#236 ⟨a, b | aa=a, aaa=b⟩

Up:Monoid enumeration
Prev:#235 ⟨a, b | aa=a, aaa=a⟩
Next:#239 ⟨a, b | aa=a, aba=a⟩

Contents

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

Properties

Commutative structure

Finite structure

The zero element z satisfies zy = yz = z 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.

Staircase diagram

Cayley table

Idempotents are shown in bold.

1a
11a
aaa

Right Cayley graph

Idempotents are shown in bold.

Rewriting system

Format:
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:
Result: 1 
1
#RuleProof
1. b ⇒ a [2]
2. a2 ⇒ a [1]
# ab:aa=a,aaa=b ab - -
b=a
aa=a

Same cardinality

1 unique, 1991 total

Σ#PresentationPropertiesDescriptionφ
41⟨a, b | aa=1, ab=1⟩Grp Com 2ℤ21990

Isomorphic instances

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

46 total

Σ#PresentationMapping
7266⟨a, b | ab=a, abb=b⟩φ(a) = b, φ(b) = b
8881⟨a, b | aa=a, aaaa=b⟩φ(a) = b, φ(b) = b
8935⟨a, b | ab=a, abbb=b⟩φ(a) = b, φ(b) = b
92027⟨a, b | aab=a, aabb=b⟩φ(a) = b, φ(b) = b
92031⟨a, b | aab=a, abab=b⟩φ(a) = b, φ(b) = b
92033⟨a, b | aab=a, abba=b⟩φ(a) = b, φ(b) = b
92035⟨a, b | aab=a, abbb=b⟩φ(a) = b, φ(b) = b
92095⟨a, b | aba=a, abba=b⟩φ(a) = b, φ(b) = b
92857⟨a, b | aa=a, aaaaa=b⟩φ(a) = b, φ(b) = b
92967⟨a, b | ab=a, abbbb=b⟩φ(a) = b, φ(b) = b
108569⟨a, b | aa=a, aaaaaa=b⟩φ(a) = b, φ(b) = b
108775⟨a, b | ab=a, abbbbb=b⟩φ(a) = b, φ(b) = b
1114416⟨a, b | aaab=a, aaabb=b⟩φ(a) = b, φ(b) = b
1114420⟨a, b | aaab=a, aabab=b⟩φ(a) = b, φ(b) = b
1114422⟨a, b | aaab=a, aabba=b⟩φ(a) = b, φ(b) = b
1114428⟨a, b | aaab=a, abaab=b⟩φ(a) = b, φ(b) = b
1114430⟨a, b | aaab=a, ababa=b⟩φ(a) = b, φ(b) = b
1114434⟨a, b | aaab=a, abbaa=b⟩φ(a) = b, φ(b) = b
1114550⟨a, b | aaba=a, aabba=b⟩φ(a) = b, φ(b) = b
1114558⟨a, b | aaba=a, ababa=b⟩φ(a) = b, φ(b) = b
1114562⟨a, b | aaba=a, abbaa=b⟩φ(a) = b, φ(b) = b
1114696⟨a, b | aabb=a, abbbb=b⟩φ(a) = b, φ(b) = b
1114760⟨a, b | abab=a, abbbb=b⟩φ(a) = b, φ(b) = b
1114835⟨a, b | abba=b, aaaab=a⟩φ(a) = b, φ(b) = b
1114837⟨a, b | abba=b, aaaba=a⟩φ(a) = b, φ(b) = b
1114841⟨a, b | abba=b, aabaa=a⟩φ(a) = b, φ(b) = b
1118847⟨a, b | aab=a, aaabbb=b⟩φ(a) = b, φ(b) = b
1118855⟨a, b | aab=a, aababb=b⟩φ(a) = b, φ(b) = b
1118859⟨a, b | aab=a, aabbab=b⟩φ(a) = b, φ(b) = b
1118861⟨a, b | aab=a, aabbba=b⟩φ(a) = b, φ(b) = b
1118863⟨a, b | aab=a, aabbbb=b⟩φ(a) = b, φ(b) = b
1118871⟨a, b | aab=a, abaabb=b⟩φ(a) = b, φ(b) = b
1118875⟨a, b | aab=a, ababab=b⟩φ(a) = b, φ(b) = b
1118877⟨a, b | aab=a, ababba=b⟩φ(a) = b, φ(b) = b
1118879⟨a, b | aab=a, ababbb=b⟩φ(a) = b, φ(b) = b
1118883⟨a, b | aab=a, abbaab=b⟩φ(a) = b, φ(b) = b
1118885⟨a, b | aab=a, abbaba=b⟩φ(a) = b, φ(b) = b
1118887⟨a, b | aab=a, abbabb=b⟩φ(a) = b, φ(b) = b
1118889⟨a, b | aab=a, abbbaa=b⟩φ(a) = b, φ(b) = b
1118891⟨a, b | aab=a, abbbab=b⟩φ(a) = b, φ(b) = b
1118893⟨a, b | aab=a, abbbba=b⟩φ(a) = b, φ(b) = b
1119115⟨a, b | aba=a, aabbba=b⟩φ(a) = b, φ(b) = b
1119127⟨a, b | aba=a, ababba=b⟩φ(a) = b, φ(b) = b
1119137⟨a, b | aba=a, abbbba=b⟩φ(a) = b, φ(b) = b
1124045⟨a, b | aa=a, aaaaaaa=b⟩φ(a) = b, φ(b) = b
1124459⟨a, b | ab=a, abbbbbb=b⟩φ(a) = b, φ(b) = b