#1632 ⟨a, b | aab=ab, bbb=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.

1bb2b3b4b5b6
11bb2b3b4b5b6
bbb2b3b4b5b6b4
b2b2b3b4b5b6b4b5
b3b3b4b5b6b4b5b6
b4b4b5b6b4b5b6b4
b5b5b6b4b5b6b4b5
b6b6b4b5b6b4b5b6

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. b7 ⇒ b4 [4]
2. a ⇒ b3 [2]
# ab:aab=ab,bbb=a b/a
bbbbbbb=bbbb
a=bbb

Same cardinality

22 unique, 1173 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
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
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.

50 total

Σ#PresentationMapping
91644⟨a, b | aab=ba, bbb=a⟩φ(a) = bbb, φ(b) = b
91674⟨a, b | aba=ab, bbb=a⟩φ(a) = bbb, φ(b) = b
93061⟨a, b | aa=b, abbb=bb⟩φ(a) = b, φ(b) = bb
93068⟨a, b | aa=b, babb=bb⟩φ(a) = b, φ(b) = bb
104633⟨a, b | aaaa=b, aaab=b⟩φ(a) = b, φ(b) = bbbb
104635⟨a, b | aaaa=b, aaba=b⟩φ(a) = b, φ(b) = bbbb
108926⟨a, b | aa=b, aaabb=bb⟩φ(a) = b, φ(b) = bb
108933⟨a, b | aa=b, aabab=bb⟩φ(a) = b, φ(b) = bb
108937⟨a, b | aa=b, aabba=bb⟩φ(a) = b, φ(b) = bb
108945⟨a, b | aa=b, abaab=bb⟩φ(a) = b, φ(b) = bb
108948⟨a, b | aa=b, ababa=bb⟩φ(a) = b, φ(b) = bb
108966⟨a, b | aa=b, baaab=bb⟩φ(a) = b, φ(b) = bb
109229⟨a, b | aa=b, abbb=aab⟩φ(a) = b, φ(b) = bb
109230⟨a, b | aa=b, abbb=aba⟩φ(a) = b, φ(b) = bb
109232⟨a, b | aa=b, abbb=baa⟩φ(a) = b, φ(b) = bb
109243⟨a, b | aa=b, babb=aab⟩φ(a) = b, φ(b) = bb
109244⟨a, b | aa=b, babb=aba⟩φ(a) = b, φ(b) = bb
109246⟨a, b | aa=b, babb=baa⟩φ(a) = b, φ(b) = bb
1112167⟨a, b | aaaa=ab, aaab=b⟩φ(a) = b, φ(b) = bbbbbb
1112169⟨a, b | aaaa=ab, aaba=b⟩φ(a) = b, φ(b) = bbbbbb
1112173⟨a, b | aaaa=ab, abaa=b⟩φ(a) = b, φ(b) = bbbbbb
1119308⟨a, b | aaa=b, aaaab=ab⟩φ(a) = b, φ(b) = bbb
1119309⟨a, b | aaa=b, aaaab=ba⟩φ(a) = b, φ(b) = bbb
1119312⟨a, b | aaa=b, aaaba=ab⟩φ(a) = b, φ(b) = bbb
1119313⟨a, b | aaa=b, aaaba=ba⟩φ(a) = b, φ(b) = bbb
1119320⟨a, b | aaa=b, aabaa=ab⟩φ(a) = b, φ(b) = bbb
1119851⟨a, b | aaa=b, aaaa=abb⟩φ(a) = b, φ(b) = bbb
1119852⟨a, b | aaa=b, aaaa=bab⟩φ(a) = b, φ(b) = bbb
1120043⟨a, b | aab=a, bbbb=abb⟩φ(a) = bbbbb, φ(b) = b
1124730⟨a, b | aa=b, aaaaab=bb⟩φ(a) = b, φ(b) = bb
1124734⟨a, b | aa=b, aaaaba=bb⟩φ(a) = b, φ(b) = bb
1124742⟨a, b | aa=b, aaabaa=bb⟩φ(a) = b, φ(b) = bb
1125283⟨a, b | aa=b, aaabb=aab⟩φ(a) = b, φ(b) = bb
1125284⟨a, b | aa=b, aaabb=aba⟩φ(a) = b, φ(b) = bb
1125286⟨a, b | aa=b, aaabb=baa⟩φ(a) = b, φ(b) = bb
1125297⟨a, b | aa=b, aabab=aab⟩φ(a) = b, φ(b) = bb
1125298⟨a, b | aa=b, aabab=aba⟩φ(a) = b, φ(b) = bb
1125300⟨a, b | aa=b, aabab=baa⟩φ(a) = b, φ(b) = bb
1125305⟨a, b | aa=b, aabba=aab⟩φ(a) = b, φ(b) = bb
1125306⟨a, b | aa=b, aabba=aba⟩φ(a) = b, φ(b) = bb
1125308⟨a, b | aa=b, aabba=baa⟩φ(a) = b, φ(b) = bb
1125321⟨a, b | aa=b, abaab=aab⟩φ(a) = b, φ(b) = bb
1125322⟨a, b | aa=b, abaab=aba⟩φ(a) = b, φ(b) = bb
1125324⟨a, b | aa=b, abaab=baa⟩φ(a) = b, φ(b) = bb
1125329⟨a, b | aa=b, ababa=aab⟩φ(a) = b, φ(b) = bb
1125330⟨a, b | aa=b, ababa=aba⟩φ(a) = b, φ(b) = bb
1125365⟨a, b | aa=b, baaab=aab⟩φ(a) = b, φ(b) = bb
1125366⟨a, b | aa=b, baaab=aba⟩φ(a) = b, φ(b) = bb
1125745⟨a, b | aa=b, abbb=aaaa⟩φ(a) = b, φ(b) = bb
1125766⟨a, b | aa=b, babb=aaaa⟩φ(a) = b, φ(b) = bb