#21039 ⟨a, b | ab=aa, bbba=bbb⟩

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Quick links

  1. Properties
  2. Elements
  3. Cayley table
  4. Right Cayley graph
  5. Left Cayley graph
  6. Rewriting system
  7. Same cardinality

Properties

Elements

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) = xm.

Cayley table

Idempotents are shown in bold.

1aba2bab2a3ba2b2ab3a4ba3b2a2ba4b2a3b2a4
11aba2bab2a3ba2b2ab3a4ba3b2a2ba4b2a3b2a4
aaa2a2a3a3a3a4a4a4a4a4a4a4a4a4a4
bbbab2ba2b2ab3ba3b2a2b3b3ba4b2a3b3b2a4b3b3
a2a2a3a3a4a4a4a4a4a4a4a4a4a4a4a4a4
bababa2ba2ba3ba3ba3ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4
b2b2b2ab3b2a2b3b3b2a3b3b3b3b2a4b3b3b3b3b3
a3a3a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4
ba2ba2ba3ba3ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4
b2ab2ab2a2b2a2b2a3b2a3b2a3b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4
b3b3b3b3b3b3b3b3b3b3b3b3b3b3b3b3b3
a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4
ba3ba3ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4
b2a2b2a2b2a3b2a3b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4
ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4
b2a3b2a3b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4
b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Rewriting system

Format:
Word to reduce:
Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Reduction strategy:
Path to normal form: 1 
1
#RuleProof
1. ab ⇒ a2 [1]
2. b3a ⇒ b3 [2]
3. b4 ⇒ b3 [4]
4. a5 ⇒ a4 [3]
# ab:ab=aa,bbba=bbb ab
ab=aa
bbba=bbb
bbbb=bbb
aaaaa=aaaa

Same cardinality

20 unique, 175 total

Σ#PresentationDescriptionRelated
8628⟨a, b | bb=aa, abab=1⟩Finite non-Abelian group with 16 elements58 iso
91331⟨a, b | aaaa=b, bbbb=1⟩Isomorphic to ℤ1667 iso
92051⟨a, b | aab=a, bbbb=b⟩Finite non-commutative monoid with 16 elements4 anti-iso
103808⟨a, b | aaab=ba, abab=1⟩Finite non-Abelian group with 16 elements7 iso
104630⟨a, b | aaaa=a, bbbb=a⟩Isomorphic to ℕ(16 = 4)1 iso
104648⟨a, b | aaaa=b, bbbb=a⟩Isomorphic to ℕ(16 = 1)5 iso
106205⟨a, b | aba=b, aaaabb=1⟩Finite non-Abelian group with 16 elements3 iso
1112164⟨a, b | aaaa=aa, bbbb=a⟩Isomorphic to ℕ(16 = 8)
1112194⟨a, b | aaaa=ab, bbbb=a⟩Isomorphic to ℕ(16 = 5)2 iso
1112212⟨a, b | aaaa=bb, bbbb=a⟩Isomorphic to ℕ(16 = 2)
1112306⟨a, b | aaab=bb, abba=a⟩Finite non-commutative monoid with 16 elements6 iso
1113251⟨a, b | bab=aab, bbb=aa⟩Finite non-commutative monoid with 16 elements
1113259⟨a, b | bab=aba, bbb=aa⟩Finite non-commutative monoid with 16 elements
1116028⟨a, b | aaa=ab, babb=bb⟩Finite non-commutative monoid with 16 elements
1116032⟨a, b | aaa=ab, bbaa=bb⟩Finite non-commutative monoid with 16 elements
1116060⟨a, b | aaa=bb, abab=aa⟩Finite non-commutative monoid with 16 elements
1116371⟨a, b | aba=aa, bbbb=ab⟩Finite non-commutative monoid with 16 elements
1118811⟨a, b | aaa=b, abbbbb=b⟩Isomorphic to ℕ(16 = 3)2 iso
1120251⟨a, b | aba=b, aaaa=abb⟩Finite non-commutative monoid with 16 elements
1124660⟨a, b | aa=a, ababab=bb⟩Finite non-commutative monoid with 16 elements