#20686 ⟨a, b | bb=aa, aaaaab=a⟩

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Contents

  1. Properties
  2. Commutative structure
  3. Finite structure
  4. Staircase diagram
  5. Cayley table
  6. Right Cayley graph
  7. Rewriting system
  8. Same cardinality
  9. Isomorphic instances

Properties

Commutative structure

Finite structure

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.

Staircase diagram

Cayley table

Idempotents are shown in bold.

1aba2aba3a2ba4a3ba5a4ba6
11aba2aba3a2ba4a3ba5a4ba6
aaa2aba3a2ba4a3ba5a4ba6aab
bbaba2a2ba3a3ba4a4ba5aa6a2
a2a2a3a2ba4a3ba5a4ba6aaba2a2b
ababa2ba3a3ba4a4ba5aa6a2aba3
a3a3a4a3ba5a4ba6aaba2a2ba3a3b
a2ba2ba3ba4a4ba5aa6a2aba3a2ba4
a4a4a5a4ba6aaba2a2ba3a3ba4a4b
a3ba3ba4ba5aa6a2aba3a2ba4a3ba5
a5a5a6aaba2a2ba3a3ba4a4ba5a
a4ba4baa6a2aba3a2ba4a3ba5a4ba6
a6a6aba2a2ba3a3ba4a4ba5aa6a2

Right 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. ba ⇒ ab [7]
2. b2 ⇒ a2 [1]
3. a5b ⇒ a [2]
4. a7 ⇒ ab [4]
# ab:bb=aa,aaaaab=a ab - -
ba=ab
bb=aa
aaaaab=a
aaaaaaa=ab

Same cardinality

37 unique, 644 total

Σ#PresentationPropertiesDescriptionφ
7124⟨a, b | aab=a, bbb=1⟩Noc 1261
8458⟨a, b | aaaa=1, abbb=1⟩Grp Com 12ℤ12325
8645⟨a, b | ab=aa, bbb=b⟩Noc 121
91646⟨a, b | aab=bb, aba=a⟩Noc 122
91963⟨a, b | aba=b, aaabb=1⟩Grp 1230
91998⟨a, b | aaa=a, bbbb=a⟩Noc Com 12ℕ(12 = 4)6
92018⟨a, b | aaa=b, bbbb=a⟩Noc Com 12ℕ(12 = 1)27
92019⟨a, b | aaa=b, bbbb=b⟩Noc Com 12ℕ(12 = 3)23
92259⟨a, b | ab=aa, bbb=bb⟩Noc 12
105040⟨a, b | aaa=aa, bbbb=a⟩Noc Com 12ℕ(12 = 8)
105072⟨a, b | aaa=ab, bbbb=a⟩Noc Com 12ℕ(12 = 5)7
105092⟨a, b | aaa=bb, bbbb=a⟩Noc Com 12ℕ(12 = 2)43
105202⟨a, b | aab=bb, abba=a⟩Noc 1241
105330⟨a, b | aaa=ab, bab=bb⟩Noc 121
106597⟨a, b | aaa=b, bbbb=bb⟩Noc Com 12ℕ(12 = 6)4
106660⟨a, b | aab=a, bbbb=ba⟩Noc 121
106661⟨a, b | aab=a, bbbb=bb⟩Noc 121
107105⟨a, b | ab=aa, bbaa=bb⟩Noc 122
1112897⟨a, b | aab=aaa, baaa=b⟩Noc 125
1112910⟨a, b | aab=aaa, bbbb=a⟩Noc Com 12ℕ(12 = 9)2
1115428⟨a, b | aaa=aa, abbbb=b⟩Noc 12
1115996⟨a, b | aaa=ab, aabb=bb⟩Noc 122
1116057⟨a, b | aaa=bb, aabb=ab⟩Noc 124
1116104⟨a, b | aab=aa, abab=bb⟩Noc 12
1116274⟨a, b | aab=bb, aaaa=ab⟩Noc 121
1116275⟨a, b | aab=bb, aaaa=ba⟩Noc 12
1116448⟨a, b | aba=bb, aabb=aa⟩Noc 122
1119773⟨a, b | aba=b, bbbbb=aa⟩Noc Com 12
1119917⟨a, b | aaa=b, bbbb=abb⟩Noc Com 12ℕ(12 = 7)2
1120054⟨a, b | aab=b, aaaa=bba⟩Noc 12
1120787⟨a, b | ab=aa, baaaa=bb⟩Noc 127
1121086⟨a, b | bb=aa, aaaa=aba⟩Noc 124
1121092⟨a, b | bb=aa, aaab=aba⟩Noc 123
1121112⟨a, b | bb=aa, abab=aba⟩Noc 12
1124147⟨a, b | aa=a, abbbbbb=b⟩Noc 12
1124991⟨a, b | ab=a, baaaaa=bb⟩Noc 12
1125603⟨a, b | ab=a, bbaaa=bbb⟩Noc 12

Isomorphic instances

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

15 total

Σ#PresentationMapping
1120687⟨a, b | bb=aa, aaaaab=b⟩φ(a) = b, φ(b) = a
1120688⟨a, b | bb=aa, aaaaba=a⟩φ(a) = a, φ(b) = b
1120689⟨a, b | bb=aa, aaaaba=b⟩φ(a) = b, φ(b) = a
1120692⟨a, b | bb=aa, aaabaa=a⟩φ(a) = a, φ(b) = b
1120693⟨a, b | bb=aa, aaabaa=b⟩φ(a) = b, φ(b) = a
1120698⟨a, b | bb=aa, aaabbb=a⟩φ(a) = a, φ(b) = b
1120703⟨a, b | bb=aa, aababb=a⟩φ(a) = a, φ(b) = b
1120706⟨a, b | bb=aa, aabbab=a⟩φ(a) = a, φ(b) = b
1120707⟨a, b | bb=aa, aabbab=b⟩φ(a) = b, φ(b) = a
1120708⟨a, b | bb=aa, aabbba=a⟩φ(a) = a, φ(b) = b
1120709⟨a, b | bb=aa, aabbba=b⟩φ(a) = b, φ(b) = a
1120714⟨a, b | bb=aa, ababab=a⟩φ(a) = a, φ(b) = b
1120715⟨a, b | bb=aa, ababba=a⟩φ(a) = a, φ(b) = b
1120716⟨a, b | bb=aa, ababba=b⟩φ(a) = b, φ(b) = a
1120717⟨a, b | bb=aa, abbaab=a⟩φ(a) = a, φ(b) = b