#236 ⟨a, b | aa=a, aaa=b⟩

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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

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

Same cardinality

1 unique, 1991 total

Σ#PresentationDescriptionRelated
41⟨a, b | aa=1, ab=1⟩Isomorphic to ℤ21990 iso

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