#5079 ⟨a, b | aaa=bb, aaba=b⟩

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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.

1aba2abb2a2bab2a2b2
11aba2abb2a2bab2a2b2
aaa2abb2a2bab2ba2b2b2
bbabb2a2bab2ba2b2aba2b
a2a2b2a2bab2ba2b2abb2ab2
ababa2bab2ba2b2abb2a2bb
b2b2ab2ba2b2abb2a2bab2a2b2
a2ba2bba2b2abb2a2bab2bab
ab2ab2a2b2abb2a2bab2ba2b2b2
a2b2a2b2b2a2bab2ba2b2abb2ab2

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 [12]
2. a3 ⇒ b2 [1]
3. b3 ⇒ b [13]
# ab:aaa=bb,aaba=b ab - -
ba=ab
aaa=bb
bbb=b

Same cardinality

26 unique, 934 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
105124⟨a, b | aab=aa, bbbb=a⟩Noc Com 9ℕ(9 = 8)5
105339⟨a, b | aaa=bb, aab=ab⟩Noc 91
106824⟨a, b | aab=b, bbb=aaa⟩Noc Com 95
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.

8 total

Σ#PresentationMapping
1112967⟨a, b | abb=aaa, aaab=b⟩φ(a) = a, φ(b) = ab
1112969⟨a, b | abb=aaa, aaba=b⟩φ(a) = a, φ(b) = ab
1112973⟨a, b | abb=aaa, abaa=b⟩φ(a) = a, φ(b) = ab
1112979⟨a, b | abb=aaa, abbb=b⟩φ(a) = a, φ(b) = ab
1113083⟨a, b | bab=aaa, aaab=b⟩φ(a) = a, φ(b) = ab
1113085⟨a, b | bab=aaa, aaba=b⟩φ(a) = a, φ(b) = ab
1113093⟨a, b | bab=aaa, abbb=b⟩φ(a) = a, φ(b) = ab
1113097⟨a, b | bab=aaa, babb=b⟩φ(a) = a, φ(b) = ab