#402 ⟨a, b | aaa=ab, bbb=1⟩

Up:Monoids with two generators and two relations
Prev:#395 ⟨a, b | aaa=aa, bbb=1⟩
Next:#413 ⟨a, b | aab=aa, bbb=1⟩

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. a7 ⇒ a
  2. ab ⇒ a3
  3. b3 ⇒ 1 
# ab:aaa=ab,bbb=1 a/b
aaaaaaa=a
ab=aaa
bbb=1

Cayley table

Idempotents are shown in bold.

1aba2bab2a3ba2b2aa4ba3b2a2a5ba4b2a3a6ba5b2a4ba6b2a5b2a6
11aba2bab2a3ba2b2aa4ba3b2a2a5ba4b2a3a6ba5b2a4ba6b2a5b2a6
aaa2a3a3a4a5a4a5a6a5a6aa6aa2aa2a3a3a4a5
bbbab2ba2b2a1ba3b2a2aba4b2a3a2ba5b2a4a3ba6b2a5a4b2a6a5a6
a2a2a3a4a4a5a6a5a6aa6aa2aa2a3a2a3a4a4a5a6
bababa2ba3ba3ba4ba5ba4ba5ba6ba5ba6baba6baba2baba2ba3ba3ba4ba5
b2b2b2a1b2a2abb2a3a2bab2a4a3ba2b2a5a4ba3b2a6a5ba4a6ba5ba6
a3a3a4a5a5a6aa6aa2aa2a3a2a3a4a3a4a5a5a6a
ba2ba2ba3ba4ba4ba5ba6ba5ba6baba6baba2baba2ba3ba2ba3ba4ba4ba5ba6
b2ab2ab2a2b2a3b2a3b2a4b2a5b2a4b2a5b2a6b2a5b2a6b2ab2a6b2ab2a2b2ab2a2b2a3b2a3b2a4b2a5
a4a4a5a6a6aa2aa2a3a2a3a4a3a4a5a4a5a6a6aa2
ba3ba3ba4ba5ba5ba6baba6baba2baba2ba3ba2ba3ba4ba3ba4ba5ba5ba6ba
b2a2b2a2b2a3b2a4b2a4b2a5b2a6b2a5b2a6b2ab2a6b2ab2a2b2ab2a2b2a3b2a2b2a3b2a4b2a4b2a5b2a6
a5a5a6aaa2a3a2a3a4a3a4a5a4a5a6a5a6aaa2a3
ba4ba4ba5ba6ba6baba2baba2ba3ba2ba3ba4ba3ba4ba5ba4ba5ba6ba6baba2
b2a3b2a3b2a4b2a5b2a5b2a6b2ab2a6b2ab2a2b2ab2a2b2a3b2a2b2a3b2a4b2a3b2a4b2a5b2a5b2a6b2a
a6a6aa2a2a3a4a3a4a5a4a5a6a5a6aa6aa2a2a3a4
ba5ba5ba6bababa2ba3ba2ba3ba4ba3ba4ba5ba4ba5ba6ba5ba6bababa2ba3
b2a4b2a4b2a5b2a6b2a6b2ab2a2b2ab2a2b2a3b2a2b2a3b2a4b2a3b2a4b2a5b2a4b2a5b2a6b2a6b2ab2a2
ba6ba6baba2ba2ba3ba4ba3ba4ba5ba4ba5ba6ba5ba6baba6baba2ba2ba3ba4
b2a5b2a5b2a6b2ab2ab2a2b2a3b2a2b2a3b2a4b2a3b2a4b2a5b2a4b2a5b2a6b2a5b2a6b2ab2ab2a2b2a3
b2a6b2a6b2ab2a2b2a2b2a3b2a4b2a3b2a4b2a5b2a4b2a5b2a6b2a5b2a6b2ab2a6b2ab2a2b2a2b2a3b2a4

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

8 unique, 25 total

Σ#PresentationDescriptionRelated
91599⟨a, b | aaa=ab, bbb=b⟩Finite non-commutative monoid with 21 elements1 anti-iso
105309⟨a, b | aaa=aa, bbb=ab⟩Finite non-commutative monoid with 21 elements
106265⟨a, b | aaa=a, abbbb=b⟩Finite non-commutative monoid with 21 elements2 iso
1111127⟨a, b | aaaaa=b, abbbb=1⟩Isomorphic to ℤ2113 iso
1113091⟨a, b | bab=aaa, abba=b⟩Finite non-commutative monoid with 21 elements
1113248⟨a, b | bab=aaa, bbb=aa⟩Finite non-commutative monoid with 21 elements
1115158⟨a, b | aab=ba, ababab=1⟩Finite non-Abelian group with 21 elements1 iso
1119212⟨a, b | aba=b, baaaab=a⟩Finite non-commutative monoid with 21 elements

Other isomorphic instances

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

12 total

Σ#PresentationMapping
91292⟨a, b | abb=aaa, bbb=1⟩φ(a) = bba, φ(b) = b
107798⟨a, b | aaa=1, abbba=ab⟩φ(a) = b, φ(b) = bba
107809⟨a, b | aaa=1, baabb=ba⟩φ(a) = b, φ(b) = bba
108070⟨a, b | aaa=1, abbb=aba⟩φ(a) = b, φ(b) = a
108086⟨a, b | aaa=1, babb=baa⟩φ(a) = b, φ(b) = a
1121677⟨a, b | aaa=1, aaabbba=b⟩φ(a) = b, φ(b) = bba
1122221⟨a, b | aaa=1, aaabbb=ba⟩φ(a) = b, φ(b) = a
1122266⟨a, b | aaa=1, ababba=ab⟩φ(a) = b, φ(b) = a
1122297⟨a, b | aaa=1, baaabb=ba⟩φ(a) = b, φ(b) = a
1122741⟨a, b | aaa=1, aaaba=bbb⟩φ(a) = b, φ(b) = a
1122804⟨a, b | aaa=1, abbab=aba⟩φ(a) = b, φ(b) = ba
1123287⟨a, b | aaa=1, abbb=abaa⟩φ(a) = b, φ(b) = bba

Other anti-isomorphic instances

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

27 total

Σ#PresentationMapping
8749⟨a, b | aaa=1, abbb=b⟩φ(a) = b, φ(b) = bba
92431⟨a, b | aaa=1, aabbb=b⟩φ(a) = b, φ(b) = a
92439⟨a, b | aaa=1, abbab=b⟩φ(a) = b, φ(b) = a
92590⟨a, b | aaa=1, babb=ab⟩φ(a) = b, φ(b) = ba
107501⟨a, b | aaa=1, aababb=b⟩φ(a) = b, φ(b) = ba
107515⟨a, b | aaa=1, abaabb=b⟩φ(a) = b, φ(b) = ba
107517⟨a, b | aaa=1, ababab=b⟩φ(a) = b, φ(b) = ba
107779⟨a, b | aaa=1, aabbb=ab⟩φ(a) = b, φ(b) = bba
107812⟨a, b | aaa=1, babab=ab⟩φ(a) = b, φ(b) = bba
108069⟨a, b | aaa=1, abbb=aab⟩φ(a) = b, φ(b) = a
1121663⟨a, b | aaa=1, aaaabbb=b⟩φ(a) = b, φ(b) = bba
1121689⟨a, b | aaa=1, aababab=b⟩φ(a) = b, φ(b) = bba
1121695⟨a, b | aaa=1, aabbaab=b⟩φ(a) = b, φ(b) = bba
1121713⟨a, b | aaa=1, abaaabb=b⟩φ(a) = b, φ(b) = bba
1121721⟨a, b | aaa=1, ababaab=b⟩φ(a) = b, φ(b) = bba
1121733⟨a, b | aaa=1, abbaaab=b⟩φ(a) = b, φ(b) = bba
1122220⟨a, b | aaa=1, aaabbb=ab⟩φ(a) = b, φ(b) = a
1122239⟨a, b | aaa=1, aabbab=ab⟩φ(a) = b, φ(b) = a
1122244⟨a, b | aaa=1, aabbba=ba⟩φ(a) = b, φ(b) = a
1122296⟨a, b | aaa=1, baaabb=ab⟩φ(a) = b, φ(b) = a
1122300⟨a, b | aaa=1, baabab=ab⟩φ(a) = b, φ(b) = a
1122733⟨a, b | aaa=1, aaaab=bbb⟩φ(a) = b, φ(b) = a
1122795⟨a, b | aaa=1, ababb=aab⟩φ(a) = b, φ(b) = ba
1122831⟨a, b | aaa=1, baabb=aab⟩φ(a) = b, φ(b) = ba
1122839⟨a, b | aaa=1, babab=aab⟩φ(a) = b, φ(b) = ba
1123284⟨a, b | aaa=1, abbb=aaab⟩φ(a) = b, φ(b) = bba
1123293⟨a, b | aaa=1, baaa=abbb⟩φ(a) = b, φ(b) = bba