#577 ⟨a, b | aaa=b, abb=a⟩

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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
aaa2a3a4a5a6a
a2a2a3a4a5a6aa2
a3a3a4a5a6aa2a3
a4a4a5a6aa2a3a4
a5a5a6aa2a3a4a5
a6a6aa2a3a4a5a6

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

Same cardinality

22 unique, 1164 total

Σ#PresentationDescriptionRelated
7116⟨a, b | aaa=b, abb=1⟩Isomorphic to ℤ7763 iso
8578⟨a, b | aaa=b, abb=b⟩Isomorphic to ℕ(7 = 3)44 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.

59 total

Σ#PresentationMapping
8579⟨a, b | aaa=b, bab=a⟩φ(a) = a, φ(b) = aaa
8912⟨a, b | aa=b, abbb=a⟩φ(a) = a, φ(b) = aa
8916⟨a, b | aa=b, babb=a⟩φ(a) = a, φ(b) = aa
92902⟨a, b | aa=b, aaabb=a⟩φ(a) = a, φ(b) = aa
92906⟨a, b | aa=b, aabab=a⟩φ(a) = a, φ(b) = aa
92908⟨a, b | aa=b, aabba=a⟩φ(a) = a, φ(b) = aa
92912⟨a, b | aa=b, abaab=a⟩φ(a) = a, φ(b) = aa
92914⟨a, b | aa=b, ababa=a⟩φ(a) = a, φ(b) = aa
92924⟨a, b | aa=b, baaab=a⟩φ(a) = a, φ(b) = aa
104632⟨a, b | aaaa=b, aaab=a⟩φ(a) = a, φ(b) = aaaa
104634⟨a, b | aaaa=b, aaba=a⟩φ(a) = a, φ(b) = aaaa
104677⟨a, b | aaab=b, aabb=a⟩φ(a) = aaaa, φ(b) = a
104681⟨a, b | aaab=b, abab=a⟩φ(a) = aaaa, φ(b) = a
104716⟨a, b | aaba=b, aabb=a⟩φ(a) = aaaa, φ(b) = a
104719⟨a, b | aaba=b, abab=a⟩φ(a) = aaaa, φ(b) = a
104721⟨a, b | aaba=b, abba=a⟩φ(a) = aaaa, φ(b) = a
104723⟨a, b | aaba=b, baab=a⟩φ(a) = aaaa, φ(b) = a
104725⟨a, b | aaba=b, baba=a⟩φ(a) = aaaa, φ(b) = a
104728⟨a, b | aaba=b, bbaa=a⟩φ(a) = aaaa, φ(b) = a
106280⟨a, b | aaa=b, aaaab=a⟩φ(a) = a, φ(b) = aaa
106282⟨a, b | aaa=b, aaaba=a⟩φ(a) = a, φ(b) = aaa
106286⟨a, b | aaa=b, aabaa=a⟩φ(a) = a, φ(b) = aaa
106319⟨a, b | aab=a, aaaaa=b⟩φ(a) = aaaaa, φ(b) = a
106396⟨a, b | aab=b, aabbb=a⟩φ(a) = aaa, φ(b) = a
106404⟨a, b | aab=b, ababb=a⟩φ(a) = aaa, φ(b) = a
106408⟨a, b | aab=b, abbab=a⟩φ(a) = aaa, φ(b) = a
106420⟨a, b | aab=b, baabb=a⟩φ(a) = aaa, φ(b) = a
106432⟨a, b | aab=b, bbaab=a⟩φ(a) = aaa, φ(b) = a
106447⟨a, b | aba=a, aaaaa=b⟩φ(a) = aaaaa, φ(b) = a
106500⟨a, b | aba=b, aabbb=a⟩φ(a) = aaa, φ(b) = a
106506⟨a, b | aba=b, ababb=a⟩φ(a) = aaa, φ(b) = a
106508⟨a, b | aba=b, abbab=a⟩φ(a) = aaa, φ(b) = a
106510⟨a, b | aba=b, abbba=a⟩φ(a) = aaa, φ(b) = a
106516⟨a, b | aba=b, baabb=a⟩φ(a) = aaa, φ(b) = a
106518⟨a, b | aba=b, babab=a⟩φ(a) = aaa, φ(b) = a
108642⟨a, b | aa=b, aaaaab=a⟩φ(a) = a, φ(b) = aa
108644⟨a, b | aa=b, aaaaba=a⟩φ(a) = a, φ(b) = aa
108648⟨a, b | aa=b, aaabaa=a⟩φ(a) = a, φ(b) = aa
108713⟨a, b | ab=a, aaaaaa=b⟩φ(a) = a, φ(b) = aaaaaa
1114479⟨a, b | aaab=b, aaabb=a⟩φ(a) = aa, φ(b) = a
1114491⟨a, b | aaab=b, abaab=a⟩φ(a) = aa, φ(b) = a
1114507⟨a, b | aaab=b, baaab=a⟩φ(a) = aa, φ(b) = a
1114607⟨a, b | aaba=b, aaabb=a⟩φ(a) = aa, φ(b) = a
1114611⟨a, b | aaba=b, aabab=a⟩φ(a) = aa, φ(b) = a
1114613⟨a, b | aaba=b, aabba=a⟩φ(a) = aa, φ(b) = a
1114619⟨a, b | aaba=b, abaab=a⟩φ(a) = aa, φ(b) = a
1114621⟨a, b | aaba=b, ababa=a⟩φ(a) = aa, φ(b) = a
1114625⟨a, b | aaba=b, abbaa=a⟩φ(a) = aa, φ(b) = a
1114635⟨a, b | aaba=b, baaab=a⟩φ(a) = aa, φ(b) = a
1114637⟨a, b | aaba=b, baaba=a⟩φ(a) = aa, φ(b) = a
1114641⟨a, b | aaba=b, babaa=a⟩φ(a) = aa, φ(b) = a
1114649⟨a, b | aaba=b, bbaaa=a⟩φ(a) = aa, φ(b) = a
1124188⟨a, b | aa=b, aaaaaaa=a⟩φ(a) = a, φ(b) = aa
1124335⟨a, b | ab=a, aaaaaab=b⟩φ(a) = a, φ(b) = aaaaaa
1124337⟨a, b | ab=a, aaaaaba=b⟩φ(a) = a, φ(b) = aaaaaa
1124341⟨a, b | ab=a, aaaabaa=b⟩φ(a) = a, φ(b) = aaaaaa
1124349⟨a, b | ab=a, aaabaaa=b⟩φ(a) = a, φ(b) = aaaaaa
1124365⟨a, b | ab=a, aabaaaa=b⟩φ(a) = a, φ(b) = aaaaaa
1124397⟨a, b | ab=a, abaaaaa=b⟩φ(a) = a, φ(b) = aaaaaa