#5335 ⟨a, b | aaa=ab, bbb=aa⟩

Up:Monoid enumeration
Prev:#5334 ⟨a, b | aaa=ab, bba=bb⟩
Next:#5336 ⟨a, b | aaa=ab, bbb=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
  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.

1aba2bab2a3b2aa4a5a6
11aba2bab2a3b2aa4a5a6
aaa2a3a3a4a5a4a6a5a6a3
bbbab2a4b2aa2a5a3a6a3a4
a2a2a3a4a4a5a6a5a3a6a3a4
babaa4a5a5a6a3a6a4a3a4a5
b2b2b2aa2a6a3a4a3a5a4a5a6
a3a3a4a5a5a6a3a6a4a3a4a5
b2ab2aa6a3a3a4a5a4a6a5a6a3
a4a4a5a6a6a3a4a3a5a4a5a6
a5a5a6a3a3a4a5a4a6a5a6a3
a6a6a3a4a4a5a6a5a3a6a3a4

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. a7 ⇒ a3 [3]
2. ba2 ⇒ a4 [4]
3. ab ⇒ a3 [1]
4. b3 ⇒ a2 [2]
# ab:aaa=ab,bbb=aa a/b - -
aaaaaaa=aaa
baa=aaaa
ab=aaa
bbb=aa

Same cardinality

29 unique, 477 total

Σ#PresentationPropertiesDescriptionφ
91472⟨a, b | aaa=bb, abbb=1⟩Grp Com 11ℤ11259
104155⟨a, b | abb=aaa, bba=b⟩Noc 114
105086⟨a, b | aaa=bb, abbb=a⟩Noc Com 1111
105087⟨a, b | aaa=bb, abbb=b⟩Noc Com 113
105111⟨a, b | aab=aa, baaa=b⟩Noc 117
105322⟨a, b | aaa=ab, abb=bb⟩Noc 113
105410⟨a, b | aab=bb, bab=aa⟩Noc 112
106292⟨a, b | aaa=b, aabbb=a⟩Noc Com 11ℕ(11 = 1)35
106293⟨a, b | aaa=b, aabbb=b⟩Noc Com 11ℕ(11 = 3)11
106380⟨a, b | aab=a, bbbbb=a⟩Noc Com 11ℕ(11 = 5)1
106834⟨a, b | aba=b, abb=aaa⟩Noc 11
107114⟨a, b | ab=aa, bbbb=aa⟩Noc 111
107116⟨a, b | ab=aa, bbbb=ba⟩Noc 11
108691⟨a, b | aa=b, abbbbb=b⟩Noc Com 11ℕ(11 = 2)32
108777⟨a, b | ab=a, baaaaa=b⟩Noc 1125
109083⟨a, b | ab=a, bbaaa=bb⟩Noc 114
1112179⟨a, b | aaaa=ab, abbb=b⟩Noc Com 11ℕ(11 = 4)20
1115607⟨a, b | aab=aa, bbbbb=a⟩Noc Com 11ℕ(11 = 10)1
1115671⟨a, b | aab=ab, bbbbb=a⟩Noc Com 11ℕ(11 = 6)8
1116148⟨a, b | aab=ab, aaaa=bb⟩Noc 111
1116212⟨a, b | aab=ba, aaaa=bb⟩Noc 11
1118715⟨a, b | aaa=a, aabbba=b⟩Noc 11
1119070⟨a, b | aab=b, bbabbb=a⟩Noc Com 114
1120096⟨a, b | aab=b, abba=aaa⟩Noc 111
1120112⟨a, b | aab=b, baaa=aaa⟩Noc 11
1120168⟨a, b | aab=b, bbbb=aaa⟩Noc Com 11
1120885⟨a, b | bb=aa, aaaaa=ab⟩Noc 1115
1125539⟨a, b | ab=a, baaaa=bbb⟩Noc 11
1125635⟨a, b | ab=a, bbbaa=bbb⟩Noc 11

Isomorphic instances

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

2 total

Σ#PresentationMapping
1116050⟨a, b | aaa=bb, aaab=ba⟩φ(a) = b, φ(b) = a
1116054⟨a, b | aaa=bb, aaba=ba⟩φ(a) = b, φ(b) = a

Anti-isomorphic instances

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

1 total

Σ#PresentationMapping
1116049⟨a, b | aaa=bb, aaab=ab⟩φ(a) = b, φ(b) = a