#21046 ⟨a, b | ab=aa, bbbb=bba⟩

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Properties
Elements
Cayley table
Right Cayley graph
Left Cayley graph
Rewriting system
Same cardinality

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 14 elements
Not cancellative, because multiplication by a⁴ is not injective:
- a⁴ ⋅ a = a⁴ and a⁴ ⋅ 1 = a⁴, however a ≠ 1

Elements in the center commute with all other elements.
A left zero element x satisfies xy = x for all y.
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 left zero elements:
- ab³, bab³, b⁵
3 non-trivial idempotents:
- ab³, bab³, b⁵
Order of generators:
- a: index 4, period 1
- b: index 5, period 1
Histogram:

index 1, period 1 3 elements ab³, bab³, b⁵

index 2, period 1 6 elements ab, ab², bab, b³, bab², ...

index 3, period 1 2 elements ba, b²

index 4, period 1 1 element a

index 5, period 1 1 element b

Cayley table

Idempotents are shown in bold.

	1	a	b	ab	ba	b²	ab²	bab	b³	ab³	bab²	b⁴	bab³	b⁵
1	1	a	b	ab	ba	b²	ab²	bab	b³	ab³	bab²	b⁴	bab³	b⁵
a	a	ab	ab	ab²	ab²	ab²	ab³	ab³	ab³	ab³	ab³	ab³	ab³	ab³
b	b	ba	b²	bab	b⁴	b³	bab²	b⁵	b⁴	bab³	b⁵	b⁵	b⁵	b⁵
ab	ab	ab²	ab²	ab³	ab³	ab³	ab³	ab³	ab³	ab³	ab³	ab³	ab³	ab³
ba	ba	bab	bab	bab²	bab²	bab²	bab³	bab³	bab³	bab³	bab³	bab³	bab³	bab³
b²	b²	b⁴	b³	b⁵	b⁵	b⁴	b⁵	b⁵	b⁵	b⁵	b⁵	b⁵	b⁵	b⁵
ab²	ab²	ab³	ab³	ab³	ab³	ab³	ab³	ab³	ab³	ab³	ab³	ab³	ab³	ab³
bab	bab	bab²	bab²	bab³	bab³	bab³	bab³	bab³	bab³	bab³	bab³	bab³	bab³	bab³
b³	b³	b⁵	b⁴	b⁵	b⁵	b⁵	b⁵	b⁵	b⁵	b⁵	b⁵	b⁵	b⁵	b⁵
ab³	ab³	ab³	ab³	ab³	ab³	ab³	ab³	ab³	ab³	ab³	ab³	ab³	ab³	ab³
bab²	bab²	bab³	bab³	bab³	bab³	bab³	bab³	bab³	bab³	bab³	bab³	bab³	bab³	bab³
b⁴	b⁴	b⁵	b⁵	b⁵	b⁵	b⁵	b⁵	b⁵	b⁵	b⁵	b⁵	b⁵	b⁵	b⁵
bab³	bab³	bab³	bab³	bab³	bab³	bab³	bab³	bab³	bab³	bab³	bab³	bab³	bab³	bab³
b⁵	b⁵	b⁵	b⁵	b⁵	b⁵	b⁵	b⁵	b⁵	b⁵	b⁵	b⁵	b⁵	b⁵	b⁵

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

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.	b⁶ ⇒ b⁵	[5]
2.	ab⁴ ⇒ ab³	[4]
3.	b²a ⇒ b⁴	[2]
4.	a² ⇒ ab	[1]
5.	aba ⇒ ab²	[3]

# ab:ab=aa,bbbb=bba b/a
bbbbbb=bbbbb
abbbb=abbb
bba=bbbb
aa=ab
aba=abb

Same cardinality

30 unique, 316 total

Σ	#	Presentation	Description	Related
8	1123	⟨a, b \| aa=1, abbbb=b⟩	Finite non-commutative monoid with 14 elements	42 iso, 17 anti-iso
9	1682	⟨a, b \| aba=bb, bab=a⟩	Finite non-commutative monoid with 14 elements	1 iso
9	3034	⟨a, b \| aa=a, bbbb=ab⟩	Finite non-commutative monoid with 14 elements	2 iso
10	3773	⟨a, b \| aaaa=bb, abbb=1⟩	Isomorphic to ℤ₁₄	165 iso
10	5218	⟨a, b \| aab=bb, bbba=a⟩	Finite non-commutative monoid with 14 elements	5 iso, 1 anti-iso
10	5404	⟨a, b \| aab=bb, aba=aa⟩	Finite non-commutative monoid with 14 elements	1 iso
10	6718	⟨a, b \| aab=b, bbba=aa⟩	Finite non-commutative monoid with 14 elements	2 iso
11	12183	⟨a, b \| aaaa=ab, baab=b⟩	Finite non-commutative monoid with 14 elements	3 iso
11	12206	⟨a, b \| aaaa=bb, abbb=a⟩	Finite commutative monoid with 14 elements	1 iso
11	12207	⟨a, b \| aaaa=bb, abbb=b⟩	Finite commutative monoid with 14 elements	1 iso
11	12441	⟨a, b \| aabb=aa, baab=b⟩	Finite non-commutative monoid with 14 elements	3 iso
11	12499	⟨a, b \| abab=aa, abba=b⟩	Finite non-commutative monoid with 14 elements
11	14383	⟨a, b \| aaaa=b, aabbb=a⟩	Isomorphic to ℕ_{(14 = 1)}	15 iso
11	14384	⟨a, b \| aaaa=b, aabbb=b⟩	Isomorphic to ℕ_{(14 = 4)}	5 iso
11	15532	⟨a, b \| aaa=bb, abbbb=b⟩	Isomorphic to ℕ_{(14 = 3)}	11 iso
11	15539	⟨a, b \| aaa=bb, babbb=a⟩	Finite commutative monoid with 14 elements
11	16020	⟨a, b \| aaa=ab, baab=bb⟩	Finite non-commutative monoid with 14 elements	1 iso
11	16079	⟨a, b \| aaa=bb, bbbb=ab⟩	Finite non-commutative monoid with 14 elements
11	16293	⟨a, b \| aab=bb, abab=aa⟩	Finite non-commutative monoid with 14 elements
11	16470	⟨a, b \| aba=bb, bbbb=aa⟩	Finite non-commutative monoid with 14 elements
11	19552	⟨a, b \| aab=b, abbaa=aa⟩	Finite non-commutative monoid with 14 elements	2 iso
11	20844	⟨a, b \| ab=aa, bbbbb=aa⟩	Finite non-commutative monoid with 14 elements	1 iso
11	20846	⟨a, b \| ab=aa, bbbbb=ba⟩	Finite non-commutative monoid with 14 elements
11	21023	⟨a, b \| ab=aa, bbaa=bbb⟩	Finite non-commutative monoid with 14 elements	1 iso
11	21040	⟨a, b \| ab=aa, bbbb=aaa⟩	Finite non-commutative monoid with 14 elements	3 iso
11	21044	⟨a, b \| ab=aa, bbbb=baa⟩	Finite non-commutative monoid with 14 elements	1 iso
11	21110	⟨a, b \| bb=aa, abab=aaa⟩	Finite non-commutative monoid with 14 elements	2 anti-iso
11	24186	⟨a, b \| aa=a, bbbbbbb=a⟩	Isomorphic to ℕ_{(14 = 7)}
11	24331	⟨a, b \| aa=b, bbbbbbb=b⟩	Isomorphic to ℕ_{(14 = 2)}
11	25055	⟨a, b \| ab=a, bbaaaa=bb⟩	Finite non-commutative monoid with 14 elements

Up:	Monoids with two generators and two relations
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Next:	#21047 ⟨a, b \| ab=aa, bbbb=bbb⟩

index 1, period 1	3 elements	ab³, bab³, b⁵
index 2, period 1	6 elements	ab, ab², bab, b³, bab², ...
index 3, period 1	2 elements	ba, b²
index 4, period 1	1 element	a
index 5, period 1	1 element	b