#12441 ⟨a, b | aabb=aa, baab=b⟩

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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

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.

1aba2abbaa3ababa2a4aba2ba3a5aba3
11aba2abbaa3ababa2a4aba2ba3a5aba3
aaa2aba3a4abaa4a5aba2a5a2aba3a2a3
bbbaba2ba2ba3ba3ba3bbbbabababa2
a2a2a3a4a4a5a5a5a2a2a2a3a3a3a4
abababaaba2aba2aba3aba3aba3ababababaabaabaaba2
bababa2ba3ba3bbbbabababa2ba2ba2ba3
a3a3a4a5a5a2a2a2a3a3a3a4a4a4a5
abaabaaba2aba3aba3ababababaabaabaaba2aba2aba2aba3
ba2ba2ba3bbbabababa2ba2ba2ba3ba3ba3b
a4a4a5a2a2a3a3a3a4a4a4a5a5a5a2
aba2aba2aba3abababaabaabaaba2aba2aba2aba3aba3aba3ab
ba3ba3bbababa2ba2ba2ba3ba3ba3bbbba
a5a5a2a3a3a4a4a4a5a5a5a2a2a2a3
aba3aba3ababaabaaba2aba2aba2aba3aba3aba3ababababa

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. a6 ⇒ a2 [7]
2. ba4 ⇒ b [5]
3. a2b ⇒ a4 [6]
4. b2 ⇒ ba2 [4]
5. bab ⇒ ba3 [8]
# ab:aabb=aa,baab=b a/b - -
aaaaaa=aa
baaaa=b
aab=aaaa
bb=baa
bab=baaa

Same cardinality

30 unique, 313 total

Σ#PresentationPropertiesDescriptionφ
81123⟨a, b | aa=1, abbbb=b⟩Noc 1459
91682⟨a, b | aba=bb, bab=a⟩Noc 141
93034⟨a, b | aa=a, bbbb=ab⟩Noc 142
103773⟨a, b | aaaa=bb, abbb=1⟩Grp Com 14ℤ14165
105218⟨a, b | aab=bb, bbba=a⟩Noc 146
105404⟨a, b | aab=bb, aba=aa⟩Noc 141
106718⟨a, b | aab=b, bbba=aa⟩Noc 142
1112183⟨a, b | aaaa=ab, baab=b⟩Noc 143
1112206⟨a, b | aaaa=bb, abbb=a⟩Noc Com 141
1112207⟨a, b | aaaa=bb, abbb=b⟩Noc Com 141
1112499⟨a, b | abab=aa, abba=b⟩Noc 14
1114383⟨a, b | aaaa=b, aabbb=a⟩Noc Com 14ℕ(14 = 1)15
1114384⟨a, b | aaaa=b, aabbb=b⟩Noc Com 14ℕ(14 = 4)5
1115532⟨a, b | aaa=bb, abbbb=b⟩Noc Com 14ℕ(14 = 3)11
1115539⟨a, b | aaa=bb, babbb=a⟩Noc Com 14
1116020⟨a, b | aaa=ab, baab=bb⟩Noc 141
1116079⟨a, b | aaa=bb, bbbb=ab⟩Noc 14
1116293⟨a, b | aab=bb, abab=aa⟩Noc 14
1116470⟨a, b | aba=bb, bbbb=aa⟩Noc 14
1119552⟨a, b | aab=b, abbaa=aa⟩Noc 142
1120844⟨a, b | ab=aa, bbbbb=aa⟩Noc 141
1120846⟨a, b | ab=aa, bbbbb=ba⟩Noc 14
1121023⟨a, b | ab=aa, bbaa=bbb⟩Noc 141
1121040⟨a, b | ab=aa, bbbb=aaa⟩Noc 143
1121044⟨a, b | ab=aa, bbbb=baa⟩Noc 141
1121046⟨a, b | ab=aa, bbbb=bba⟩Noc 14
1121110⟨a, b | bb=aa, abab=aaa⟩Noc 142
1124186⟨a, b | aa=a, bbbbbbb=a⟩Noc Com 14ℕ(14 = 7)
1124331⟨a, b | aa=b, bbbbbbb=b⟩Noc Com 14ℕ(14 = 2)
1125055⟨a, b | ab=a, bbaaaa=bb⟩Noc 14

Isomorphic instances

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

3 total

Σ#PresentationMapping
1112443⟨a, b | aabb=aa, baba=b⟩φ(a) = a, φ(b) = b
1112447⟨a, b | aabb=aa, bbaa=b⟩φ(a) = a, φ(b) = b
1115578⟨a, b | aab=aa, baaaa=b⟩φ(a) = a, φ(b) = baa