#4162 ⟨a, b | abb=aab, bbb=a⟩

Up:Monoids with two generators and two relations
Prev:#4161 ⟨a, b | abb=aab, bab=b⟩
Next:#4163 ⟨a, b | abb=aab, bbb=b⟩

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. b7 ⇒ b5
  2. a ⇒ b3 
# ab:abb=aab,bbb=a b/a
bbbbbbb=bbbbb
a=bbb

Staircase diagram

Cayley table

Idempotents are shown in bold.

1bb2b3b4b5b6
11bb2b3b4b5b6
bbb2b3b4b5b6b5
b2b2b3b4b5b6b5b6
b3b3b4b5b6b5b6b5
b4b4b5b6b5b6b5b6
b5b5b6b5b6b5b6b5
b6b6b5b6b5b6b5b6

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

22 unique, 1181 total

Σ#PresentationDescriptionRelated
7116⟨a, b | aaa=b, abb=1⟩Isomorphic to ℤ7763 iso
8577⟨a, b | aaa=b, abb=a⟩Isomorphic to ℕ(7 = 1)59 iso
8578⟨a, b | aaa=b, abb=b⟩Isomorphic to ℕ(7 = 3)44 iso
8913⟨a, b | aa=b, abbb=b⟩Isomorphic to ℕ(7 = 2)82 iso
8937⟨a, b | ab=a, baaa=b⟩Finite non-commutative monoid with 7 elements54 iso, 12 anti-iso
91593⟨a, b | aaa=ab, baa=b⟩Finite non-commutative monoid with 7 elements1 iso, 2 anti-iso
91601⟨a, b | aaa=bb, aab=b⟩Finite commutative monoid with 7 elements5 iso
91620⟨a, b | aab=aa, bbb=a⟩Isomorphic to ℕ(7 = 6)35 iso
91632⟨a, b | aab=ab, bbb=a⟩Isomorphic to ℕ(7 = 4)50 iso
92074⟨a, b | aab=b, babb=a⟩Finite commutative monoid with 7 elements20 iso
92231⟨a, b | ab=aa, aaa=bb⟩Finite non-commutative monoid with 7 elements3 iso
92271⟨a, b | bb=aa, aaa=ab⟩Finite non-commutative monoid with 7 elements1 iso, 2 anti-iso
93197⟨a, b | ab=a, bbb=baa⟩Finite non-commutative monoid with 7 elements5 iso
106251⟨a, b | aaa=a, aabba=b⟩Finite non-commutative monoid with 7 elements1 iso
108987⟨a, b | ab=a, aaaaa=bb⟩Finite commutative monoid with 7 elements5 iso
109108⟨a, b | ab=a, bbbbb=aa⟩Finite commutative monoid with 7 elements2 iso
109110⟨a, b | ab=a, bbbbb=ba⟩Finite non-commutative monoid with 7 elements1 iso
109263⟨a, b | ab=a, aaaa=bbb⟩Finite commutative monoid with 7 elements4 iso
109375⟨a, b | ab=a, bbba=bbb⟩Finite non-commutative monoid with 7 elements3 iso
109376⟨a, b | ab=a, bbbb=aaa⟩Finite commutative monoid with 7 elements3 iso
109382⟨a, b | ab=a, bbbb=bba⟩Finite non-commutative monoid with 7 elements1 iso
1114843⟨a, b | abba=b, aabab=a⟩Finite non-commutative monoid with 7 elements1 iso

Other isomorphic instances

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

42 total

Σ#PresentationMapping
104170⟨a, b | abb=aba, bbb=a⟩φ(a) = bbb, φ(b) = b
104182⟨a, b | baa=abb, bbb=a⟩φ(a) = bbb, φ(b) = b
104188⟨a, b | bab=aab, bbb=a⟩φ(a) = bbb, φ(b) = b
104192⟨a, b | bab=aba, bbb=a⟩φ(a) = bbb, φ(b) = b
106383⟨a, b | aab=b, aaaaa=b⟩φ(a) = b, φ(b) = bbbbb
106487⟨a, b | aba=b, aaaaa=b⟩φ(a) = b, φ(b) = bbbbb
109231⟨a, b | aa=b, abbb=abb⟩φ(a) = b, φ(b) = bb
109233⟨a, b | aa=b, abbb=bab⟩φ(a) = b, φ(b) = bb
109234⟨a, b | aa=b, abbb=bba⟩φ(a) = b, φ(b) = bb
109245⟨a, b | aa=b, babb=abb⟩φ(a) = b, φ(b) = bb
109247⟨a, b | aa=b, babb=bab⟩φ(a) = b, φ(b) = bb
109248⟨a, b | aa=b, babb=bba⟩φ(a) = b, φ(b) = bb
1112608⟨a, b | abbb=ab, bbbb=a⟩φ(a) = bbbb, φ(b) = b
1112610⟨a, b | abbb=ba, bbbb=a⟩φ(a) = bbbb, φ(b) = b
1112634⟨a, b | babb=ab, bbbb=a⟩φ(a) = bbbb, φ(b) = b
1112636⟨a, b | babb=ba, bbbb=a⟩φ(a) = bbbb, φ(b) = b
1119505⟨a, b | aab=b, aaaaa=ab⟩φ(a) = b, φ(b) = bbbbbb
1119705⟨a, b | aba=b, aaaaa=ab⟩φ(a) = b, φ(b) = bbbbbb
1125285⟨a, b | aa=b, aaabb=abb⟩φ(a) = b, φ(b) = bb
1125287⟨a, b | aa=b, aaabb=bab⟩φ(a) = b, φ(b) = bb
1125288⟨a, b | aa=b, aaabb=bba⟩φ(a) = b, φ(b) = bb
1125299⟨a, b | aa=b, aabab=abb⟩φ(a) = b, φ(b) = bb
1125301⟨a, b | aa=b, aabab=bab⟩φ(a) = b, φ(b) = bb
1125302⟨a, b | aa=b, aabab=bba⟩φ(a) = b, φ(b) = bb
1125307⟨a, b | aa=b, aabba=abb⟩φ(a) = b, φ(b) = bb
1125309⟨a, b | aa=b, aabba=bab⟩φ(a) = b, φ(b) = bb
1125310⟨a, b | aa=b, aabba=bba⟩φ(a) = b, φ(b) = bb
1125323⟨a, b | aa=b, abaab=abb⟩φ(a) = b, φ(b) = bb
1125325⟨a, b | aa=b, abaab=bab⟩φ(a) = b, φ(b) = bb
1125326⟨a, b | aa=b, abaab=bba⟩φ(a) = b, φ(b) = bb
1125331⟨a, b | aa=b, ababa=abb⟩φ(a) = b, φ(b) = bb
1125332⟨a, b | aa=b, ababa=bab⟩φ(a) = b, φ(b) = bb
1125367⟨a, b | aa=b, baaab=abb⟩φ(a) = b, φ(b) = bb
1125368⟨a, b | aa=b, baaab=bab⟩φ(a) = b, φ(b) = bb
1125746⟨a, b | aa=b, abbb=aaab⟩φ(a) = b, φ(b) = bb
1125747⟨a, b | aa=b, abbb=aaba⟩φ(a) = b, φ(b) = bb
1125749⟨a, b | aa=b, abbb=abaa⟩φ(a) = b, φ(b) = bb
1125755⟨a, b | aa=b, baaa=abbb⟩φ(a) = b, φ(b) = bb
1125767⟨a, b | aa=b, babb=aaab⟩φ(a) = b, φ(b) = bb
1125768⟨a, b | aa=b, babb=aaba⟩φ(a) = b, φ(b) = bb
1125770⟨a, b | aa=b, babb=abaa⟩φ(a) = b, φ(b) = bb
1125774⟨a, b | aa=b, babb=baaa⟩φ(a) = b, φ(b) = bb