#5404 ⟨a, b | aab=bb, aba=aa⟩

Up:Monoids with two generators and two relations
Prev:#5401 ⟨a, b | aab=ba, bab=bb⟩
Next:#5405 ⟨a, b | aab=bb, aba=ab⟩

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. a6 ⇒ a4
  2. ba3 ⇒ a5
  3. aba ⇒ a2
  4. a5b ⇒ a3b
  5. b2 ⇒ a2b
  6. ba2b ⇒ a4b 
# ab:aab=bb,aba=aa a/b
aaaaaa=aaaa
baaa=aaaaa
aba=aa
aaaaab=aaab
bb=aab
baab=aaaab

Cayley table

Idempotents are shown in bold.

1aba2abbaa3a2bba2baba4a3ba5a4b
11aba2abbaa3a2bba2baba4a3ba5a4b
aaa2aba3a2ba2a4a3ba3a2ba5a4ba4a3b
bbbaa2bba2baba3a5a4ba4a3ba4a3ba5a4b
a2a2a3a2ba4a3ba3a5a4ba4a3ba4a3ba5a4b
ababa2a3ba3a2ba4a4a3ba5a4ba5a4ba4a3b
bababa2baba5a4bba2a4a3ba5a4ba5a4ba4a3b
a3a3a4a3ba5a4ba4a4a3ba5a4ba5a4ba4a3b
a2ba2ba3a4ba4a3ba5a5a4ba4a3ba4a3ba5a4b
ba2ba2a5a4ba4a3ba5a5a4ba4a3ba4a3ba5a4b
babbabba2a3ba5a4ba4a4a3ba5a4ba5a4ba4a3b
a4a4a5a4ba4a3ba5a5a4ba4a3ba4a3ba5a4b
a3ba3ba4a3ba5a4ba4a4a3ba5a4ba5a4ba4a3b
a5a5a4a3ba5a4ba4a4a3ba5a4ba5a4ba4a3b
a4ba4ba5a4ba4a3ba5a5a4ba4a3ba4a3ba5a4b

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

30 unique, 315 total

Σ#PresentationDescriptionRelated
81123⟨a, b | aa=1, abbbb=b⟩Finite non-commutative monoid with 14 elements42 iso, 17 anti-iso
91682⟨a, b | aba=bb, bab=a⟩Finite non-commutative monoid with 14 elements1 iso
93034⟨a, b | aa=a, bbbb=ab⟩Finite non-commutative monoid with 14 elements2 iso
103773⟨a, b | aaaa=bb, abbb=1⟩Isomorphic to ℤ14165 iso
105218⟨a, b | aab=bb, bbba=a⟩Finite non-commutative monoid with 14 elements5 iso, 1 anti-iso
106718⟨a, b | aab=b, bbba=aa⟩Finite non-commutative monoid with 14 elements2 iso
1112183⟨a, b | aaaa=ab, baab=b⟩Finite non-commutative monoid with 14 elements3 iso
1112206⟨a, b | aaaa=bb, abbb=a⟩Finite commutative monoid with 14 elements1 iso
1112207⟨a, b | aaaa=bb, abbb=b⟩Finite commutative monoid with 14 elements1 iso
1112441⟨a, b | aabb=aa, baab=b⟩Finite non-commutative monoid with 14 elements3 iso
1112499⟨a, b | abab=aa, abba=b⟩Finite non-commutative monoid with 14 elements
1114383⟨a, b | aaaa=b, aabbb=a⟩Isomorphic to ℕ(14 = 1)15 iso
1114384⟨a, b | aaaa=b, aabbb=b⟩Isomorphic to ℕ(14 = 4)5 iso
1115532⟨a, b | aaa=bb, abbbb=b⟩Isomorphic to ℕ(14 = 3)11 iso
1115539⟨a, b | aaa=bb, babbb=a⟩Finite commutative monoid with 14 elements
1116020⟨a, b | aaa=ab, baab=bb⟩Finite non-commutative monoid with 14 elements1 iso
1116079⟨a, b | aaa=bb, bbbb=ab⟩Finite non-commutative monoid with 14 elements
1116293⟨a, b | aab=bb, abab=aa⟩Finite non-commutative monoid with 14 elements
1116470⟨a, b | aba=bb, bbbb=aa⟩Finite non-commutative monoid with 14 elements
1119552⟨a, b | aab=b, abbaa=aa⟩Finite non-commutative monoid with 14 elements2 iso
1120844⟨a, b | ab=aa, bbbbb=aa⟩Finite non-commutative monoid with 14 elements1 iso
1120846⟨a, b | ab=aa, bbbbb=ba⟩Finite non-commutative monoid with 14 elements
1121023⟨a, b | ab=aa, bbaa=bbb⟩Finite non-commutative monoid with 14 elements1 iso
1121040⟨a, b | ab=aa, bbbb=aaa⟩Finite non-commutative monoid with 14 elements3 iso
1121044⟨a, b | ab=aa, bbbb=baa⟩Finite non-commutative monoid with 14 elements1 iso
1121046⟨a, b | ab=aa, bbbb=bba⟩Finite non-commutative monoid with 14 elements
1121110⟨a, b | bb=aa, abab=aaa⟩Finite non-commutative monoid with 14 elements2 anti-iso
1124186⟨a, b | aa=a, bbbbbbb=a⟩Isomorphic to ℕ(14 = 7)
1124331⟨a, b | aa=b, bbbbbbb=b⟩Isomorphic to ℕ(14 = 2)
1125055⟨a, b | ab=a, bbaaaa=bb⟩Finite non-commutative monoid with 14 elements

Other isomorphic instances

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

1 total

Σ#PresentationMapping
1116355⟨a, b | aba=aa, abab=bb⟩φ(a) = a, φ(b) = b