#4375 ⟨a, b | aaaa=b, abbbb=1⟩

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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 order of x is the least n such that xn = 1.

Staircase diagram

Cayley table

1aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16
11aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16
aaa2a3a4a5a6a7a8a9a10a11a12a13a14a15a161
a2a2a3a4a5a6a7a8a9a10a11a12a13a14a15a161a
a3a3a4a5a6a7a8a9a10a11a12a13a14a15a161aa2
a4a4a5a6a7a8a9a10a11a12a13a14a15a161aa2a3
a5a5a6a7a8a9a10a11a12a13a14a15a161aa2a3a4
a6a6a7a8a9a10a11a12a13a14a15a161aa2a3a4a5
a7a7a8a9a10a11a12a13a14a15a161aa2a3a4a5a6
a8a8a9a10a11a12a13a14a15a161aa2a3a4a5a6a7
a9a9a10a11a12a13a14a15a161aa2a3a4a5a6a7a8
a10a10a11a12a13a14a15a161aa2a3a4a5a6a7a8a9
a11a11a12a13a14a15a161aa2a3a4a5a6a7a8a9a10
a12a12a13a14a15a161aa2a3a4a5a6a7a8a9a10a11
a13a13a14a15a161aa2a3a4a5a6a7a8a9a10a11a12
a14a14a15a161aa2a3a4a5a6a7a8a9a10a11a12a13
a15a15a161aa2a3a4a5a6a7a8a9a10a11a12a13a14
a16a161aa2a3a4a5a6a7a8a9a10a11a12a13a14a15

Right Cayley graph

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. a17 ⇒ 1 [2]
2. b ⇒ a4 [1]
# ab:aaaa=b,abbbb=1 a/b
aaaaaaaaaaaaaaaaa=1
b=aaaa

Same cardinality

16 unique, 32 total

Σ#PresentationDescriptionRelated
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
1115488⟨a, b | aaa=ab, babbb=b⟩Finite non-commutative monoid with 17 elements3 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.

39 total

Σ#PresentationMapping
104379⟨a, b | aaaa=b, babbb=1⟩φ(a) = bbbbbbbbbbbbb, φ(b) = b
104380⟨a, b | aaaa=b, bbabb=1⟩φ(a) = bbbbbbbbbbbbb, φ(b) = b
1111121⟨a, b | aaaaa=b, aabbb=1⟩φ(a) = bbbbbbb, φ(b) = b
1111124⟨a, b | aaaaa=b, ababb=1⟩φ(a) = bbbbbbb, φ(b) = b
1111125⟨a, b | aaaaa=b, abbab=1⟩φ(a) = bbbbbbb, φ(b) = b
1111126⟨a, b | aaaaa=b, abbba=1⟩φ(a) = bbbbbbb, φ(b) = b
1111129⟨a, b | aaaaa=b, baabb=1⟩φ(a) = bbbbbbb, φ(b) = b
1111130⟨a, b | aaaaa=b, babab=1⟩φ(a) = bbbbbbb, φ(b) = b
1114998⟨a, b | aaa=bb, abbbbb=1⟩φ(a) = bbbbbbbbbbbb, φ(b) = b
1115005⟨a, b | aaa=bb, babbbb=1⟩φ(a) = bbbbbbbbbbbb, φ(b) = b
1115007⟨a, b | aaa=bb, bbabbb=1⟩φ(a) = bbbbbbbbbbbb, φ(b) = b
1116695⟨a, b | aaab=1, abbbbbb=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1116727⟨a, b | aaab=1, babbbbb=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1116743⟨a, b | aaab=1, bbabbbb=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1116751⟨a, b | aaab=1, bbbabbb=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1116755⟨a, b | aaab=1, bbbbabb=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1116757⟨a, b | aaab=1, bbbbbab=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1116758⟨a, b | aaab=1, bbbbbba=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1116823⟨a, b | aaba=1, abbbbbb=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1116855⟨a, b | aaba=1, babbbbb=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1116871⟨a, b | aaba=1, bbabbbb=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1116879⟨a, b | aaba=1, bbbabbb=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1116883⟨a, b | aaba=1, bbbbabb=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1116885⟨a, b | aaba=1, bbbbbab=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1116886⟨a, b | aaba=1, bbbbbba=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1117812⟨a, b | aaab=1, bbbbb=aa⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1117940⟨a, b | aaba=1, bbbbb=aa⟩φ(a) = bbbbbbbbbbb, φ(b) = b
1118245⟨a, b | aaa=b, aabbbbb=1⟩φ(a) = bbbbbb, φ(b) = b
1118257⟨a, b | aaa=b, ababbbb=1⟩φ(a) = bbbbbb, φ(b) = b
1118262⟨a, b | aaa=b, abbabbb=1⟩φ(a) = bbbbbb, φ(b) = b
1118264⟨a, b | aaa=b, abbbabb=1⟩φ(a) = bbbbbb, φ(b) = b
1118265⟨a, b | aaa=b, abbbbab=1⟩φ(a) = bbbbbb, φ(b) = b
1118266⟨a, b | aaa=b, abbbbba=1⟩φ(a) = bbbbbb, φ(b) = b
1118275⟨a, b | aaa=b, baabbbb=1⟩φ(a) = bbbbbb, φ(b) = b
1118278⟨a, b | aaa=b, bababbb=1⟩φ(a) = bbbbbb, φ(b) = b
1118279⟨a, b | aaa=b, babbabb=1⟩φ(a) = bbbbbb, φ(b) = b
1118280⟨a, b | aaa=b, babbbab=1⟩φ(a) = bbbbbb, φ(b) = b
1118283⟨a, b | aaa=b, bbaabbb=1⟩φ(a) = bbbbbb, φ(b) = b
1118284⟨a, b | aaa=b, bbababb=1⟩φ(a) = bbbbbb, φ(b) = b