#5404 ⟨a, b | aab=bb, aba=aa⟩

Up:Monoid enumeration
Prev:#5401 ⟨a, b | aab=ba, bab=bb⟩
Next:#5405 ⟨a, b | aab=bb, aba=ab⟩

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.

1aba2abbaa3a2bba2baba4a3ba5a4b
11aba2abbaa3a2bba2baba4a3ba5a4b
aaa2aba3a2ba2a4a3ba3a2ba5a4ba4a3b
bbbaa2bba2baba3a5a4ba4a3ba4a3ba5a4b
a2a2a3a2ba4a3ba3a5a4ba4a3ba4a3ba5a4b
ababa2a3ba3a2ba4a4a3ba5a4ba5a4ba4a3b
bababa2baba5a4bba2a4a3ba5a4ba5a4ba4a3b
a3a3a4a3ba5a4ba4a4a3ba5a4ba5a4ba4a3b
a2ba2ba3a4ba4a3ba5a5a4ba4a3ba4a3ba5a4b
ba2ba2a5a4ba4a3ba5a5a4ba4a3ba4a3ba5a4b
babbabba2a3ba5a4ba4a4a3ba5a4ba5a4ba4a3b
a4a4a5a4ba4a3ba5a5a4ba4a3ba4a3ba5a4b
a3ba3ba4a3ba5a4ba4a4a3ba5a4ba5a4ba4a3b
a5a5a4a3ba5a4ba4a4a3ba5a4ba5a4ba4a3b
a4ba4ba5a4ba4a3ba5a5a4ba4a3ba4a3ba5a4b

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 ⇒ a4 [6]
2. ba3 ⇒ a5 [5]
3. aba ⇒ a2 [2]
4. a5b ⇒ a3b [4]
5. b2 ⇒ a2b [1]
6. ba2b ⇒ a4b [3]
# ab:aab=bb,aba=aa a/b - -
aaaaaa=aaaa
baaa=aaaaa
aba=aa
aaaaab=aaab
bb=aab
baab=aaaab

Same cardinality

30 unique, 315 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
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
1112441⟨a, b | aabb=aa, baab=b⟩Noc 143
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.

1 total

Σ#PresentationMapping
1116355⟨a, b | aba=aa, abab=bb⟩φ(a) = a, φ(b) = b