#1648 ⟨a, b | aab=bb, abb=a⟩

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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
aaa2a3a3a4a
bba3a4a4aa2
a2a2a3a4a4aa2
a3a3a4aaa2a3
a4a4aa2a2a3a4

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 [7]
2. ba ⇒ a3 [8]
3. ab ⇒ a3 [4]
4. b2 ⇒ a4 [9]
# ab:aab=bb,abb=a a/b
aaaaa=a
ba=aaa
ab=aaa
bb=aaaa

Same cardinality

19 unique, 1912 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
91686⟨a, b | abb=aa, bbb=a⟩Isomorphic to ℕ(6 = 5)49 iso
93075⟨a, b | ab=a, aaaa=bb⟩Finite commutative monoid with 6 elements14 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.

13 total

Σ#PresentationMapping
91652⟨a, b | aab=bb, bab=a⟩φ(a) = a, φ(b) = b
91678⟨a, b | aba=bb, abb=a⟩φ(a) = a, φ(b) = b
105192⟨a, b | aab=bb, aaab=a⟩φ(a) = a, φ(b) = b
105276⟨a, b | aba=bb, aaab=a⟩φ(a) = a, φ(b) = b
105278⟨a, b | aba=bb, aaba=a⟩φ(a) = a, φ(b) = b
1119516⟨a, b | aab=b, aaabb=aa⟩φ(a) = b, φ(b) = a
1119524⟨a, b | aab=b, aabab=aa⟩φ(a) = b, φ(b) = a
1119540⟨a, b | aab=b, abaab=aa⟩φ(a) = b, φ(b) = a
1119572⟨a, b | aab=b, baaab=aa⟩φ(a) = b, φ(b) = a
1119715⟨a, b | aba=b, aaabb=aa⟩φ(a) = b, φ(b) = a
1119722⟨a, b | aba=b, aabab=aa⟩φ(a) = b, φ(b) = a
1119726⟨a, b | aba=b, aabba=aa⟩φ(a) = b, φ(b) = a
1119738⟨a, b | aba=b, ababa=aa⟩φ(a) = b, φ(b) = a