#59 ⟨a, b | aa=b, ab=a⟩

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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. a3 ⇒ a
  2. b ⇒ a2 
# ab:aa=b,ab=a a/b
aaa=a
b=aa

Staircase diagram

Cayley table

Idempotents are shown in bold.

1aa2
11aa2
aaa2a
a2a2aa2

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

4 unique, 2146 total

Σ#PresentationDescriptionRelated
56⟨a, b | aa=b, ab=1⟩Isomorphic to ℤ32029 iso
660⟨a, b | aa=b, ab=b⟩Isomorphic to ℕ(3 = 2)23 iso
663⟨a, b | ab=a, ba=b⟩Finite non-commutative monoid with 3 elements61 iso, 17 anti-iso
8891⟨a, b | aa=a, abba=b⟩Finite commutative monoid with 3 elements12 iso

Other isomorphic instances

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

61 total

Σ#PresentationMapping
7247⟨a, b | aa=b, aaa=a⟩φ(a) = a, φ(b) = aa
7262⟨a, b | ab=a, aab=b⟩φ(a) = a, φ(b) = aa
7264⟨a, b | ab=a, aba=b⟩φ(a) = a, φ(b) = aa
8583⟨a, b | aab=a, aba=b⟩φ(a) = a, φ(b) = a
8927⟨a, b | ab=a, aabb=b⟩φ(a) = a, φ(b) = aa
8931⟨a, b | ab=a, abab=b⟩φ(a) = a, φ(b) = aa
8933⟨a, b | ab=a, abba=b⟩φ(a) = a, φ(b) = aa
91981⟨a, b | aaa=a, aaaa=b⟩φ(a) = a, φ(b) = aa
92054⟨a, b | aab=b, aaab=a⟩φ(a) = a, φ(b) = aa
92106⟨a, b | aba=b, aaab=a⟩φ(a) = a, φ(b) = aa
92108⟨a, b | aba=b, aaba=a⟩φ(a) = a, φ(b) = aa
92951⟨a, b | ab=a, aabbb=b⟩φ(a) = a, φ(b) = aa
92959⟨a, b | ab=a, ababb=b⟩φ(a) = a, φ(b) = aa
92963⟨a, b | ab=a, abbab=b⟩φ(a) = a, φ(b) = aa
92965⟨a, b | ab=a, abbba=b⟩φ(a) = a, φ(b) = aa
106239⟨a, b | aaa=a, aaaaa=b⟩φ(a) = a, φ(b) = a
106325⟨a, b | aab=a, aaabb=b⟩φ(a) = a, φ(b) = a
106329⟨a, b | aab=a, aabab=b⟩φ(a) = a, φ(b) = a
106331⟨a, b | aab=a, aabba=b⟩φ(a) = a, φ(b) = a
106337⟨a, b | aab=a, abaab=b⟩φ(a) = a, φ(b) = a
106339⟨a, b | aab=a, ababa=b⟩φ(a) = a, φ(b) = a
106343⟨a, b | aab=a, abbaa=b⟩φ(a) = a, φ(b) = a
106349⟨a, b | aab=a, abbbb=b⟩φ(a) = a, φ(b) = a
106384⟨a, b | aab=b, aaaab=a⟩φ(a) = a, φ(b) = a
106459⟨a, b | aba=a, aabba=b⟩φ(a) = a, φ(b) = a
106465⟨a, b | aba=a, ababa=b⟩φ(a) = a, φ(b) = a
106488⟨a, b | aba=b, aaaab=a⟩φ(a) = a, φ(b) = a
106490⟨a, b | aba=b, aaaba=a⟩φ(a) = a, φ(b) = a
106494⟨a, b | aba=b, aabaa=a⟩φ(a) = a, φ(b) = a
108743⟨a, b | ab=a, aabbbb=b⟩φ(a) = a, φ(b) = aa
108759⟨a, b | ab=a, ababbb=b⟩φ(a) = a, φ(b) = aa
108767⟨a, b | ab=a, abbabb=b⟩φ(a) = a, φ(b) = aa
108771⟨a, b | ab=a, abbbab=b⟩φ(a) = a, φ(b) = aa
108773⟨a, b | ab=a, abbbba=b⟩φ(a) = a, φ(b) = aa
1114424⟨a, b | aaab=a, aabbb=b⟩φ(a) = a, φ(b) = aa
1114432⟨a, b | aaab=a, ababb=b⟩φ(a) = a, φ(b) = aa
1114436⟨a, b | aaab=a, abbab=b⟩φ(a) = a, φ(b) = aa
1114438⟨a, b | aaab=a, abbba=b⟩φ(a) = a, φ(b) = aa
1114566⟨a, b | aaba=a, abbba=b⟩φ(a) = a, φ(b) = aa
1114680⟨a, b | aabb=a, aabbb=b⟩φ(a) = aa, φ(b) = a
1114692⟨a, b | aabb=a, abbab=b⟩φ(a) = aa, φ(b) = a
1114694⟨a, b | aabb=a, abbba=b⟩φ(a) = aa, φ(b) = a
1114744⟨a, b | abab=a, aabbb=b⟩φ(a) = aa, φ(b) = a
1114752⟨a, b | abab=a, ababb=b⟩φ(a) = aa, φ(b) = a
1114756⟨a, b | abab=a, abbab=b⟩φ(a) = aa, φ(b) = a
1114758⟨a, b | abab=a, abbba=b⟩φ(a) = aa, φ(b) = a
1114818⟨a, b | abba=a, abbba=b⟩φ(a) = aa, φ(b) = a
1114839⟨a, b | abba=b, aaabb=a⟩φ(a) = a, φ(b) = aa
1114845⟨a, b | abba=b, aabba=a⟩φ(a) = a, φ(b) = aa
1114849⟨a, b | abba=b, abaab=a⟩φ(a) = a, φ(b) = aa
1118689⟨a, b | aaa=a, aaaaaa=b⟩φ(a) = a, φ(b) = aa
1118962⟨a, b | aab=b, aaaaab=a⟩φ(a) = a, φ(b) = aa
1119162⟨a, b | aba=b, aaaaab=a⟩φ(a) = a, φ(b) = aa
1119164⟨a, b | aba=b, aaaaba=a⟩φ(a) = a, φ(b) = aa
1119168⟨a, b | aba=b, aaabaa=a⟩φ(a) = a, φ(b) = aa
1124395⟨a, b | ab=a, aabbbbb=b⟩φ(a) = a, φ(b) = aa
1124427⟨a, b | ab=a, ababbbb=b⟩φ(a) = a, φ(b) = aa
1124443⟨a, b | ab=a, abbabbb=b⟩φ(a) = a, φ(b) = aa
1124451⟨a, b | ab=a, abbbabb=b⟩φ(a) = a, φ(b) = aa
1124455⟨a, b | ab=a, abbbbab=b⟩φ(a) = a, φ(b) = aa
1124457⟨a, b | ab=a, abbbbba=b⟩φ(a) = a, φ(b) = aa