#574 ⟨a, b | aaa=b, aab=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
aaa2a3a4a3
a2a2a3a4a3a4
a3a3a4a3a4a3
a4a4a3a4a3a4

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 ⇒ a3 [2]
2. b ⇒ a3 [1]
# ab:aaa=b,aab=b a/b - -
aaaaa=aaa
b=aaa

Same cardinality

11 unique, 1435 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
7254⟨a, b | aa=b, abb=b⟩Noc Com 5ℕ(5 = 2)43
7268⟨a, b | ab=a, baa=b⟩Noc 586
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.

27 total

Σ#PresentationMapping
8576⟨a, b | aaa=b, aba=b⟩φ(a) = a, φ(b) = aaa
8983⟨a, b | aa=b, abb=ab⟩φ(a) = a, φ(b) = aa
8984⟨a, b | aa=b, abb=ba⟩φ(a) = a, φ(b) = aa
8987⟨a, b | aa=b, bab=ab⟩φ(a) = a, φ(b) = aa
91587⟨a, b | aaa=ab, aab=b⟩φ(a) = a, φ(b) = aaaa
91589⟨a, b | aaa=ab, aba=b⟩φ(a) = a, φ(b) = aaaa
93040⟨a, b | aa=b, aaab=ab⟩φ(a) = a, φ(b) = aa
93041⟨a, b | aa=b, aaab=ba⟩φ(a) = a, φ(b) = aa
93044⟨a, b | aa=b, aaba=ab⟩φ(a) = a, φ(b) = aa
93045⟨a, b | aa=b, aaba=ba⟩φ(a) = a, φ(b) = aa
93157⟨a, b | aa=b, abb=aaa⟩φ(a) = a, φ(b) = aa
93162⟨a, b | aa=b, bab=aaa⟩φ(a) = a, φ(b) = aa
104121⟨a, b | aab=aaa, aba=b⟩φ(a) = a, φ(b) = aaa
104123⟨a, b | aab=aaa, abb=b⟩φ(a) = a, φ(b) = aaa
105043⟨a, b | aaa=ab, aaaa=b⟩φ(a) = a, φ(b) = aaaa
106279⟨a, b | aaa=b, aaaaa=b⟩φ(a) = a, φ(b) = aaa
106803⟨a, b | aaa=b, aab=aaa⟩φ(a) = a, φ(b) = aaa
106804⟨a, b | aaa=b, aba=aaa⟩φ(a) = a, φ(b) = aaa
106842⟨a, b | abb=a, bbb=abb⟩φ(a) = aaa, φ(b) = a
106844⟨a, b | bab=a, bbb=bab⟩φ(a) = aaa, φ(b) = a
108913⟨a, b | aa=b, aaaaa=ab⟩φ(a) = a, φ(b) = aa
109190⟨a, b | aa=b, aaab=aaa⟩φ(a) = a, φ(b) = aa
109198⟨a, b | aa=b, aaba=aaa⟩φ(a) = a, φ(b) = aa
1120056⟨a, b | aab=b, aaab=aaa⟩φ(a) = a, φ(b) = aaaa
1120254⟨a, b | aba=b, aaab=aaa⟩φ(a) = a, φ(b) = aaaa
1120262⟨a, b | aba=b, aaba=aaa⟩φ(a) = a, φ(b) = aaaa
1125260⟨a, b | aa=b, aaaaa=aaa⟩φ(a) = a, φ(b) = aa