#6633 ⟨a, b | aab=a, baaa=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.

1aba2baa3ba2a4ba3ba4
11aba2baa3ba2a4ba3ba4
aaa2a4a3aa4a2aa3a4
bbbaba3ba2ba4ba3baba4ba2ba3
a2a2a3aa4a2aa3a2a4a
bababa2ba4ba3baba4ba2baba3ba4
a3a3a4a2aa3a2a4a3aa2
ba2ba2ba3baba4ba2baba3ba2ba4ba
a4a4aa3a2a4a3aa4a2a3
ba3ba3ba4ba2baba3ba2ba4ba3baba2
ba4ba4baba3ba2ba4ba3baba4ba2ba3

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. a5 ⇒ a [6]
2. ab ⇒ a4 [3]
3. b2 ⇒ ba3 [2]
# ab:aab=a,baaa=bb a/b - -
aaaaa=a
ab=aaaa
bb=baaa

Same cardinality

31 unique, 898 total

Σ#PresentationPropertiesDescriptionφ
7332⟨a, b | aa=1, abbb=b⟩Noc 1085
8507⟨a, b | aaa=b, abbb=1⟩Grp Com 10ℤ10536
8656⟨a, b | bb=aa, aba=a⟩Noc 101
8657⟨a, b | bb=aa, aba=b⟩Noc 101
8970⟨a, b | aa=a, bbb=ab⟩Noc 103
91654⟨a, b | aab=bb, bba=a⟩Noc 106
92012⟨a, b | aaa=b, abbb=a⟩Noc Com 10ℕ(10 = 1)59
92013⟨a, b | aaa=b, abbb=b⟩Noc Com 10ℕ(10 = 3)35
92277⟨a, b | bb=aa, aba=aa⟩Noc 101
92894⟨a, b | aa=a, bbbbb=a⟩Noc Com 10ℕ(10 = 5)17
92935⟨a, b | aa=b, bbbbb=b⟩Noc Com 10ℕ(10 = 2)57
104637⟨a, b | aaaa=b, aabb=b⟩Noc Com 10ℕ(10 = 4)20
105346⟨a, b | aaa=bb, abb=ab⟩Noc 103
105356⟨a, b | aab=aa, aba=bb⟩Noc 10
106587⟨a, b | aaa=b, abbb=bb⟩Noc Com 10ℕ(10 = 6)3
107089⟨a, b | ab=aa, baaa=bb⟩Noc 103
108619⟨a, b | aa=a, abbbbb=b⟩Noc 105
109051⟨a, b | ab=a, baaaa=bb⟩Noc 104
1110729⟨a, b | aaab=baa, abab=1⟩Grp 103
1112157⟨a, b | aaaa=aa, abba=b⟩Noc 10
1112181⟨a, b | aaaa=ab, baaa=b⟩Noc 101
1112196⟨a, b | aaaa=bb, aaab=a⟩Noc Com 107
1112197⟨a, b | aaaa=bb, aaab=b⟩Noc Com 101
1112452⟨a, b | aabb=aa, bbbb=a⟩Noc Com 10ℕ(10 = 8)3
1115738⟨a, b | aab=bb, aaaaa=b⟩Noc Com 10ℕ(10 = 7)7
1119564⟨a, b | aab=b, abbbb=aa⟩Noc Com 105
1119899⟨a, b | aaa=b, abbb=bbb⟩Noc Com 10ℕ(10 = 9)1
1120052⟨a, b | aab=b, aaaa=baa⟩Noc 10
1120253⟨a, b | aba=b, aaaa=bbb⟩Noc Com 10
1125905⟨a, b | ab=a, bbbb=baaa⟩Noc 10
1125909⟨a, b | ab=a, bbbb=bbaa⟩Noc 10

Isomorphic instances

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

20 total

Σ#PresentationMapping
106983⟨a, b | ab=aa, baaaa=b⟩φ(a) = b, φ(b) = aaa
106985⟨a, b | ab=aa, baaab=b⟩φ(a) = b, φ(b) = aaa
106987⟨a, b | ab=aa, baaba=b⟩φ(a) = b, φ(b) = aaa
106989⟨a, b | ab=aa, baabb=b⟩φ(a) = b, φ(b) = aaa
106991⟨a, b | ab=aa, babaa=b⟩φ(a) = b, φ(b) = aaa
106993⟨a, b | ab=aa, babab=b⟩φ(a) = b, φ(b) = aaa
106995⟨a, b | ab=aa, babba=b⟩φ(a) = b, φ(b) = aaa
106997⟨a, b | ab=aa, babbb=b⟩φ(a) = b, φ(b) = aaa
106999⟨a, b | ab=aa, bbaaa=b⟩φ(a) = b, φ(b) = aaa
107001⟨a, b | ab=aa, bbaab=b⟩φ(a) = b, φ(b) = aaa
107003⟨a, b | ab=aa, bbaba=b⟩φ(a) = b, φ(b) = aaa
107005⟨a, b | ab=aa, bbabb=b⟩φ(a) = b, φ(b) = aaa
107007⟨a, b | ab=aa, bbbaa=b⟩φ(a) = b, φ(b) = aaa
107009⟨a, b | ab=aa, bbbab=b⟩φ(a) = b, φ(b) = aaa
107011⟨a, b | ab=aa, bbbba=b⟩φ(a) = b, φ(b) = aaa
1112505⟨a, b | abab=aa, baab=b⟩φ(a) = b, φ(b) = aa
1112507⟨a, b | abab=aa, baba=b⟩φ(a) = b, φ(b) = aa
1112511⟨a, b | abab=aa, bbaa=b⟩φ(a) = b, φ(b) = aa
1112561⟨a, b | abba=aa, baba=b⟩φ(a) = b, φ(b) = aa
1112565⟨a, b | abba=aa, bbaa=b⟩φ(a) = b, φ(b) = aa

Anti-isomorphic instances

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

1 total

Σ#PresentationMapping
1115804⟨a, b | aba=aa, aaaab=b⟩φ(a) = b, φ(b) = aaaa