#59 ⟨a, b | aa=b, ab=a⟩

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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

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.

1aa2
11aa2
aaa2a
a2a2aa2

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. a3 ⇒ a [2]
2. b ⇒ a2 [1]
# ab:aa=b,ab=a a/b - -
aaa=a
b=aa

Same cardinality

4 unique, 2146 total

Σ#PresentationPropertiesDescriptionφ
56⟨a, b | aa=b, ab=1⟩Grp Com 3ℤ32029
660⟨a, b | aa=b, ab=b⟩Noc Com 3ℕ(3 = 2)23
663⟨a, b | ab=a, ba=b⟩Noc 378
8891⟨a, b | aa=a, abba=b⟩Noc Com 312

Isomorphic instances

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

61 total

Σ#PresentationMapping
7247⟨a, b | aa=b, aaa=a⟩φ(a) = a, φ(b) = aa
7262⟨a, b | ab=a, aab=b⟩φ(a) = a, φ(b) = aa
7264⟨a, b | ab=a, aba=b⟩φ(a) = a, φ(b) = aa
8583⟨a, b | aab=a, aba=b⟩φ(a) = a, φ(b) = a
8927⟨a, b | ab=a, aabb=b⟩φ(a) = a, φ(b) = aa
8931⟨a, b | ab=a, abab=b⟩φ(a) = a, φ(b) = aa
8933⟨a, b | ab=a, abba=b⟩φ(a) = a, φ(b) = aa
91981⟨a, b | aaa=a, aaaa=b⟩φ(a) = a, φ(b) = aa
92054⟨a, b | aab=b, aaab=a⟩φ(a) = a, φ(b) = aa
92106⟨a, b | aba=b, aaab=a⟩φ(a) = a, φ(b) = aa
92108⟨a, b | aba=b, aaba=a⟩φ(a) = a, φ(b) = aa
92951⟨a, b | ab=a, aabbb=b⟩φ(a) = a, φ(b) = aa
92959⟨a, b | ab=a, ababb=b⟩φ(a) = a, φ(b) = aa
92963⟨a, b | ab=a, abbab=b⟩φ(a) = a, φ(b) = aa
92965⟨a, b | ab=a, abbba=b⟩φ(a) = a, φ(b) = aa
106239⟨a, b | aaa=a, aaaaa=b⟩φ(a) = a, φ(b) = a
106325⟨a, b | aab=a, aaabb=b⟩φ(a) = a, φ(b) = a
106329⟨a, b | aab=a, aabab=b⟩φ(a) = a, φ(b) = a
106331⟨a, b | aab=a, aabba=b⟩φ(a) = a, φ(b) = a
106337⟨a, b | aab=a, abaab=b⟩φ(a) = a, φ(b) = a
106339⟨a, b | aab=a, ababa=b⟩φ(a) = a, φ(b) = a
106343⟨a, b | aab=a, abbaa=b⟩φ(a) = a, φ(b) = a
106349⟨a, b | aab=a, abbbb=b⟩φ(a) = a, φ(b) = a
106384⟨a, b | aab=b, aaaab=a⟩φ(a) = a, φ(b) = a
106459⟨a, b | aba=a, aabba=b⟩φ(a) = a, φ(b) = a
106465⟨a, b | aba=a, ababa=b⟩φ(a) = a, φ(b) = a
106488⟨a, b | aba=b, aaaab=a⟩φ(a) = a, φ(b) = a
106490⟨a, b | aba=b, aaaba=a⟩φ(a) = a, φ(b) = a
106494⟨a, b | aba=b, aabaa=a⟩φ(a) = a, φ(b) = a
108743⟨a, b | ab=a, aabbbb=b⟩φ(a) = a, φ(b) = aa
108759⟨a, b | ab=a, ababbb=b⟩φ(a) = a, φ(b) = aa
108767⟨a, b | ab=a, abbabb=b⟩φ(a) = a, φ(b) = aa
108771⟨a, b | ab=a, abbbab=b⟩φ(a) = a, φ(b) = aa
108773⟨a, b | ab=a, abbbba=b⟩φ(a) = a, φ(b) = aa
1114424⟨a, b | aaab=a, aabbb=b⟩φ(a) = a, φ(b) = aa
1114432⟨a, b | aaab=a, ababb=b⟩φ(a) = a, φ(b) = aa
1114436⟨a, b | aaab=a, abbab=b⟩φ(a) = a, φ(b) = aa
1114438⟨a, b | aaab=a, abbba=b⟩φ(a) = a, φ(b) = aa
1114566⟨a, b | aaba=a, abbba=b⟩φ(a) = a, φ(b) = aa
1114680⟨a, b | aabb=a, aabbb=b⟩φ(a) = aa, φ(b) = a
1114692⟨a, b | aabb=a, abbab=b⟩φ(a) = aa, φ(b) = a
1114694⟨a, b | aabb=a, abbba=b⟩φ(a) = aa, φ(b) = a
1114744⟨a, b | abab=a, aabbb=b⟩φ(a) = aa, φ(b) = a
1114752⟨a, b | abab=a, ababb=b⟩φ(a) = aa, φ(b) = a
1114756⟨a, b | abab=a, abbab=b⟩φ(a) = aa, φ(b) = a
1114758⟨a, b | abab=a, abbba=b⟩φ(a) = aa, φ(b) = a
1114818⟨a, b | abba=a, abbba=b⟩φ(a) = aa, φ(b) = a
1114839⟨a, b | abba=b, aaabb=a⟩φ(a) = a, φ(b) = aa
1114845⟨a, b | abba=b, aabba=a⟩φ(a) = a, φ(b) = aa
1114849⟨a, b | abba=b, abaab=a⟩φ(a) = a, φ(b) = aa
1118689⟨a, b | aaa=a, aaaaaa=b⟩φ(a) = a, φ(b) = aa
1118962⟨a, b | aab=b, aaaaab=a⟩φ(a) = a, φ(b) = aa
1119162⟨a, b | aba=b, aaaaab=a⟩φ(a) = a, φ(b) = aa
1119164⟨a, b | aba=b, aaaaba=a⟩φ(a) = a, φ(b) = aa
1119168⟨a, b | aba=b, aaabaa=a⟩φ(a) = a, φ(b) = aa
1124395⟨a, b | ab=a, aabbbbb=b⟩φ(a) = a, φ(b) = aa
1124427⟨a, b | ab=a, ababbbb=b⟩φ(a) = a, φ(b) = aa
1124443⟨a, b | ab=a, abbabbb=b⟩φ(a) = a, φ(b) = aa
1124451⟨a, b | ab=a, abbbabb=b⟩φ(a) = a, φ(b) = aa
1124455⟨a, b | ab=a, abbbbab=b⟩φ(a) = a, φ(b) = aa
1124457⟨a, b | ab=a, abbbbba=b⟩φ(a) = a, φ(b) = aa