#12196 ⟨a, b | aaaa=bb, aaab=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.

1aba2abb2a3a2bab2a2b2
11aba2abb2a3a2bab2a2b2
aaa2aba3a2bab2b2aa2b2ab
bbabb2a2bab2a2aa2b2a3b2
a2a2a3a2bb2aa2b2ab2a2aba2b
ababa2bab2aa2b2a3a2abb2ab2
b2b2ab2a2a2b2a3a2babb2aa2
a3a3b2aab2a2aba2b2a3a2ba
a2ba2baa2b2a2abb2a3a2bab2a2b2
ab2ab2a2b2a3abb2aa2bab2a2a3
a2b2a2b2abb2a2bab2a2aa2b2a3b2

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 [8]
2. b3 ⇒ a2 [4]
3. a4 ⇒ b2 [1]
4. a3b ⇒ a [2]
# ab:aaaa=bb,aaab=a ab - -
ba=ab
bbb=aa
aaaa=bb
aaab=a

Same cardinality

31 unique, 912 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
106633⟨a, b | aab=a, baaa=bb⟩Noc 1021
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
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.

7 total

Σ#PresentationMapping
1112198⟨a, b | aaaa=bb, aaba=a⟩φ(a) = a, φ(b) = b
1115767⟨a, b | aab=bb, abbbb=a⟩φ(a) = ab, φ(b) = b
1115783⟨a, b | aab=bb, babbb=a⟩φ(a) = ab, φ(b) = b
1115791⟨a, b | aab=bb, bbabb=a⟩φ(a) = ab, φ(b) = b
1115795⟨a, b | aab=bb, bbbab=a⟩φ(a) = ab, φ(b) = b
1115931⟨a, b | aba=bb, abbbb=a⟩φ(a) = ab, φ(b) = b
1115939⟨a, b | aba=bb, babbb=a⟩φ(a) = ab, φ(b) = b