#20927 ⟨a, b | ab=aa, aaaa=bbb⟩

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Properties

Element profile

Complete 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
  1. ab ⇒ a2
  2. a4 ⇒ b3
  3. b3a ⇒ b3
  4. b4 ⇒ b3 
# ab:ab=aa,aaaa=bbb ab
ab=aa
aaaa=bbb
bbba=bbb
bbbb=bbb

Cayley table

Idempotents are shown in bold.

1aba2bab2a3ba2b2ab3ba3b2a2b2a3
11aba2bab2a3ba2b2ab3ba3b2a2b2a3
aaa2a2a3a3a3b3b3b3b3b3b3b3
bbbab2ba2b2ab3ba3b2a2b3b3b2a3b3b3
a2a2a3a3b3b3b3b3b3b3b3b3b3b3
bababa2ba2ba3ba3ba3b3b3b3b3b3b3b3
b2b2b2ab3b2a2b3b3b2a3b3b3b3b3b3b3
a3a3b3b3b3b3b3b3b3b3b3b3b3b3
ba2ba2ba3ba3b3b3b3b3b3b3b3b3b3b3
b2ab2ab2a2b2a2b2a3b2a3b2a3b3b3b3b3b3b3b3
b3b3b3b3b3b3b3b3b3b3b3b3b3b3
ba3ba3b3b3b3b3b3b3b3b3b3b3b3b3
b2a2b2a2b2a3b2a3b3b3b3b3b3b3b3b3b3b3
b2a3b2a3b3b3b3b3b3b3b3b3b3b3b3b3

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

26 unique, 296 total

Σ#PresentationDescriptionRelated
91328⟨a, b | aaaa=b, abbb=1⟩Isomorphic to ℤ13189 iso
92118⟨a, b | aba=b, baab=a⟩Finite non-commutative monoid with 13 elements4 iso
104642⟨a, b | aaaa=b, abbb=a⟩Isomorphic to ℕ(13 = 1)10 iso
104643⟨a, b | aaaa=b, abbb=b⟩Isomorphic to ℕ(13 = 4)6 iso
104683⟨a, b | aaab=b, abba=a⟩Finite non-commutative monoid with 13 elements15 iso, 1 anti-iso
105065⟨a, b | aaa=ab, babb=b⟩Finite non-commutative monoid with 13 elements7 iso
105334⟨a, b | aaa=ab, bba=bb⟩Finite non-commutative monoid with 13 elements
105336⟨a, b | aaa=ab, bbb=ab⟩Finite non-commutative monoid with 13 elements1 iso
105337⟨a, b | aaa=ab, bbb=ba⟩Finite non-commutative monoid with 13 elements
105340⟨a, b | aaa=bb, aab=ba⟩Finite non-commutative monoid with 13 elements
106305⟨a, b | aaa=b, abbbb=b⟩Isomorphic to ℕ(13 = 3)2 iso
107143⟨a, b | bb=aa, aaab=ba⟩Finite non-commutative monoid with 13 elements1 iso
1112240⟨a, b | aaab=aa, bbbb=a⟩Isomorphic to ℕ(13 = 8)1 iso
1112268⟨a, b | aaab=ab, bbbb=a⟩Isomorphic to ℕ(13 = 5)3 iso
1112324⟨a, b | aaab=bb, bbbb=a⟩Isomorphic to ℕ(13 = 2)14 iso
1114647⟨a, b | aaba=b, babbb=a⟩Finite commutative monoid with 13 elements2 iso
1115520⟨a, b | aaa=bb, aabbb=b⟩Finite commutative monoid with 13 elements2 iso
1116012⟨a, b | aaa=ab, abbb=bb⟩Finite non-commutative monoid with 13 elements
1116069⟨a, b | aaa=bb, abbb=ba⟩Finite non-commutative monoid with 13 elements1 anti-iso
1116205⟨a, b | aab=ab, bbbb=aa⟩Finite non-commutative monoid with 13 elements1 iso, 1 anti-iso
1116459⟨a, b | aba=bb, abbb=aa⟩Finite non-commutative monoid with 13 elements1 iso
1116506⟨a, b | aab=ab, bbb=aaa⟩Finite non-commutative monoid with 13 elements1 iso
1116515⟨a, b | aab=ba, bbb=aaa⟩Finite non-commutative monoid with 13 elements
1118958⟨a, b | aab=a, bbbbbb=a⟩Isomorphic to ℕ(13 = 6)4 iso
1120046⟨a, b | aab=a, bbbb=bba⟩Finite non-commutative monoid with 13 elements
1120991⟨a, b | ab=aa, baaa=bbb⟩Finite non-commutative monoid with 13 elements3 iso

Other isomorphic instances

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

7 total

Σ#PresentationMapping
1120935⟨a, b | ab=aa, aaab=bbb⟩φ(a) = a, φ(b) = b
1120943⟨a, b | ab=aa, aaba=bbb⟩φ(a) = a, φ(b) = b
1120951⟨a, b | ab=aa, aabb=bbb⟩φ(a) = a, φ(b) = b
1120959⟨a, b | ab=aa, abaa=bbb⟩φ(a) = a, φ(b) = b
1120967⟨a, b | ab=aa, abab=bbb⟩φ(a) = a, φ(b) = b
1120975⟨a, b | ab=aa, abba=bbb⟩φ(a) = a, φ(b) = b
1120983⟨a, b | ab=aa, abbb=bbb⟩φ(a) = a, φ(b) = b