#574 ⟨a, b | aaa=b, aab=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 ⇒ a3
  2. b ⇒ a3 
# ab:aaa=b,aab=b a/b
aaaaa=aaa
b=aaa

Staircase diagram

Cayley table

Idempotents are shown in bold.

1aa2a3a4
11aa2a3a4
aaa2a3a4a3
a2a2a3a4a3a4
a3a3a4a3a4a3
a4a4a3a4a3a4

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

11 unique, 1435 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
7254⟨a, b | aa=b, abb=b⟩Isomorphic to ℕ(5 = 2)43 iso
7268⟨a, b | ab=a, baa=b⟩Finite non-commutative monoid with 5 elements63 iso, 23 anti-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.

27 total

Σ#PresentationMapping
8576⟨a, b | aaa=b, aba=b⟩φ(a) = a, φ(b) = aaa
8983⟨a, b | aa=b, abb=ab⟩φ(a) = a, φ(b) = aa
8984⟨a, b | aa=b, abb=ba⟩φ(a) = a, φ(b) = aa
8987⟨a, b | aa=b, bab=ab⟩φ(a) = a, φ(b) = aa
91587⟨a, b | aaa=ab, aab=b⟩φ(a) = a, φ(b) = aaaa
91589⟨a, b | aaa=ab, aba=b⟩φ(a) = a, φ(b) = aaaa
93040⟨a, b | aa=b, aaab=ab⟩φ(a) = a, φ(b) = aa
93041⟨a, b | aa=b, aaab=ba⟩φ(a) = a, φ(b) = aa
93044⟨a, b | aa=b, aaba=ab⟩φ(a) = a, φ(b) = aa
93045⟨a, b | aa=b, aaba=ba⟩φ(a) = a, φ(b) = aa
93157⟨a, b | aa=b, abb=aaa⟩φ(a) = a, φ(b) = aa
93162⟨a, b | aa=b, bab=aaa⟩φ(a) = a, φ(b) = aa
104121⟨a, b | aab=aaa, aba=b⟩φ(a) = a, φ(b) = aaa
104123⟨a, b | aab=aaa, abb=b⟩φ(a) = a, φ(b) = aaa
105043⟨a, b | aaa=ab, aaaa=b⟩φ(a) = a, φ(b) = aaaa
106279⟨a, b | aaa=b, aaaaa=b⟩φ(a) = a, φ(b) = aaa
106803⟨a, b | aaa=b, aab=aaa⟩φ(a) = a, φ(b) = aaa
106804⟨a, b | aaa=b, aba=aaa⟩φ(a) = a, φ(b) = aaa
106842⟨a, b | abb=a, bbb=abb⟩φ(a) = aaa, φ(b) = a
106844⟨a, b | bab=a, bbb=bab⟩φ(a) = aaa, φ(b) = a
108913⟨a, b | aa=b, aaaaa=ab⟩φ(a) = a, φ(b) = aa
109190⟨a, b | aa=b, aaab=aaa⟩φ(a) = a, φ(b) = aa
109198⟨a, b | aa=b, aaba=aaa⟩φ(a) = a, φ(b) = aa
1120056⟨a, b | aab=b, aaab=aaa⟩φ(a) = a, φ(b) = aaaa
1120254⟨a, b | aba=b, aaab=aaa⟩φ(a) = a, φ(b) = aaaa
1120262⟨a, b | aba=b, aaba=aaa⟩φ(a) = a, φ(b) = aaaa
1125260⟨a, b | aa=b, aaaaa=aaa⟩φ(a) = a, φ(b) = aa