#245 ⟨a, b | aa=a, 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.

1bb2b3b4b5
11bb2b3b4b5
bbb2b3b4b5b3
b2b2b3b4b5b3b4
b3b3b4b5b3b4b5
b4b4b5b3b4b5b3
b5b5b3b4b5b3b4

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 ⇒ b3 [4]
2. a ⇒ b3 [2]
# ab:aa=a,bbb=a b/a
bbbbbb=bbb
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
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
91686⟨a, b | abb=aa, bbb=a⟩Isomorphic to ℕ(6 = 5)49 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
8969⟨a, b | aa=a, bbb=aa⟩φ(a) = bbb, φ(b) = b
8990⟨a, b | aa=b, bbb=ab⟩φ(a) = b, φ(b) = bb
91591⟨a, b | aaa=ab, abb=b⟩φ(a) = b, φ(b) = bbbbb
92003⟨a, b | aaa=b, aaab=b⟩φ(a) = b, φ(b) = bbb
92005⟨a, b | aaa=b, aaba=b⟩φ(a) = b, φ(b) = bbb
93048⟨a, b | aa=b, aabb=ab⟩φ(a) = b, φ(b) = bb
93049⟨a, b | aa=b, aabb=ba⟩φ(a) = b, φ(b) = bb
93052⟨a, b | aa=b, abab=ab⟩φ(a) = b, φ(b) = bb
93053⟨a, b | aa=b, abab=ba⟩φ(a) = b, φ(b) = bb
93056⟨a, b | aa=b, abba=ab⟩φ(a) = b, φ(b) = bb
93063⟨a, b | aa=b, baab=ab⟩φ(a) = b, φ(b) = bb
93149⟨a, b | aa=a, bbb=aaa⟩φ(a) = bbb, φ(b) = b
93167⟨a, b | aa=b, bbb=aaa⟩φ(a) = b, φ(b) = bb
105045⟨a, b | aaa=ab, aaab=b⟩φ(a) = b, φ(b) = bbbbb
105047⟨a, b | aaa=ab, aaba=b⟩φ(a) = b, φ(b) = bbbbb
105051⟨a, b | aaa=ab, abaa=b⟩φ(a) = b, φ(b) = bbbbb
108916⟨a, b | aa=b, aaaab=ab⟩φ(a) = b, φ(b) = bb
108917⟨a, b | aa=b, aaaab=ba⟩φ(a) = b, φ(b) = bb
108920⟨a, b | aa=b, aaaba=ab⟩φ(a) = b, φ(b) = bb
108921⟨a, b | aa=b, aaaba=ba⟩φ(a) = b, φ(b) = bb
108928⟨a, b | aa=b, aabaa=ab⟩φ(a) = b, φ(b) = bb
109117⟨a, b | aa=a, aaaa=bbb⟩φ(a) = bbb, φ(b) = b
109206⟨a, b | aa=b, aabb=aaa⟩φ(a) = b, φ(b) = bb
109214⟨a, b | aa=b, abab=aaa⟩φ(a) = b, φ(b) = bb
109222⟨a, b | aa=b, abba=aaa⟩φ(a) = b, φ(b) = bb
109236⟨a, b | aa=b, baab=aaa⟩φ(a) = b, φ(b) = bb
1112881⟨a, b | aab=aaa, aaaa=b⟩φ(a) = b, φ(b) = bbbb
1112883⟨a, b | aab=aaa, aaab=b⟩φ(a) = b, φ(b) = bbbb
1112885⟨a, b | aab=aaa, aaba=b⟩φ(a) = b, φ(b) = bbbb
1112887⟨a, b | aab=aaa, aabb=b⟩φ(a) = b, φ(b) = bbbb
1112889⟨a, b | aab=aaa, abaa=b⟩φ(a) = b, φ(b) = bbbb
1112891⟨a, b | aab=aaa, abab=b⟩φ(a) = b, φ(b) = bbbb
1112893⟨a, b | aab=aaa, abba=b⟩φ(a) = b, φ(b) = bbbb
1112895⟨a, b | aab=aaa, abbb=b⟩φ(a) = b, φ(b) = bbbb
1112913⟨a, b | aba=aaa, aaaa=b⟩φ(a) = b, φ(b) = bbbb
1112917⟨a, b | aba=aaa, aaba=b⟩φ(a) = b, φ(b) = bbbb
1112923⟨a, b | aba=aaa, abba=b⟩φ(a) = b, φ(b) = bbbb
1115442⟨a, b | aaa=ab, aaaaa=b⟩φ(a) = b, φ(b) = bbbbb
1118761⟨a, b | aaa=b, aaaaaa=b⟩φ(a) = b, φ(b) = bbb
1119854⟨a, b | aaa=b, aaab=aaa⟩φ(a) = b, φ(b) = bbb
1119862⟨a, b | aaa=b, aaba=aaa⟩φ(a) = b, φ(b) = bbb
1119951⟨a, b | aab=a, aabb=bbb⟩φ(a) = bbbbb, φ(b) = b
1119967⟨a, b | aab=a, abab=bbb⟩φ(a) = bbbbb, φ(b) = b
1119975⟨a, b | aab=a, abba=bbb⟩φ(a) = bbbbb, φ(b) = b
1124725⟨a, b | aa=b, aaaaaa=ab⟩φ(a) = b, φ(b) = bb
1125121⟨a, b | aa=a, aaaaa=bbb⟩φ(a) = bbb, φ(b) = b
1125266⟨a, b | aa=b, aaaab=aaa⟩φ(a) = b, φ(b) = bb
1125274⟨a, b | aa=b, aaaba=aaa⟩φ(a) = b, φ(b) = bb
1125290⟨a, b | aa=b, aabaa=aaa⟩φ(a) = b, φ(b) = bb