#12183 ⟨a, b | aaaa=ab, 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.

1aba2abb2a3ab2b3ab3b4ab4b5ab5
11aba2abb2a3ab2b3ab3b4ab4b5ab5
aaa2aba3ab3ab2abab4ab3ab5ab4abab5ab2
bbb3b2b5b4b3b2b5b4bb5b2bb3
a2a2a3ab3abab5ab4ab3abab5ab2abab3ab2ab4
ababab3ab2ab5ab4ab3ab2ab5ab4abab5ab2abab3
b2b2b4b3bb5b4b3bb5b2bb3b2b4
a3a3abab5ab3ab2abab5ab3ab2ab4ab3ab5ab4ab
ab2ab2ab4ab3abab5ab4ab3abab5ab2abab3ab2ab4
b3b3b5b4b2bb5b4b2bb3b2b4b3b5
ab3ab3ab5ab4ab2abab5ab4ab2abab3ab2ab4ab3ab5
b4b4bb5b3b2bb5b3b2b4b3b5b4b
ab4ab4abab5ab3ab2abab5ab3ab2ab4ab3ab5ab4ab
b5b5b2bb4b3b2bb4b3b5b4bb5b2
ab5ab5ab2abab4ab3ab2abab4ab3ab5ab4abab5ab2

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. b6 ⇒ b [8]
2. ba ⇒ b3 [7]
3. a2b ⇒ ab3 [6]
4. a4 ⇒ ab [1]
# ab:aaaa=ab,baab=b b/a - -
bbbbbb=b
ba=bbb
aab=abbb
aaaa=ab

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
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.

3 total

Σ#PresentationMapping
1112185⟨a, b | aaaa=ab, baba=b⟩φ(a) = a, φ(b) = b
1112189⟨a, b | aaaa=ab, bbaa=b⟩φ(a) = a, φ(b) = b
1120044⟨a, b | aab=a, bbbb=baa⟩φ(a) = bbb, φ(b) = a