#15520 ⟨a, b | aaa=bb, aabbb=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.

1aba2abb2a2bab2b3a2b2ab3b4ab4
11aba2abb2a2bab2b3a2b2ab3b4ab4
aaa2abb2a2bab2b3a2b2ab3b4bab4b2
bbabb2a2bab2b3a2b2ab3b4bab4aba2b
a2a2b2a2bab2b3a2b2ab3b4bab4abb2ab2
ababa2bab2b3a2b2ab3b4bab4abb2a2bb3
b2b2ab2b3a2b2ab3b4bab4abb2a2bab2a2b2
a2ba2bb3a2b2ab3b4bab4abb2a2bab2b3ab3
ab2ab2a2b2ab3b4bab4abb2a2bab2b3a2b2b4
b3b3ab3b4bab4abb2a2bab2b3a2b2ab3b
a2b2a2b2b4bab4abb2a2bab2b3a2b2ab3b4ab4
ab3ab3bab4abb2a2bab2b3a2b2ab3b4bab
b4b4ab4abb2a2bab2b3a2b2ab3b4bab4b2
ab4ab4b2a2bab2b3a2b2ab3b4bab4abb2ab2

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. a3 ⇒ b2 [1]
3. a2b3 ⇒ b [2]
4. b5 ⇒ ab [4]
# ab:aaa=bb,aabbb=b ab - -
ba=ab
aaa=bb
aabbb=b
bbbbb=ab

Same cardinality

26 unique, 301 total

Σ#PresentationPropertiesDescriptionφ
91328⟨a, b | aaaa=b, abbb=1⟩Grp Com 13ℤ13189
92118⟨a, b | aba=b, baab=a⟩Noc 134
104642⟨a, b | aaaa=b, abbb=a⟩Noc Com 13ℕ(13 = 1)10
104643⟨a, b | aaaa=b, abbb=b⟩Noc Com 13ℕ(13 = 4)6
104683⟨a, b | aaab=b, abba=a⟩Noc 1316
105065⟨a, b | aaa=ab, babb=b⟩Noc 137
105334⟨a, b | aaa=ab, bba=bb⟩Noc 13
105336⟨a, b | aaa=ab, bbb=ab⟩Noc 131
105337⟨a, b | aaa=ab, bbb=ba⟩Noc 13
105340⟨a, b | aaa=bb, aab=ba⟩Noc 13
106305⟨a, b | aaa=b, abbbb=b⟩Noc Com 13ℕ(13 = 3)2
107143⟨a, b | bb=aa, aaab=ba⟩Noc 131
1112240⟨a, b | aaab=aa, bbbb=a⟩Noc Com 13ℕ(13 = 8)1
1112268⟨a, b | aaab=ab, bbbb=a⟩Noc Com 13ℕ(13 = 5)3
1112324⟨a, b | aaab=bb, bbbb=a⟩Noc Com 13ℕ(13 = 2)14
1114647⟨a, b | aaba=b, babbb=a⟩Noc Com 132
1116012⟨a, b | aaa=ab, abbb=bb⟩Noc 13
1116069⟨a, b | aaa=bb, abbb=ba⟩Noc 131
1116205⟨a, b | aab=ab, bbbb=aa⟩Noc 132
1116459⟨a, b | aba=bb, abbb=aa⟩Noc 131
1116506⟨a, b | aab=ab, bbb=aaa⟩Noc 131
1116515⟨a, b | aab=ba, bbb=aaa⟩Noc 13
1118958⟨a, b | aab=a, bbbbbb=a⟩Noc Com 13ℕ(13 = 6)4
1120046⟨a, b | aab=a, bbbb=bba⟩Noc 13
1120927⟨a, b | ab=aa, aaaa=bbb⟩Noc 137
1120991⟨a, b | ab=aa, baaa=bbb⟩Noc 133

Isomorphic instances

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

2 total

Σ#PresentationMapping
1115528⟨a, b | aaa=bb, abbab=b⟩φ(a) = a, φ(b) = b
1115536⟨a, b | aaa=bb, baabb=b⟩φ(a) = a, φ(b) = b