#254 ⟨a, b | aa=b, abb=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.

1aa2a3a4
11aa2a3a4
aaa2a3a4a2
a2a2a3a4a2a3
a3a3a4a2a3a4
a4a4a2a3a4a2

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. a5 ⇒ a2 [2]
2. b ⇒ a2 [1]
# ab:aa=b,abb=b a/b - -
aaaaa=aa
b=aa

Same cardinality

11 unique, 1419 total

Σ#PresentationPropertiesDescriptionφ
644⟨a, b | aa=b, abb=1⟩Grp Com 5ℤ51132
7253⟨a, b | aa=b, abb=a⟩Noc Com 5ℕ(5 = 1)71
7268⟨a, b | ab=a, baa=b⟩Noc 586
8574⟨a, b | aaa=b, aab=b⟩Noc Com 5ℕ(5 = 3)27
8950⟨a, b | ab=a, bbbb=a⟩Noc Com 5ℕ(5 = 4)32
8995⟨a, b | ab=a, aaa=bb⟩Noc Com 519
81019⟨a, b | ab=a, bba=bb⟩Noc 525
81020⟨a, b | ab=a, bbb=aa⟩Noc Com 59
81022⟨a, b | ab=a, bbb=ba⟩Noc 54
108617⟨a, b | aa=a, abbbba=b⟩Noc Com 53
1115426⟨a, b | aaa=aa, abbba=b⟩Noc Com 5

Isomorphic instances

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

43 total

Σ#PresentationMapping
7256⟨a, b | aa=b, bab=b⟩φ(a) = a, φ(b) = aa
8903⟨a, b | aa=b, aaab=b⟩φ(a) = a, φ(b) = aa
8905⟨a, b | aa=b, aaba=b⟩φ(a) = a, φ(b) = aa
8982⟨a, b | aa=b, abb=aa⟩φ(a) = a, φ(b) = aa
8986⟨a, b | aa=b, bab=aa⟩φ(a) = a, φ(b) = aa
91680⟨a, b | aba=bb, baa=a⟩φ(a) = aa, φ(b) = a
91692⟨a, b | abb=bb, bbb=a⟩φ(a) = aaa, φ(b) = a
91698⟨a, b | bab=bb, bbb=a⟩φ(a) = aaa, φ(b) = a
92177⟨a, b | ab=aa, aaaa=b⟩φ(a) = a, φ(b) = aaaa
92179⟨a, b | ab=aa, aaab=b⟩φ(a) = a, φ(b) = aaaa
92181⟨a, b | ab=aa, aaba=b⟩φ(a) = a, φ(b) = aaaa
92183⟨a, b | ab=aa, aabb=b⟩φ(a) = a, φ(b) = aaaa
92185⟨a, b | ab=aa, abaa=b⟩φ(a) = a, φ(b) = aaaa
92187⟨a, b | ab=aa, abab=b⟩φ(a) = a, φ(b) = aaaa
92189⟨a, b | ab=aa, abba=b⟩φ(a) = a, φ(b) = aaaa
92191⟨a, b | ab=aa, abbb=b⟩φ(a) = a, φ(b) = aaaa
92897⟨a, b | aa=b, aaaaa=b⟩φ(a) = a, φ(b) = aa
93039⟨a, b | aa=b, aaab=aa⟩φ(a) = a, φ(b) = aa
93043⟨a, b | aa=b, aaba=aa⟩φ(a) = a, φ(b) = aa
105097⟨a, b | aab=aa, aaab=b⟩φ(a) = a, φ(b) = aaa
105099⟨a, b | aab=aa, aaba=b⟩φ(a) = a, φ(b) = aaa
105103⟨a, b | aab=aa, abaa=b⟩φ(a) = a, φ(b) = aaa
105227⟨a, b | aba=aa, aaba=b⟩φ(a) = a, φ(b) = aaa
108912⟨a, b | aa=b, aaaaa=aa⟩φ(a) = a, φ(b) = aa
1112215⟨a, b | aaab=aa, aaba=b⟩φ(a) = a, φ(b) = aa
1112219⟨a, b | aaab=aa, abaa=b⟩φ(a) = a, φ(b) = aa
1112329⟨a, b | aaba=aa, abaa=b⟩φ(a) = a, φ(b) = aa
1115552⟨a, b | aab=aa, aaabb=b⟩φ(a) = a, φ(b) = aaa
1115556⟨a, b | aab=aa, aabab=b⟩φ(a) = a, φ(b) = aaa
1115558⟨a, b | aab=aa, aabba=b⟩φ(a) = a, φ(b) = aaa
1115564⟨a, b | aab=aa, abaab=b⟩φ(a) = a, φ(b) = aaa
1115566⟨a, b | aab=aa, ababa=b⟩φ(a) = a, φ(b) = aaa
1115570⟨a, b | aab=aa, abbaa=b⟩φ(a) = a, φ(b) = aaa
1115814⟨a, b | aba=aa, aabba=b⟩φ(a) = a, φ(b) = aaa
1115820⟨a, b | aba=aa, ababa=b⟩φ(a) = a, φ(b) = aaa
1119304⟨a, b | aaa=b, aaaaa=aa⟩φ(a) = a, φ(b) = aaa
1119391⟨a, b | aab=a, aaabb=bb⟩φ(a) = aa, φ(b) = a
1119399⟨a, b | aab=a, aabab=bb⟩φ(a) = aa, φ(b) = a
1119403⟨a, b | aab=a, aabba=bb⟩φ(a) = aa, φ(b) = a
1119415⟨a, b | aab=a, abaab=bb⟩φ(a) = aa, φ(b) = a
1119419⟨a, b | aab=a, ababa=bb⟩φ(a) = aa, φ(b) = a
1119427⟨a, b | aab=a, abbaa=bb⟩φ(a) = aa, φ(b) = a
1119668⟨a, b | aba=a, ababa=bb⟩φ(a) = aa, φ(b) = a