#4162 ⟨a, b | abb=aab, 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
bbb2b3b4b5b6b5
b2b2b3b4b5b6b5b6
b3b3b4b5b6b5b6b5
b4b4b5b6b5b6b5b6
b5b5b6b5b6b5b6b5
b6b6b5b6b5b6b5b6

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 ⇒ b5 [4]
2. a ⇒ b3 [2]
# ab:abb=aab,bbb=a b/a
bbbbbbb=bbbbb
a=bbb

Same cardinality

22 unique, 1181 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
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
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.

42 total

Σ#PresentationMapping
104170⟨a, b | abb=aba, bbb=a⟩φ(a) = bbb, φ(b) = b
104182⟨a, b | baa=abb, bbb=a⟩φ(a) = bbb, φ(b) = b
104188⟨a, b | bab=aab, bbb=a⟩φ(a) = bbb, φ(b) = b
104192⟨a, b | bab=aba, bbb=a⟩φ(a) = bbb, φ(b) = b
106383⟨a, b | aab=b, aaaaa=b⟩φ(a) = b, φ(b) = bbbbb
106487⟨a, b | aba=b, aaaaa=b⟩φ(a) = b, φ(b) = bbbbb
109231⟨a, b | aa=b, abbb=abb⟩φ(a) = b, φ(b) = bb
109233⟨a, b | aa=b, abbb=bab⟩φ(a) = b, φ(b) = bb
109234⟨a, b | aa=b, abbb=bba⟩φ(a) = b, φ(b) = bb
109245⟨a, b | aa=b, babb=abb⟩φ(a) = b, φ(b) = bb
109247⟨a, b | aa=b, babb=bab⟩φ(a) = b, φ(b) = bb
109248⟨a, b | aa=b, babb=bba⟩φ(a) = b, φ(b) = bb
1112608⟨a, b | abbb=ab, bbbb=a⟩φ(a) = bbbb, φ(b) = b
1112610⟨a, b | abbb=ba, bbbb=a⟩φ(a) = bbbb, φ(b) = b
1112634⟨a, b | babb=ab, bbbb=a⟩φ(a) = bbbb, φ(b) = b
1112636⟨a, b | babb=ba, bbbb=a⟩φ(a) = bbbb, φ(b) = b
1119505⟨a, b | aab=b, aaaaa=ab⟩φ(a) = b, φ(b) = bbbbbb
1119705⟨a, b | aba=b, aaaaa=ab⟩φ(a) = b, φ(b) = bbbbbb
1125285⟨a, b | aa=b, aaabb=abb⟩φ(a) = b, φ(b) = bb
1125287⟨a, b | aa=b, aaabb=bab⟩φ(a) = b, φ(b) = bb
1125288⟨a, b | aa=b, aaabb=bba⟩φ(a) = b, φ(b) = bb
1125299⟨a, b | aa=b, aabab=abb⟩φ(a) = b, φ(b) = bb
1125301⟨a, b | aa=b, aabab=bab⟩φ(a) = b, φ(b) = bb
1125302⟨a, b | aa=b, aabab=bba⟩φ(a) = b, φ(b) = bb
1125307⟨a, b | aa=b, aabba=abb⟩φ(a) = b, φ(b) = bb
1125309⟨a, b | aa=b, aabba=bab⟩φ(a) = b, φ(b) = bb
1125310⟨a, b | aa=b, aabba=bba⟩φ(a) = b, φ(b) = bb
1125323⟨a, b | aa=b, abaab=abb⟩φ(a) = b, φ(b) = bb
1125325⟨a, b | aa=b, abaab=bab⟩φ(a) = b, φ(b) = bb
1125326⟨a, b | aa=b, abaab=bba⟩φ(a) = b, φ(b) = bb
1125331⟨a, b | aa=b, ababa=abb⟩φ(a) = b, φ(b) = bb
1125332⟨a, b | aa=b, ababa=bab⟩φ(a) = b, φ(b) = bb
1125367⟨a, b | aa=b, baaab=abb⟩φ(a) = b, φ(b) = bb
1125368⟨a, b | aa=b, baaab=bab⟩φ(a) = b, φ(b) = bb
1125746⟨a, b | aa=b, abbb=aaab⟩φ(a) = b, φ(b) = bb
1125747⟨a, b | aa=b, abbb=aaba⟩φ(a) = b, φ(b) = bb
1125749⟨a, b | aa=b, abbb=abaa⟩φ(a) = b, φ(b) = bb
1125755⟨a, b | aa=b, baaa=abbb⟩φ(a) = b, φ(b) = bb
1125767⟨a, b | aa=b, babb=aaab⟩φ(a) = b, φ(b) = bb
1125768⟨a, b | aa=b, babb=aaba⟩φ(a) = b, φ(b) = bb
1125770⟨a, b | aa=b, babb=abaa⟩φ(a) = b, φ(b) = bb
1125774⟨a, b | aa=b, babb=baaa⟩φ(a) = b, φ(b) = bb