#1595 ⟨a, b | aaa=ab, bab=b⟩

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Contents

  1. Properties
  2. Finite structure
  3. Cayley table
  4. Right Cayley graph
  5. Left Cayley graph
  6. Rewriting system
  7. Same cardinality
  8. Isomorphic instances
  9. Anti-isomorphic instances

Properties

Finite structure

Elements in the center commute with all other elements.
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.

Cayley table

Idempotents are shown in bold.

1aba2abb2ab2b3ab3
11aba2abb2ab2b3ab3
aaa2ababab3ab2abab3ab2
bbb3b2b2bb3b2bb3
a2a2abab3ab3ab2abab3ab2ab
ababab3ab2ab2abab3ab2abab3
b2b2bb3b3b2bb3b2b
ab2ab2abab3ab3ab2abab3ab2ab
b3b3b2bbb3b2bb3b2
ab3ab3ab2ababab3ab2abab3ab2

Right Cayley graph

Idempotents are shown in bold.

Left 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. b4 ⇒ b [8]
2. ba ⇒ b3 [7]
3. a2b ⇒ ab3 [6]
4. a3 ⇒ ab [1]
# ab:aaa=ab,bab=b b/a - -
bbbb=b
ba=bbb
aab=abbb
aaa=ab

Same cardinality

26 unique, 933 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
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
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.

4 total

Σ#PresentationMapping
91597⟨a, b | aaa=ab, bba=b⟩φ(a) = a, φ(b) = b
105059⟨a, b | aaa=ab, baaa=b⟩φ(a) = a, φ(b) = b
1119999⟨a, b | aab=a, baab=bbb⟩φ(a) = b, φ(b) = a
1120007⟨a, b | aab=a, baba=bbb⟩φ(a) = b, φ(b) = a

Anti-isomorphic instances

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

5 total

Σ#PresentationMapping
1112915⟨a, b | aba=aaa, aaab=b⟩φ(a) = a, φ(b) = bb
1112919⟨a, b | aba=aaa, aabb=b⟩φ(a) = a, φ(b) = bb
1112921⟨a, b | aba=aaa, abab=b⟩φ(a) = a, φ(b) = bb
1112925⟨a, b | aba=aaa, abbb=b⟩φ(a) = a, φ(b) = bb
1120213⟨a, b | aba=a, abab=bbb⟩φ(a) = b, φ(b) = a