#1686 ⟨a, b | abb=aa, 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

The zero element z satisfies zy = yz = z for all y.
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.

1bb2b3b4b5
11bb2b3b4b5
bbb2b3b4b5b5
b2b2b3b4b5b5b5
b3b3b4b5b5b5b5
b4b4b5b5b5b5b5
b5b5b5b5b5b5b5

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. b6 ⇒ b5 [4]
2. a ⇒ b3 [2]
# ab:abb=aa,bbb=a b/a
bbbbbb=bbbbb
a=bbb

Same cardinality

19 unique, 1876 total

Σ#PresentationDescriptionRelated
622⟨a, b | ab=aa, bb=1⟩Finite non-commutative monoid with 6 elements44 iso, 63 anti-iso
626⟨a, b | aaa=1, abb=1⟩Isomorphic to ℤ61373 iso
7160⟨a, b | ab=aa, bb=b⟩Finite non-commutative monoid with 6 elements4 anti-iso
7245⟨a, b | aa=a, bbb=a⟩Isomorphic to ℕ(6 = 3)49 iso
7257⟨a, b | aa=b, bbb=a⟩Isomorphic to ℕ(6 = 1)61 iso
7258⟨a, b | aa=b, bbb=b⟩Isomorphic to ℕ(6 = 2)55 iso
8639⟨a, b | ab=aa, baa=b⟩Finite non-commutative monoid with 6 elements5 iso, 8 anti-iso
8644⟨a, b | ab=aa, bbb=a⟩Isomorphic to ℕ(6 = 4)46 iso
8893⟨a, b | aa=a, abbb=b⟩Finite non-commutative monoid with 6 elements19 iso
81011⟨a, b | ab=a, baa=bb⟩Finite non-commutative monoid with 6 elements12 iso, 1 anti-iso
91427⟨a, b | abba=b, baba=1⟩Finite non-Abelian group with 6 elements66 iso
91581⟨a, b | aaa=aa, abb=b⟩Finite non-commutative monoid with 6 elements5 iso
91648⟨a, b | aab=bb, abb=a⟩Finite commutative monoid with 6 elements13 iso
93075⟨a, b | ab=a, aaaa=bb⟩Finite commutative monoid with 6 elements14 iso
93132⟨a, b | ab=a, bbbb=aa⟩Finite commutative monoid with 6 elements5 iso
93134⟨a, b | ab=a, bbbb=ba⟩Finite non-commutative monoid with 6 elements2 iso
93193⟨a, b | ab=a, bbb=aaa⟩Finite commutative monoid with 6 elements9 iso
93199⟨a, b | ab=a, bbb=bba⟩Finite non-commutative monoid with 6 elements3 iso
1124145⟨a, b | aa=a, abbbbba=b⟩Finite commutative monoid with 6 elements

Isomorphic instances

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

49 total

Σ#PresentationMapping
91694⟨a, b | bab=aa, bbb=a⟩φ(a) = bbb, φ(b) = b
92998⟨a, b | ab=a, bbbbb=a⟩φ(a) = bbbbb, φ(b) = b
93170⟨a, b | aa=b, bbb=abb⟩φ(a) = b, φ(b) = bb
93171⟨a, b | aa=b, bbb=bab⟩φ(a) = b, φ(b) = bb
105127⟨a, b | aab=ab, aaaa=b⟩φ(a) = b, φ(b) = bbbb
105159⟨a, b | aab=ba, aaaa=b⟩φ(a) = b, φ(b) = bbbb
105243⟨a, b | aba=ab, aaaa=b⟩φ(a) = b, φ(b) = bbbb
109109⟨a, b | ab=a, bbbbb=ab⟩φ(a) = bbbbb, φ(b) = b
109197⟨a, b | aa=b, aaab=bbb⟩φ(a) = b, φ(b) = bb
109205⟨a, b | aa=b, aaba=bbb⟩φ(a) = b, φ(b) = bb
109209⟨a, b | aa=b, aabb=abb⟩φ(a) = b, φ(b) = bb
109211⟨a, b | aa=b, aabb=bab⟩φ(a) = b, φ(b) = bb
109212⟨a, b | aa=b, aabb=bba⟩φ(a) = b, φ(b) = bb
109217⟨a, b | aa=b, abab=abb⟩φ(a) = b, φ(b) = bb
109219⟨a, b | aa=b, abab=bab⟩φ(a) = b, φ(b) = bb
109220⟨a, b | aa=b, abab=bba⟩φ(a) = b, φ(b) = bb
109225⟨a, b | aa=b, abba=abb⟩φ(a) = b, φ(b) = bb
109226⟨a, b | aa=b, abba=bab⟩φ(a) = b, φ(b) = bb
109239⟨a, b | aa=b, baab=abb⟩φ(a) = b, φ(b) = bb
109240⟨a, b | aa=b, baab=bab⟩φ(a) = b, φ(b) = bb
1119306⟨a, b | aaa=b, aaaaa=bb⟩φ(a) = b, φ(b) = bbb
1119855⟨a, b | aaa=b, aaab=aab⟩φ(a) = b, φ(b) = bbb
1119856⟨a, b | aaa=b, aaab=aba⟩φ(a) = b, φ(b) = bbb
1119858⟨a, b | aaa=b, aaab=baa⟩φ(a) = b, φ(b) = bbb
1119863⟨a, b | aaa=b, aaba=aab⟩φ(a) = b, φ(b) = bbb
1119864⟨a, b | aaa=b, aaba=aba⟩φ(a) = b, φ(b) = bbb
1119866⟨a, b | aaa=b, aaba=baa⟩φ(a) = b, φ(b) = bbb
1125265⟨a, b | aa=b, aaaaa=bbb⟩φ(a) = b, φ(b) = bb
1125269⟨a, b | aa=b, aaaab=abb⟩φ(a) = b, φ(b) = bb
1125271⟨a, b | aa=b, aaaab=bab⟩φ(a) = b, φ(b) = bb
1125272⟨a, b | aa=b, aaaab=bba⟩φ(a) = b, φ(b) = bb
1125277⟨a, b | aa=b, aaaba=abb⟩φ(a) = b, φ(b) = bb
1125279⟨a, b | aa=b, aaaba=bab⟩φ(a) = b, φ(b) = bb
1125280⟨a, b | aa=b, aaaba=bba⟩φ(a) = b, φ(b) = bb
1125293⟨a, b | aa=b, aabaa=abb⟩φ(a) = b, φ(b) = bb
1125294⟨a, b | aa=b, aabaa=bab⟩φ(a) = b, φ(b) = bb
1125655⟨a, b | ab=a, bbbbb=abb⟩φ(a) = bbbbb, φ(b) = b
1125730⟨a, b | aa=b, aabb=aaab⟩φ(a) = b, φ(b) = bb
1125731⟨a, b | aa=b, aabb=aaba⟩φ(a) = b, φ(b) = bb
1125734⟨a, b | aa=b, abaa=aabb⟩φ(a) = b, φ(b) = bb
1125736⟨a, b | aa=b, abab=aaab⟩φ(a) = b, φ(b) = bb
1125737⟨a, b | aa=b, abab=aaba⟩φ(a) = b, φ(b) = bb
1125739⟨a, b | aa=b, abab=abaa⟩φ(a) = b, φ(b) = bb
1125741⟨a, b | aa=b, abba=aaab⟩φ(a) = b, φ(b) = bb
1125742⟨a, b | aa=b, abba=aaba⟩φ(a) = b, φ(b) = bb
1125753⟨a, b | aa=b, baaa=aabb⟩φ(a) = b, φ(b) = bb
1125754⟨a, b | aa=b, baaa=abab⟩φ(a) = b, φ(b) = bb
1125757⟨a, b | aa=b, baab=aaab⟩φ(a) = b, φ(b) = bb
1125758⟨a, b | aa=b, baab=aaba⟩φ(a) = b, φ(b) = bb