#2881 ⟨a, b | aa=a, abbba=b⟩

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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. a2 ⇒ a
  2. ab ⇒ b
  3. ba ⇒ b
  4. b3 ⇒ b 
# ab:aa=a,abbba=b ab
aa=a
ab=b
ba=b
bbb=b

Staircase diagram

Cayley table

Idempotents are shown in bold.

1abb2
11abb2
aaabb2
bbbb2b
b2b2b2bb2

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

8 unique, 1599 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
7280⟨a, b | ab=a, bb=aa⟩Finite commutative monoid with 4 elements17 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.

8 total

Σ#PresentationMapping
108595⟨a, b | aa=a, aabbba=b⟩φ(a) = a, φ(b) = b
108607⟨a, b | aa=a, ababba=b⟩φ(a) = a, φ(b) = b
1124073⟨a, b | aa=a, aaabbba=b⟩φ(a) = a, φ(b) = b
1124087⟨a, b | aa=a, aababba=b⟩φ(a) = a, φ(b) = b
1124093⟨a, b | aa=a, aabbaba=b⟩φ(a) = a, φ(b) = b
1124097⟨a, b | aa=a, aabbbaa=b⟩φ(a) = a, φ(b) = b
1124113⟨a, b | aa=a, abaabba=b⟩φ(a) = a, φ(b) = b
1124119⟨a, b | aa=a, abababa=b⟩φ(a) = a, φ(b) = b