#61 ⟨a, b | aa=b, bb=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.

1aa2a3
11aa2a3
aaa2a3a
a2a2a3aa2
a3a3aa2a3

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. a4 ⇒ a [2]
2. b ⇒ a2 [1]
# ab:aa=b,bb=a a/b - -
aaaa=a
b=aa

Same cardinality

8 unique, 1535 total

Σ#PresentationPropertiesDescriptionφ
57⟨a, b | aa=b, bb=1⟩Grp Com 4ℤ41419
657⟨a, b | aa=a, bb=a⟩Noc Com 4ℕ(4 = 2)37
7158⟨a, b | ab=aa, ba=b⟩Noc 414
7159⟨a, b | ab=aa, bb=a⟩Noc Com 4ℕ(4 = 3)16
7242⟨a, b | aa=a, abb=b⟩Noc 414
7280⟨a, b | ab=a, bb=aa⟩Noc Com 417
92881⟨a, b | aa=a, abbba=b⟩Noc Com 48
105033⟨a, b | aaa=aa, abba=b⟩Noc Com 42

Isomorphic instances

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

72 total

Σ#PresentationMapping
7249⟨a, b | aa=b, aab=a⟩φ(a) = bb, φ(b) = b
7251⟨a, b | aa=b, aba=a⟩φ(a) = bb, φ(b) = b
7260⟨a, b | ab=a, aaa=b⟩φ(a) = b, φ(b) = bbb
8900⟨a, b | aa=b, aaaa=a⟩φ(a) = bb, φ(b) = b
8923⟨a, b | ab=a, aaab=b⟩φ(a) = b, φ(b) = bbb
8925⟨a, b | ab=a, aaba=b⟩φ(a) = b, φ(b) = bbb
8929⟨a, b | ab=a, abaa=b⟩φ(a) = b, φ(b) = bbb
92000⟨a, b | aaa=b, aaaa=a⟩φ(a) = b, φ(b) = bbb
92023⟨a, b | aab=a, aaab=b⟩φ(a) = bb, φ(b) = b
92025⟨a, b | aab=a, aaba=b⟩φ(a) = bb, φ(b) = b
92029⟨a, b | aab=a, abaa=b⟩φ(a) = bb, φ(b) = b
92089⟨a, b | aba=a, aaba=b⟩φ(a) = bb, φ(b) = b
92943⟨a, b | ab=a, aaabb=b⟩φ(a) = b, φ(b) = bbb
92947⟨a, b | ab=a, aabab=b⟩φ(a) = b, φ(b) = bbb
92949⟨a, b | ab=a, aabba=b⟩φ(a) = b, φ(b) = bbb
92955⟨a, b | ab=a, abaab=b⟩φ(a) = b, φ(b) = bbb
92957⟨a, b | ab=a, ababa=b⟩φ(a) = b, φ(b) = bbb
92961⟨a, b | ab=a, abbaa=b⟩φ(a) = b, φ(b) = bbb
104650⟨a, b | aaab=a, aaba=b⟩φ(a) = b, φ(b) = b
104654⟨a, b | aaab=a, abaa=b⟩φ(a) = b, φ(b) = b
104660⟨a, b | aaab=a, abbb=b⟩φ(a) = b, φ(b) = b
104701⟨a, b | aaba=a, abaa=b⟩φ(a) = b, φ(b) = b
104732⟨a, b | aabb=a, abab=b⟩φ(a) = b, φ(b) = b
104734⟨a, b | aabb=a, abba=b⟩φ(a) = b, φ(b) = b
104742⟨a, b | abab=a, abba=b⟩φ(a) = b, φ(b) = b
108727⟨a, b | ab=a, aaabbb=b⟩φ(a) = b, φ(b) = bbb
108735⟨a, b | ab=a, aababb=b⟩φ(a) = b, φ(b) = bbb
108739⟨a, b | ab=a, aabbab=b⟩φ(a) = b, φ(b) = bbb
108741⟨a, b | ab=a, aabbba=b⟩φ(a) = b, φ(b) = bbb
108751⟨a, b | ab=a, abaabb=b⟩φ(a) = b, φ(b) = bbb
108755⟨a, b | ab=a, ababab=b⟩φ(a) = b, φ(b) = bbb
108757⟨a, b | ab=a, ababba=b⟩φ(a) = b, φ(b) = bbb
108763⟨a, b | ab=a, abbaab=b⟩φ(a) = b, φ(b) = bbb
108765⟨a, b | ab=a, abbaba=b⟩φ(a) = b, φ(b) = bbb
108769⟨a, b | ab=a, abbbaa=b⟩φ(a) = b, φ(b) = bbb
1114330⟨a, b | aaaa=a, aaaaa=b⟩φ(a) = bb, φ(b) = b
1114475⟨a, b | aaab=b, aaaab=a⟩φ(a) = b, φ(b) = bbb
1114603⟨a, b | aaba=b, aaaab=a⟩φ(a) = b, φ(b) = bbb
1114605⟨a, b | aaba=b, aaaba=a⟩φ(a) = b, φ(b) = bbb
1114609⟨a, b | aaba=b, aabaa=a⟩φ(a) = b, φ(b) = bbb
1114617⟨a, b | aaba=b, abaaa=a⟩φ(a) = b, φ(b) = bbb
1114633⟨a, b | aaba=b, baaaa=a⟩φ(a) = b, φ(b) = bbb
1118839⟨a, b | aab=a, aaaabb=b⟩φ(a) = bb, φ(b) = b
1118843⟨a, b | aab=a, aaabab=b⟩φ(a) = bb, φ(b) = b
1118845⟨a, b | aab=a, aaabba=b⟩φ(a) = bb, φ(b) = b
1118851⟨a, b | aab=a, aabaab=b⟩φ(a) = bb, φ(b) = b
1118853⟨a, b | aab=a, aababa=b⟩φ(a) = bb, φ(b) = b
1118857⟨a, b | aab=a, aabbaa=b⟩φ(a) = bb, φ(b) = b
1118867⟨a, b | aab=a, abaaab=b⟩φ(a) = bb, φ(b) = b
1118869⟨a, b | aab=a, abaaba=b⟩φ(a) = bb, φ(b) = b
1118873⟨a, b | aab=a, ababaa=b⟩φ(a) = bb, φ(b) = b
1118881⟨a, b | aab=a, abbaaa=b⟩φ(a) = bb, φ(b) = b
1118895⟨a, b | aab=a, abbbbb=b⟩φ(a) = bb, φ(b) = b
1119101⟨a, b | aba=a, aaabba=b⟩φ(a) = bb, φ(b) = b
1119107⟨a, b | aba=a, aababa=b⟩φ(a) = bb, φ(b) = b
1119111⟨a, b | aba=a, aabbaa=b⟩φ(a) = bb, φ(b) = b
1119121⟨a, b | aba=a, abaaba=b⟩φ(a) = bb, φ(b) = b
1124363⟨a, b | ab=a, aaabbbb=b⟩φ(a) = b, φ(b) = bbb
1124379⟨a, b | ab=a, aababbb=b⟩φ(a) = b, φ(b) = bbb
1124387⟨a, b | ab=a, aabbabb=b⟩φ(a) = b, φ(b) = bbb
1124391⟨a, b | ab=a, aabbbab=b⟩φ(a) = b, φ(b) = bbb
1124393⟨a, b | ab=a, aabbbba=b⟩φ(a) = b, φ(b) = bbb
1124411⟨a, b | ab=a, abaabbb=b⟩φ(a) = b, φ(b) = bbb
1124419⟨a, b | ab=a, abababb=b⟩φ(a) = b, φ(b) = bbb
1124423⟨a, b | ab=a, ababbab=b⟩φ(a) = b, φ(b) = bbb
1124425⟨a, b | ab=a, ababbba=b⟩φ(a) = b, φ(b) = bbb
1124435⟨a, b | ab=a, abbaabb=b⟩φ(a) = b, φ(b) = bbb
1124439⟨a, b | ab=a, abbabab=b⟩φ(a) = b, φ(b) = bbb
1124441⟨a, b | ab=a, abbabba=b⟩φ(a) = b, φ(b) = bbb
1124447⟨a, b | ab=a, abbbaab=b⟩φ(a) = b, φ(b) = bbb
1124449⟨a, b | ab=a, abbbaba=b⟩φ(a) = b, φ(b) = bbb
1124453⟨a, b | ab=a, abbbbaa=b⟩φ(a) = b, φ(b) = bbb