#1011 ⟨a, b | ab=a, baa=bb⟩

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Contents

  1. Properties
  2. Finite structure
  3. Cayley table
  4. Right Cayley graph
  5. Left Cayley graph
  6. Rewriting system
  7. Same cardinality
  8. Isomorphic instances
  9. Anti-isomorphic instances

Properties

Finite structure

Elements in the center commute with all other 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.

Cayley table

Idempotents are shown in bold.

1aba2baba2
11aba2baba2
aaa2aaa2a
bbbaba2ba2baba2
a2a2aa2a2aa2
bababa2bababa2ba
ba2ba2baba2ba2baba2

Right Cayley graph

Idempotents are shown in bold.

Left 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 [3]
2. ab ⇒ a [1]
3. b2 ⇒ ba2 [2]
# ab:ab=a,baa=bb a/b - -
aaa=a
ab=a
bb=baa

Same cardinality

19 unique, 1912 total

Σ#PresentationPropertiesDescriptionφ
622⟨a, b | ab=aa, bb=1⟩Noc 6107
626⟨a, b | aaa=1, abb=1⟩Grp Com 6ℤ61373
7160⟨a, b | ab=aa, bb=b⟩Noc 64
7245⟨a, b | aa=a, bbb=a⟩Noc Com 6ℕ(6 = 3)49
7257⟨a, b | aa=b, bbb=a⟩Noc Com 6ℕ(6 = 1)61
7258⟨a, b | aa=b, bbb=b⟩Noc Com 6ℕ(6 = 2)55
8639⟨a, b | ab=aa, baa=b⟩Noc 613
8644⟨a, b | ab=aa, bbb=a⟩Noc Com 6ℕ(6 = 4)46
8893⟨a, b | aa=a, abbb=b⟩Noc 619
91427⟨a, b | abba=b, baba=1⟩Grp 666
91581⟨a, b | aaa=aa, abb=b⟩Noc 65
91648⟨a, b | aab=bb, abb=a⟩Noc Com 613
91686⟨a, b | abb=aa, bbb=a⟩Noc Com 6ℕ(6 = 5)49
93075⟨a, b | ab=a, aaaa=bb⟩Noc Com 614
93132⟨a, b | ab=a, bbbb=aa⟩Noc Com 65
93134⟨a, b | ab=a, bbbb=ba⟩Noc 62
93193⟨a, b | ab=a, bbb=aaa⟩Noc Com 69
93199⟨a, b | ab=a, bbb=bba⟩Noc 63
1124145⟨a, b | aa=a, abbbbba=b⟩Noc Com 6

Isomorphic instances

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

12 total

Σ#PresentationMapping
93111⟨a, b | ab=a, baab=bb⟩φ(a) = a, φ(b) = b
93115⟨a, b | ab=a, baba=bb⟩φ(a) = a, φ(b) = b
109063⟨a, b | ab=a, baabb=bb⟩φ(a) = a, φ(b) = b
109071⟨a, b | ab=a, babab=bb⟩φ(a) = a, φ(b) = b
109075⟨a, b | ab=a, babba=bb⟩φ(a) = a, φ(b) = b
1112509⟨a, b | abab=aa, babb=b⟩φ(a) = b, φ(b) = a
1112513⟨a, b | abab=aa, bbab=b⟩φ(a) = b, φ(b) = a
1112515⟨a, b | abab=aa, bbba=b⟩φ(a) = b, φ(b) = a
1125019⟨a, b | ab=a, baabbb=bb⟩φ(a) = a, φ(b) = b
1125035⟨a, b | ab=a, bababb=bb⟩φ(a) = a, φ(b) = b
1125043⟨a, b | ab=a, babbab=bb⟩φ(a) = a, φ(b) = b
1125047⟨a, b | ab=a, babbba=bb⟩φ(a) = a, φ(b) = b

Anti-isomorphic instances

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

1 total

Σ#PresentationMapping
1112555⟨a, b | abba=aa, abbb=b⟩φ(a) = b, φ(b) = a