#158 ⟨a, b | ab=aa, ba=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. b2 ⇒ b
  2. ba ⇒ b
  3. a2 ⇒ ab 
# ab:ab=aa,ba=b ba
bb=b
ba=b
aa=ab

Cayley table

Idempotents are shown in bold.

1abab
11abab
aaababab
bbbbb
ababababab

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

8 unique, 1593 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
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
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.

8 total

Σ#PresentationMapping
81015⟨a, b | ab=a, bab=bb⟩φ(a) = b, φ(b) = a
93119⟨a, b | ab=a, babb=bb⟩φ(a) = b, φ(b) = a
109079⟨a, b | ab=a, babbb=bb⟩φ(a) = b, φ(b) = a
1119455⟨a, b | aab=a, baabb=bb⟩φ(a) = b, φ(b) = a
1119463⟨a, b | aab=a, babab=bb⟩φ(a) = b, φ(b) = a
1119467⟨a, b | aab=a, babba=bb⟩φ(a) = b, φ(b) = a
1119471⟨a, b | aab=a, babbb=bb⟩φ(a) = b, φ(b) = a
1125051⟨a, b | ab=a, babbbb=bb⟩φ(a) = b, φ(b) = a

Other anti-isomorphic instances

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

6 total

Σ#PresentationMapping
91659⟨a, b | aba=aa, abb=b⟩φ(a) = a, φ(b) = b
105235⟨a, b | aba=aa, abbb=b⟩φ(a) = a, φ(b) = b
1112327⟨a, b | aaba=aa, aabb=b⟩φ(a) = a, φ(b) = b
1112331⟨a, b | aaba=aa, abab=b⟩φ(a) = a, φ(b) = b
1112335⟨a, b | aaba=aa, abbb=b⟩φ(a) = a, φ(b) = b
1115828⟨a, b | aba=aa, abbbb=b⟩φ(a) = a, φ(b) = b