#280 ⟨a, b | ab=a, bb=aa⟩

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Properties

Element profile

Complete 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
  1. ab ⇒ a
  2. ba ⇒ a
  3. b2 ⇒ a2
  4. a3 ⇒ a 
# ab:ab=a,bb=aa ab
ab=a
ba=a
bb=aa
aaa=a

Staircase diagram

Cayley table

Idempotents are shown in bold.

1aba2
11aba2
aaa2aa
bbaa2a2
a2a2aa2a2

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

8 unique, 1590 total

Σ#PresentationDescriptionRelated
57⟨a, b | aa=b, bb=1⟩Isomorphic to ℤ41419 iso
657⟨a, b | aa=a, bb=a⟩Isomorphic to ℕ(4 = 2)37 iso
661⟨a, b | aa=b, bb=a⟩Isomorphic to ℕ(4 = 1)72 iso
7158⟨a, b | ab=aa, ba=b⟩Finite non-commutative monoid with 4 elements8 iso, 6 anti-iso
7159⟨a, b | ab=aa, bb=a⟩Isomorphic to ℕ(4 = 3)16 iso
7242⟨a, b | aa=a, abb=b⟩Finite non-commutative monoid with 4 elements14 iso
92881⟨a, b | aa=a, abbba=b⟩Finite commutative monoid with 4 elements8 iso
105033⟨a, b | aaa=aa, abba=b⟩Finite commutative monoid with 4 elements2 iso

Other isomorphic instances

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

17 total

Σ#PresentationMapping
8999⟨a, b | ab=a, aab=bb⟩φ(a) = a, φ(b) = b
81003⟨a, b | ab=a, aba=bb⟩φ(a) = a, φ(b) = b
93087⟨a, b | ab=a, aabb=bb⟩φ(a) = a, φ(b) = b
93095⟨a, b | ab=a, abab=bb⟩φ(a) = a, φ(b) = b
93099⟨a, b | ab=a, abba=bb⟩φ(a) = a, φ(b) = b
109015⟨a, b | ab=a, aabbb=bb⟩φ(a) = a, φ(b) = b
109031⟨a, b | ab=a, ababb=bb⟩φ(a) = a, φ(b) = b
109039⟨a, b | ab=a, abbab=bb⟩φ(a) = a, φ(b) = b
109043⟨a, b | ab=a, abbba=bb⟩φ(a) = a, φ(b) = b
1112437⟨a, b | aabb=aa, abbb=b⟩φ(a) = b, φ(b) = a
1112501⟨a, b | abab=aa, abbb=b⟩φ(a) = b, φ(b) = a
1112594⟨a, b | abba=bb, baaa=a⟩φ(a) = a, φ(b) = b
1124923⟨a, b | ab=a, aabbbb=bb⟩φ(a) = a, φ(b) = b
1124955⟨a, b | ab=a, ababbb=bb⟩φ(a) = a, φ(b) = b
1124971⟨a, b | ab=a, abbabb=bb⟩φ(a) = a, φ(b) = b
1124979⟨a, b | ab=a, abbbab=bb⟩φ(a) = a, φ(b) = b
1124983⟨a, b | ab=a, abbbba=bb⟩φ(a) = a, φ(b) = b