#3075 ⟨a, b | ab=a, aaaa=bb⟩

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Quick links

  1. Properties
  2. Elements
  3. Staircase diagram
  4. Cayley table
  5. Right Cayley graph
  6. Rewriting system
  7. Same cardinality
  8. Isomorphic instances

Properties

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) = xm.

Staircase diagram

Cayley table

Idempotents are shown in bold.

1aba2a3a4
11aba2a3a4
aaa2aa3a4a
bbaa4a2a3a4
a2a2a3a2a4aa2
a3a3a4a3aa2a3
a4a4aa4a2a3a4

Right 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. a5 ⇒ a [3]
2. ba ⇒ a [5]
3. ab ⇒ a [1]
4. b2 ⇒ a4 [2]
# ab:ab=a,aaaa=bb a/b
aaaaa=a
ba=a
ab=a
bb=aaaa

Same cardinality

19 unique, 1911 total

Σ#PresentationDescriptionRelated
622⟨a, b | ab=aa, bb=1⟩Finite non-commutative monoid with 6 elements44 iso, 63 anti-iso
626⟨a, b | aaa=1, abb=1⟩Isomorphic to ℤ61373 iso
7160⟨a, b | ab=aa, bb=b⟩Finite non-commutative monoid with 6 elements4 anti-iso
7245⟨a, b | aa=a, bbb=a⟩Isomorphic to ℕ(6 = 3)49 iso
7257⟨a, b | aa=b, bbb=a⟩Isomorphic to ℕ(6 = 1)61 iso
7258⟨a, b | aa=b, bbb=b⟩Isomorphic to ℕ(6 = 2)55 iso
8639⟨a, b | ab=aa, baa=b⟩Finite non-commutative monoid with 6 elements5 iso, 8 anti-iso
8644⟨a, b | ab=aa, bbb=a⟩Isomorphic to ℕ(6 = 4)46 iso
8893⟨a, b | aa=a, abbb=b⟩Finite non-commutative monoid with 6 elements19 iso
81011⟨a, b | ab=a, baa=bb⟩Finite non-commutative monoid with 6 elements12 iso, 1 anti-iso
91427⟨a, b | abba=b, baba=1⟩Finite non-Abelian group with 6 elements66 iso
91581⟨a, b | aaa=aa, abb=b⟩Finite non-commutative monoid with 6 elements5 iso
91648⟨a, b | aab=bb, abb=a⟩Finite commutative monoid with 6 elements13 iso
91686⟨a, b | abb=aa, bbb=a⟩Isomorphic to ℕ(6 = 5)49 iso
93132⟨a, b | ab=a, bbbb=aa⟩Finite commutative monoid with 6 elements5 iso
93134⟨a, b | ab=a, bbbb=ba⟩Finite non-commutative monoid with 6 elements2 iso
93193⟨a, b | ab=a, bbb=aaa⟩Finite commutative monoid with 6 elements9 iso
93199⟨a, b | ab=a, bbb=bba⟩Finite non-commutative monoid with 6 elements3 iso
1124145⟨a, b | aa=a, abbbbba=b⟩Finite commutative monoid with 6 elements

Isomorphic instances

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

14 total

Σ#PresentationMapping
108991⟨a, b | ab=a, aaaab=bb⟩φ(a) = a, φ(b) = b
108995⟨a, b | ab=a, aaaba=bb⟩φ(a) = a, φ(b) = b
109003⟨a, b | ab=a, aabaa=bb⟩φ(a) = a, φ(b) = b
109019⟨a, b | ab=a, abaaa=bb⟩φ(a) = a, φ(b) = b
1124875⟨a, b | ab=a, aaaabb=bb⟩φ(a) = a, φ(b) = b
1124883⟨a, b | ab=a, aaabab=bb⟩φ(a) = a, φ(b) = b
1124887⟨a, b | ab=a, aaabba=bb⟩φ(a) = a, φ(b) = b
1124899⟨a, b | ab=a, aabaab=bb⟩φ(a) = a, φ(b) = b
1124903⟨a, b | ab=a, aababa=bb⟩φ(a) = a, φ(b) = b
1124911⟨a, b | ab=a, aabbaa=bb⟩φ(a) = a, φ(b) = b
1124931⟨a, b | ab=a, abaaab=bb⟩φ(a) = a, φ(b) = b
1124935⟨a, b | ab=a, abaaba=bb⟩φ(a) = a, φ(b) = b
1124943⟨a, b | ab=a, ababaa=bb⟩φ(a) = a, φ(b) = b
1124959⟨a, b | ab=a, abbaaa=bb⟩φ(a) = a, φ(b) = b