a⁵	a⁴	a⁵	a⁶	a⁶	a³	a⁴
ba⁵	ba⁵	ba⁶	ba³	ba³	ba⁴	ba⁵	ba⁴	ba⁵	ba⁶	ba⁵	ba⁶	ba³	ba⁶	ba³	ba⁴	ba³	ba⁴	ba⁵	ba⁵	ba⁶	ba³
b²a⁴	b²a⁴	b²a⁵	b²a⁶	b²a⁶	b²a³	b²a⁴	b²a³	b²a⁴	b²a⁵	b²a⁴	b²a⁵	b²a⁶	b²a⁵	b²a⁶	b²a³	b²a⁶	b²a³	b²a⁴	b²a⁴	b²a⁵	b²a⁶
ba⁶	ba⁶	ba³	ba⁴	ba⁴	ba⁵	ba⁶	ba⁵	ba⁶	ba³	ba⁶	ba³	ba⁴	ba³	ba⁴	ba⁵	ba⁴	ba⁵	ba⁶	ba⁶	ba³	ba⁴
b²a⁵	b²a⁵	b²a⁶	b²a³	b²a³	b²a⁴	b²a⁵	b²a⁴	b²a⁵	b²a⁶	b²a⁵	b²a⁶	b²a³	b²a⁶	b²a³	b²a⁴	b²a³	b²a⁴	b²a⁵	b²a⁵	b²a⁶	b²a³
b²a⁶	b²a⁶	b²a³	b²a⁴	b²a⁴	b²a⁵	b²a⁶	b²a⁵	b²a⁶	b²a³	b²a⁶	b²a³	b²a⁴	b²a³	b²a⁴	b²a⁵	b²a⁴	b²a⁵	b²a⁶	b²a⁶	b²a³	b²a⁴

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.	a⁷ ⇒ a³	[3]
2.	ab ⇒ a³	[1]
3.	b³ ⇒ b	[2]

# ab:aaa=ab,bbb=b a/b - -
aaaaaaa=aaa
ab=aaa
bbb=b

Same cardinality

8 unique, 63 total

Σ	#	Presentation	Properties	Description	φ
8	402	⟨a, b \| aaa=ab, bbb=1⟩	Noc 21		39
10	5309	⟨a, b \| aaa=aa, bbb=ab⟩	Noc 21
10	6265	⟨a, b \| aaa=a, abbbb=b⟩	Noc 21		2
11	11127	⟨a, b \| aaaaa=b, abbbb=1⟩	Grp Com 21	ℤ₂₁	13
11	13091	⟨a, b \| bab=aaa, abba=b⟩	Noc 21
11	13248	⟨a, b \| bab=aaa, bbb=aa⟩	Noc 21
11	15158	⟨a, b \| aab=ba, ababab=1⟩	Grp 21		1
11	19212	⟨a, b \| aba=b, baaaab=a⟩	Noc 21

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	19789	⟨a, b \| aaa=a, aaab=bbb⟩	φ(a) = b, φ(b) = a

Up:	Monoid enumeration
Prev:	#1598 ⟨a, b \| aaa=ab, bbb=a⟩
Next:	#1601 ⟨a, b \| aaa=bb, aab=b⟩

#1599 ⟨a, b | aaa=ab, bbb=b⟩

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