#578 ⟨a, b | aaa=b, abb=b⟩

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Quick links

  1. Properties
  2. Elements
  3. Staircase diagram
  4. Cayley table
  5. Right Cayley graph
  6. Rewriting system
  7. Same cardinality
  8. Isomorphic instances

Properties

Elements

An idempotent element x satisfies x2 = x.
The index and period of x is the least m (index) and n (period) such that x(m+n) = xm.

Staircase diagram

Cayley table

Idempotents are shown in bold.

1aa2a3a4a5a6
11aa2a3a4a5a6
aaa2a3a4a5a6a3
a2a2a3a4a5a6a3a4
a3a3a4a5a6a3a4a5
a4a4a5a6a3a4a5a6
a5a5a6a3a4a5a6a3
a6a6a3a4a5a6a3a4

Right Cayley graph

Idempotents are shown in bold.

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
#RuleProof
1. a7 ⇒ a3 [2]
2. b ⇒ a3 [1]
# ab:aaa=b,abb=b a/b
aaaaaaa=aaa
b=aaa

Same cardinality

22 unique, 1179 total

Σ#PresentationDescriptionRelated
7116⟨a, b | aaa=b, abb=1⟩Isomorphic to ℤ7763 iso
8577⟨a, b | aaa=b, abb=a⟩Isomorphic to ℕ(7 = 1)59 iso
8913⟨a, b | aa=b, abbb=b⟩Isomorphic to ℕ(7 = 2)82 iso
8937⟨a, b | ab=a, baaa=b⟩Finite non-commutative monoid with 7 elements54 iso, 12 anti-iso
91593⟨a, b | aaa=ab, baa=b⟩Finite non-commutative monoid with 7 elements1 iso, 2 anti-iso
91601⟨a, b | aaa=bb, aab=b⟩Finite commutative monoid with 7 elements5 iso
91620⟨a, b | aab=aa, bbb=a⟩Isomorphic to ℕ(7 = 6)35 iso
91632⟨a, b | aab=ab, bbb=a⟩Isomorphic to ℕ(7 = 4)50 iso
92074⟨a, b | aab=b, babb=a⟩Finite commutative monoid with 7 elements20 iso
92231⟨a, b | ab=aa, aaa=bb⟩Finite non-commutative monoid with 7 elements3 iso
92271⟨a, b | bb=aa, aaa=ab⟩Finite non-commutative monoid with 7 elements1 iso, 2 anti-iso
93197⟨a, b | ab=a, bbb=baa⟩Finite non-commutative monoid with 7 elements5 iso
104162⟨a, b | abb=aab, bbb=a⟩Isomorphic to ℕ(7 = 5)42 iso
106251⟨a, b | aaa=a, aabba=b⟩Finite non-commutative monoid with 7 elements1 iso
108987⟨a, b | ab=a, aaaaa=bb⟩Finite commutative monoid with 7 elements5 iso
109108⟨a, b | ab=a, bbbbb=aa⟩Finite commutative monoid with 7 elements2 iso
109110⟨a, b | ab=a, bbbbb=ba⟩Finite non-commutative monoid with 7 elements1 iso
109263⟨a, b | ab=a, aaaa=bbb⟩Finite commutative monoid with 7 elements4 iso
109375⟨a, b | ab=a, bbba=bbb⟩Finite non-commutative monoid with 7 elements3 iso
109376⟨a, b | ab=a, bbbb=aaa⟩Finite commutative monoid with 7 elements3 iso
109382⟨a, b | ab=a, bbbb=bba⟩Finite non-commutative monoid with 7 elements1 iso
1114843⟨a, b | abba=b, aabab=a⟩Finite non-commutative monoid with 7 elements1 iso

Isomorphic instances

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

44 total

Σ#PresentationMapping
8580⟨a, b | aaa=b, bab=b⟩φ(a) = a, φ(b) = aaa
93059⟨a, b | aa=b, abbb=ab⟩φ(a) = a, φ(b) = aa
93060⟨a, b | aa=b, abbb=ba⟩φ(a) = a, φ(b) = aa
93066⟨a, b | aa=b, babb=ab⟩φ(a) = a, φ(b) = aa
93067⟨a, b | aa=b, babb=ba⟩φ(a) = a, φ(b) = aa
104185⟨a, b | bab=aaa, bba=b⟩φ(a) = a, φ(b) = aaa
105049⟨a, b | aaa=ab, aabb=b⟩φ(a) = a, φ(b) = aaaaaa
105053⟨a, b | aaa=ab, abab=b⟩φ(a) = a, φ(b) = aaaaaa
105055⟨a, b | aaa=ab, abba=b⟩φ(a) = a, φ(b) = aaaaaa
106281⟨a, b | aaa=b, aaaab=b⟩φ(a) = a, φ(b) = aaa
106283⟨a, b | aaa=b, aaaba=b⟩φ(a) = a, φ(b) = aaa
106287⟨a, b | aaa=b, aabaa=b⟩φ(a) = a, φ(b) = aaa
106805⟨a, b | aaa=b, abb=aaa⟩φ(a) = a, φ(b) = aaa
106806⟨a, b | aaa=b, bab=aaa⟩φ(a) = a, φ(b) = aaa
106816⟨a, b | aab=a, bbb=aab⟩φ(a) = aaa, φ(b) = a
106833⟨a, b | aba=a, bbb=aba⟩φ(a) = aaa, φ(b) = a
108924⟨a, b | aa=b, aaabb=ab⟩φ(a) = a, φ(b) = aa
108925⟨a, b | aa=b, aaabb=ba⟩φ(a) = a, φ(b) = aa
108931⟨a, b | aa=b, aabab=ab⟩φ(a) = a, φ(b) = aa
108932⟨a, b | aa=b, aabab=ba⟩φ(a) = a, φ(b) = aa
108935⟨a, b | aa=b, aabba=ab⟩φ(a) = a, φ(b) = aa
108936⟨a, b | aa=b, aabba=ba⟩φ(a) = a, φ(b) = aa
108943⟨a, b | aa=b, abaab=ab⟩φ(a) = a, φ(b) = aa
108944⟨a, b | aa=b, abaab=ba⟩φ(a) = a, φ(b) = aa
108947⟨a, b | aa=b, ababa=ab⟩φ(a) = a, φ(b) = aa
108965⟨a, b | aa=b, baaab=ab⟩φ(a) = a, φ(b) = aa
109228⟨a, b | aa=b, abbb=aaa⟩φ(a) = a, φ(b) = aa
109242⟨a, b | aa=b, babb=aaa⟩φ(a) = a, φ(b) = aa
1115444⟨a, b | aaa=ab, aaaab=b⟩φ(a) = a, φ(b) = aaaaaa
1115446⟨a, b | aaa=ab, aaaba=b⟩φ(a) = a, φ(b) = aaaaaa
1115450⟨a, b | aaa=ab, aabaa=b⟩φ(a) = a, φ(b) = aaaaaa
1115458⟨a, b | aaa=ab, abaaa=b⟩φ(a) = a, φ(b) = aaaaaa
1124728⟨a, b | aa=b, aaaaab=ab⟩φ(a) = a, φ(b) = aa
1124729⟨a, b | aa=b, aaaaab=ba⟩φ(a) = a, φ(b) = aa
1124732⟨a, b | aa=b, aaaaba=ab⟩φ(a) = a, φ(b) = aa
1124733⟨a, b | aa=b, aaaaba=ba⟩φ(a) = a, φ(b) = aa
1124740⟨a, b | aa=b, aaabaa=ab⟩φ(a) = a, φ(b) = aa
1124741⟨a, b | aa=b, aaabaa=ba⟩φ(a) = a, φ(b) = aa
1125282⟨a, b | aa=b, aaabb=aaa⟩φ(a) = a, φ(b) = aa
1125296⟨a, b | aa=b, aabab=aaa⟩φ(a) = a, φ(b) = aa
1125304⟨a, b | aa=b, aabba=aaa⟩φ(a) = a, φ(b) = aa
1125320⟨a, b | aa=b, abaab=aaa⟩φ(a) = a, φ(b) = aa
1125328⟨a, b | aa=b, ababa=aaa⟩φ(a) = a, φ(b) = aa
1125364⟨a, b | aa=b, baaab=aaa⟩φ(a) = a, φ(b) = aa