#6824 ⟨a, b | aab=b, bbb=aaa⟩

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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.

1aba2bab2b2ab3b3a
11aba2bab2b2ab3b3a
aaa2bab3bb2ab2b3ab3
bbbab2bb2ab3b3abab
a2a2b3bb3abab2b2ab3b3a
bababb2abab2b3ab3bba
b2b2b2ab3b2b3ababb2ab2
b2ab2ab2b3ab2ab3bbab2b2a
b3b3b3abab3bb2ab2b3ab3
b3ab3ab3bb3abab2b2ab3b3a

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. ab ⇒ ba [11]
2. ba2 ⇒ b [12]
3. a3 ⇒ b3 [2]
4. b4 ⇒ ba [13]
# ab:aab=b,bbb=aaa ba - -
ab=ba
baa=b
aaa=bbb
bbbb=ba

Same cardinality

26 unique, 937 total

Σ#PresentationPropertiesDescriptionφ
7118⟨a, b | aaa=b, bbb=1⟩Grp Com 9ℤ9562
8568⟨a, b | aaa=a, abb=b⟩Noc 95
8571⟨a, b | aaa=a, bbb=a⟩Noc Com 9ℕ(9 = 3)48
8581⟨a, b | aaa=b, bbb=a⟩Noc Com 9ℕ(9 = 1)62
8601⟨a, b | aba=b, bab=a⟩Noc 93
91584⟨a, b | aaa=aa, bbb=a⟩Noc Com 9ℕ(9 = 6)16
91595⟨a, b | aaa=ab, bab=b⟩Noc 99
91598⟨a, b | aaa=ab, bbb=a⟩Noc Com 9ℕ(9 = 4)40
91608⟨a, b | aaa=bb, bbb=a⟩Noc Com 9ℕ(9 = 2)70
91991⟨a, b | aaa=a, abba=b⟩Noc 93
92064⟨a, b | aab=b, abba=a⟩Noc 938
92255⟨a, b | ab=aa, bba=bb⟩Noc 9
104125⟨a, b | aab=aaa, baa=b⟩Noc 92
104130⟨a, b | aab=aaa, bbb=a⟩Noc Com 9ℕ(9 = 7)1
104156⟨a, b | abb=aaa, bbb=a⟩Noc Com 9ℕ(9 = 5)12
105035⟨a, b | aaa=aa, abbb=b⟩Noc 93
105079⟨a, b | aaa=bb, aaba=b⟩Noc Com 98
105124⟨a, b | aab=aa, bbbb=a⟩Noc Com 9ℕ(9 = 8)5
105339⟨a, b | aaa=bb, aab=ab⟩Noc 91
107139⟨a, b | bb=aa, aaaa=ab⟩Noc 93
109327⟨a, b | ab=a, baaa=bbb⟩Noc 93
109359⟨a, b | ab=a, bbaa=bbb⟩Noc 92
1119506⟨a, b | aab=b, aaaaa=ba⟩Noc 9
1119507⟨a, b | aab=b, aaaaa=bb⟩Noc Com 9
1120723⟨a, b | ab=aa, aaaaa=bb⟩Noc 915
1125656⟨a, b | ab=a, bbbbb=baa⟩Noc 9

Isomorphic instances

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

5 total

Σ#PresentationMapping
106840⟨a, b | aba=b, bbb=aaa⟩φ(a) = a, φ(b) = b
1120136⟨a, b | aab=b, babb=aaa⟩φ(a) = a, φ(b) = ba
1120152⟨a, b | aab=b, bbab=aaa⟩φ(a) = a, φ(b) = ba
1120292⟨a, b | aba=b, abbb=aaa⟩φ(a) = a, φ(b) = ba
1120306⟨a, b | aba=b, babb=aaa⟩φ(a) = a, φ(b) = ba