#254 ⟨a, b | aa=b, abb=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. a5 ⇒ a2
  2. b ⇒ a2 
# ab:aa=b,abb=b a/b
aaaaa=aa
b=aa

Staircase diagram

Cayley table

Idempotents are shown in bold.

1aa2a3a4
11aa2a3a4
aaa2a3a4a2
a2a2a3a4a2a3
a3a3a4a2a3a4
a4a4a2a3a4a2

Right Cayley graph

Idempotents are shown in bold.

Others with 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

Other 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