#2206 ⟨a, b | ab=aa, bbbb=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.

1bb2b3b4b5b6b7
11bb2b3b4b5b6b7
bbb2b3b4b5b6b7b5
b2b2b3b4b5b6b7b5b6
b3b3b4b5b6b7b5b6b7
b4b4b5b6b7b5b6b7b5
b5b5b6b7b5b6b7b5b6
b6b6b7b5b6b7b5b6b7
b7b7b5b6b7b5b6b7b5

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. b8 ⇒ b5 [4]
2. a ⇒ b4 [2]
# ab:ab=aa,bbbb=a b/a
bbbbbbbb=bbbbb
a=bbbb

Same cardinality

33 unique, 1179 total

Σ#PresentationDescriptionRelated
7188⟨a, b | aab=1, bbbb=1⟩Isomorphic to ℤ8727 iso
7330⟨a, b | aa=1, abba=b⟩Finite non-commutative monoid with 8 elements60 iso
8556⟨a, b | aba=b, aabb=1⟩Finite non-Abelian group with 8 elements28 iso
8898⟨a, b | aa=a, bbbb=a⟩Isomorphic to ℕ(8 = 4)48 iso
8918⟨a, b | aa=b, bbbb=a⟩Isomorphic to ℕ(8 = 1)34 iso
8919⟨a, b | aa=b, bbbb=b⟩Isomorphic to ℕ(8 = 2)46 iso
8961⟨a, b | aa=a, aba=bb⟩Finite non-commutative monoid with 8 elements5 iso
91605⟨a, b | aaa=bb, abb=b⟩Isomorphic to ℕ(8 = 3)55 iso
91606⟨a, b | aaa=bb, bab=a⟩Finite commutative monoid with 8 elements14 iso
91615⟨a, b | aab=aa, baa=b⟩Finite non-commutative monoid with 8 elements9 iso, 5 anti-iso
91650⟨a, b | aab=bb, baa=a⟩Finite non-commutative monoid with 8 elements4 iso, 11 anti-iso
92220⟨a, b | bb=aa, aaab=a⟩Finite commutative monoid with 8 elements22 iso
92247⟨a, b | ab=aa, baa=bb⟩Finite non-commutative monoid with 8 elements1 iso
92256⟨a, b | ab=aa, bbb=aa⟩Finite non-commutative monoid with 8 elements1 iso
92258⟨a, b | ab=aa, bbb=ba⟩Finite non-commutative monoid with 8 elements
92883⟨a, b | aa=a, abbbb=b⟩Finite non-commutative monoid with 8 elements14 iso
93107⟨a, b | ab=a, baaa=bb⟩Finite non-commutative monoid with 8 elements9 iso
93123⟨a, b | ab=a, bbaa=bb⟩Finite non-commutative monoid with 8 elements10 iso
105191⟨a, b | aab=bb, aaaa=b⟩Isomorphic to ℕ(8 = 6)18 iso
106664⟨a, b | aab=b, aaaa=ba⟩Finite non-commutative monoid with 8 elements1 iso
107057⟨a, b | ab=aa, aaaa=bb⟩Finite non-commutative monoid with 8 elements7 iso
109380⟨a, b | ab=a, bbbb=baa⟩Finite non-commutative monoid with 8 elements2 iso
1112606⟨a, b | abbb=aa, bbbb=a⟩Isomorphic to ℕ(8 = 7)14 iso
1120047⟨a, b | aab=a, bbbb=bbb⟩Finite non-commutative monoid with 8 elements
1120051⟨a, b | aab=b, aaaa=abb⟩Finite commutative monoid with 8 elements1 iso
1124863⟨a, b | ab=a, aaaaaa=bb⟩Finite commutative monoid with 8 elements
1125112⟨a, b | ab=a, bbbbbb=aa⟩Finite commutative monoid with 8 elements
1125114⟨a, b | ab=a, bbbbbb=ba⟩Finite non-commutative monoid with 8 elements
1125411⟨a, b | ab=a, aaaaa=bbb⟩Finite commutative monoid with 8 elements
1125652⟨a, b | ab=a, bbbbb=aaa⟩Finite commutative monoid with 8 elements
1125658⟨a, b | ab=a, bbbbb=bba⟩Finite non-commutative monoid with 8 elements
1125897⟨a, b | ab=a, bbbb=aaaa⟩Finite commutative monoid with 8 elements
1125911⟨a, b | ab=a, bbbb=bbba⟩Finite non-commutative monoid with 8 elements

Isomorphic instances

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

32 total

Σ#PresentationMapping
109253⟨a, b | aa=b, bbbb=abb⟩φ(a) = b, φ(b) = bb
109254⟨a, b | aa=b, bbbb=bab⟩φ(a) = b, φ(b) = bb
1114474⟨a, b | aaab=b, aaaaa=b⟩φ(a) = b, φ(b) = bbbbb
1114602⟨a, b | aaba=b, aaaaa=b⟩φ(a) = b, φ(b) = bbbbb
1119871⟨a, b | aaa=b, aabb=aab⟩φ(a) = b, φ(b) = bbb
1119872⟨a, b | aaa=b, aabb=aba⟩φ(a) = b, φ(b) = bbb
1119874⟨a, b | aaa=b, aabb=baa⟩φ(a) = b, φ(b) = bbb
1119879⟨a, b | aaa=b, abab=aab⟩φ(a) = b, φ(b) = bbb
1119880⟨a, b | aaa=b, abab=aba⟩φ(a) = b, φ(b) = bbb
1119882⟨a, b | aaa=b, abab=baa⟩φ(a) = b, φ(b) = bbb
1119887⟨a, b | aaa=b, abba=aab⟩φ(a) = b, φ(b) = bbb
1119888⟨a, b | aaa=b, abba=aba⟩φ(a) = b, φ(b) = bbb
1119901⟨a, b | aaa=b, baab=aab⟩φ(a) = b, φ(b) = bbb
1119902⟨a, b | aaa=b, baab=aba⟩φ(a) = b, φ(b) = bbb
1125315⟨a, b | aa=b, aabbb=abb⟩φ(a) = b, φ(b) = bb
1125317⟨a, b | aa=b, aabbb=bab⟩φ(a) = b, φ(b) = bb
1125318⟨a, b | aa=b, aabbb=bba⟩φ(a) = b, φ(b) = bb
1125337⟨a, b | aa=b, ababb=abb⟩φ(a) = b, φ(b) = bb
1125339⟨a, b | aa=b, ababb=bab⟩φ(a) = b, φ(b) = bb
1125340⟨a, b | aa=b, ababb=bba⟩φ(a) = b, φ(b) = bb
1125345⟨a, b | aa=b, abbab=abb⟩φ(a) = b, φ(b) = bb
1125347⟨a, b | aa=b, abbab=bab⟩φ(a) = b, φ(b) = bb
1125348⟨a, b | aa=b, abbab=bba⟩φ(a) = b, φ(b) = bb
1125353⟨a, b | aa=b, abbba=abb⟩φ(a) = b, φ(b) = bb
1125354⟨a, b | aa=b, abbba=bab⟩φ(a) = b, φ(b) = bb
1125373⟨a, b | aa=b, baabb=abb⟩φ(a) = b, φ(b) = bb
1125375⟨a, b | aa=b, baabb=bab⟩φ(a) = b, φ(b) = bb
1125376⟨a, b | aa=b, baabb=bba⟩φ(a) = b, φ(b) = bb
1125381⟨a, b | aa=b, babab=abb⟩φ(a) = b, φ(b) = bb
1125382⟨a, b | aa=b, babab=bab⟩φ(a) = b, φ(b) = bb
1125784⟨a, b | aa=b, bbbb=aaab⟩φ(a) = b, φ(b) = bb
1125785⟨a, b | aa=b, bbbb=aaba⟩φ(a) = b, φ(b) = bb