#15488 ⟨a, b | aaa=ab, babbb=b⟩

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Quick links

  1. Properties
  2. Elements
  3. Cayley table
  4. Right Cayley graph
  5. Left Cayley graph
  6. Rewriting system
  7. Same cardinality
  8. Isomorphic instances

Properties

Elements

Elements in the center commute with all other 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.

Cayley table

Idempotents are shown in bold.

1aba2abb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7
11aba2abb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7
aaa2ababab5ab2ab6ab3ab7ab4abab5ab2ab6ab3ab7ab4
bbb5b2b2b6b3b7b4bb5b2b6b3b7b4bb5
a2a2abab5ab5ab2ab6ab3ab7ab4abab5ab2ab6ab3ab7ab4ab
ababab5ab2ab2ab6ab3ab7ab4abab5ab2ab6ab3ab7ab4abab5
b2b2b6b3b3b7b4bb5b2b6b3b7b4bb5b2b6
ab2ab2ab6ab3ab3ab7ab4abab5ab2ab6ab3ab7ab4abab5ab2ab6
b3b3b7b4b4bb5b2b6b3b7b4bb5b2b6b3b7
ab3ab3ab7ab4ab4abab5ab2ab6ab3ab7ab4abab5ab2ab6ab3ab7
b4b4bb5b5b2b6b3b7b4bb5b2b6b3b7b4b
ab4ab4abab5ab5ab2ab6ab3ab7ab4abab5ab2ab6ab3ab7ab4ab
b5b5b2b6b6b3b7b4bb5b2b6b3b7b4bb5b2
ab5ab5ab2ab6ab6ab3ab7ab4abab5ab2ab6ab3ab7ab4abab5ab2
b6b6b3b7b7b4bb5b2b6b3b7b4bb5b2b6b3
ab6ab6ab3ab7ab7ab4abab5ab2ab6ab3ab7ab4abab5ab2ab6ab3
b7b7b4bbb5b2b6b3b7b4bb5b2b6b3b7b4
ab7ab7ab4ababab5ab2ab6ab3ab7ab4abab5ab2ab6ab3ab7ab4

Right Cayley graph

Idempotents are shown in bold.

Left 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 ⇒ b [9]
2. ba ⇒ b5 [8]
3. a2b ⇒ ab5 [5]
4. a3 ⇒ ab [1]
# ab:aaa=ab,babbb=b b/a
bbbbbbbb=b
ba=bbbbb
aab=abbbbb
aaa=ab

Same cardinality

16 unique, 68 total

Σ#PresentationDescriptionRelated
104375⟨a, b | aaaa=b, abbbb=1⟩Isomorphic to ℤ1739 iso
106428⟨a, b | aab=b, babbb=a⟩Finite non-commutative monoid with 17 elements2 iso
106514⟨a, b | aba=b, baaab=a⟩Finite non-commutative monoid with 17 elements
107152⟨a, b | bb=aa, abab=aa⟩Finite non-commutative monoid with 17 elements
107153⟨a, b | bb=aa, abab=ab⟩Finite non-commutative monoid with 17 elements
107154⟨a, b | bb=aa, abab=ba⟩Finite non-commutative monoid with 17 elements
1113141⟨a, b | aab=aaa, bbb=ab⟩Finite non-commutative monoid with 17 elements
1113142⟨a, b | aab=aaa, bbb=ba⟩Finite non-commutative monoid with 17 elements
1113151⟨a, b | aba=aaa, baa=bb⟩Finite non-commutative monoid with 17 elements
1113156⟨a, b | aba=aaa, bbb=ab⟩Finite non-commutative monoid with 17 elements
1113218⟨a, b | abb=aba, bbb=aa⟩Finite non-commutative monoid with 17 elements
1114395⟨a, b | aaaa=b, abbbb=a⟩Isomorphic to ℕ(17 = 1)2 iso
1114396⟨a, b | aaaa=b, abbbb=b⟩Isomorphic to ℕ(17 = 4)2 iso
1114698⟨a, b | aabb=a, baaaa=b⟩Finite non-commutative monoid with 17 elements1 iso, 6 anti-iso
1116143⟨a, b | aab=aa, bbbb=ba⟩Finite non-commutative monoid with 17 elements
1116521⟨a, b | aab=bb, bab=aaa⟩Finite non-commutative monoid with 17 elements

Isomorphic instances

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

3 total

Σ#PresentationMapping
1115496⟨a, b | aaa=ab, bbabb=b⟩φ(a) = a, φ(b) = b
1115500⟨a, b | aaa=ab, bbbab=b⟩φ(a) = a, φ(b) = b
1115502⟨a, b | aaa=ab, bbbba=b⟩φ(a) = a, φ(b) = b