#254 ⟨a, b | aa=b, abb=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
aaa2a3a4a2
a2a2a3a4a2a3
a3a3a4a2a3a4
a4a4a2a3a4a2

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

Same cardinality

11 unique, 1419 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
7268⟨a, b | ab=a, baa=b⟩Finite non-commutative monoid with 5 elements63 iso, 23 anti-iso
8574⟨a, b | aaa=b, aab=b⟩Isomorphic to ℕ(5 = 3)27 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.

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