#574 ⟨a, b | aaa=b, aab=b⟩

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Quick links

  1. Properties
  2. Elements
  3. Staircase diagram
  4. Cayley table
  5. Right Cayley graph
  6. Rewriting system
  7. Same cardinality
  8. Isomorphic instances

Properties

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.

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 to reduce:
Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Reduction strategy:
Path to normal form: 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#PresentationDescriptionRelated
644⟨a, b | aa=b, abb=1⟩Isomorphic to ℤ51132 iso
7253⟨a, b | aa=b, abb=a⟩Isomorphic to ℕ(5 = 1)71 iso
7254⟨a, b | aa=b, abb=b⟩Isomorphic to ℕ(5 = 2)43 iso
7268⟨a, b | ab=a, baa=b⟩Finite non-commutative monoid with 5 elements63 iso, 23 anti-iso
8950⟨a, b | ab=a, bbbb=a⟩Isomorphic to ℕ(5 = 4)32 iso
8995⟨a, b | ab=a, aaa=bb⟩Finite commutative monoid with 5 elements19 iso
81019⟨a, b | ab=a, bba=bb⟩Finite non-commutative monoid with 5 elements25 iso
81020⟨a, b | ab=a, bbb=aa⟩Finite commutative monoid with 5 elements9 iso
81022⟨a, b | ab=a, bbb=ba⟩Finite non-commutative monoid with 5 elements4 iso
108617⟨a, b | aa=a, abbbba=b⟩Finite commutative monoid with 5 elements3 iso
1115426⟨a, b | aaa=aa, abbba=b⟩Finite commutative monoid with 5 elements

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