#5 ⟨a, b | aba=1⟩

Up:Monoids with two generators and one relation
Prev:#4 ⟨a, b | aab=1⟩
Next:#6 ⟨a, b | aa=a⟩

Quick links

  1. Properties
  2. Staircase diagram
  3. Rewriting system
  4. Other submonoids of same group
  5. Isomorphic instances

Properties

Staircase diagram

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. ba ⇒ ab [2]
2. a2b ⇒ 1 [3]
# ab:aba=1 ab
ba=ab
aab=1

Other submonoids of same group

1 unique, 9 total

Σ#PresentationDescriptionRelated
37⟨a, b | aa=b⟩Isomorphic to ℕ8 iso

Isomorphic instances

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

67 total

Σ#PresentationMapping
411⟨a, b | aaba=1⟩φ(a) = a, φ(b) = abb
526⟨a, b | aaaba=1⟩φ(a) = a, φ(b) = bb
528⟨a, b | aabaa=1⟩φ(a) = a, φ(b) = bb
531⟨a, b | abaab=1⟩φ(a) = b, φ(b) = aaa
532⟨a, b | ababa=1⟩φ(a) = b, φ(b) = aaa
656⟨a, b | aaaaba=1⟩φ(a) = a, φ(b) = abbb
658⟨a, b | aaabaa=1⟩φ(a) = a, φ(b) = abbb
7127⟨a, b | aaaaaba=1⟩φ(a) = a, φ(b) = bbb
7129⟨a, b | aaaabaa=1⟩φ(a) = a, φ(b) = bbb
7133⟨a, b | aaabaaa=1⟩φ(a) = a, φ(b) = bbb
7140⟨a, b | aabaaab=1⟩φ(a) = b, φ(b) = aaaaa
7141⟨a, b | aabaaba=1⟩φ(a) = b, φ(b) = aaaaa
7152⟨a, b | abaaaba=1⟩φ(a) = b, φ(b) = aaaaa
7153⟨a, b | abaabab=1⟩φ(a) = abb, φ(b) = aaaa
7155⟨a, b | ababaab=1⟩φ(a) = abb, φ(b) = aaaa
7156⟨a, b | abababa=1⟩φ(a) = abb, φ(b) = aaaa
8271⟨a, b | aaaaaaba=1⟩φ(a) = a, φ(b) = abbbb
8273⟨a, b | aaaaabaa=1⟩φ(a) = a, φ(b) = abbbb
8277⟨a, b | aaaabaaa=1⟩φ(a) = a, φ(b) = abbbb
8305⟨a, b | aababaab=1⟩φ(a) = aaa, φ(b) = abbb
8325⟨a, b | abaabaab=1⟩φ(a) = aaa, φ(b) = abbb
8326⟨a, b | abaababa=1⟩φ(a) = aaa, φ(b) = abbb
9597⟨a, b | aaaaaaaba=1⟩φ(a) = a, φ(b) = bbbb
9599⟨a, b | aaaaaabaa=1⟩φ(a) = a, φ(b) = bbbb
9603⟨a, b | aaaaabaaa=1⟩φ(a) = a, φ(b) = bbbb
9611⟨a, b | aaaabaaaa=1⟩φ(a) = a, φ(b) = bbbb
9626⟨a, b | aaabaaaab=1⟩φ(a) = b, φ(b) = aaaaaaa
9627⟨a, b | aaabaaaba=1⟩φ(a) = b, φ(b) = aaaaaaa
9654⟨a, b | aabaaaaba=1⟩φ(a) = b, φ(b) = aaaaaaa
9656⟨a, b | aabaaabaa=1⟩φ(a) = b, φ(b) = aaaaaaa
9705⟨a, b | abaababab=1⟩φ(a) = bb, φ(b) = aaaaa
9711⟨a, b | ababaabab=1⟩φ(a) = bb, φ(b) = aaaaa
9713⟨a, b | abababaab=1⟩φ(a) = bb, φ(b) = aaaaa
9714⟨a, b | ababababa=1⟩φ(a) = bb, φ(b) = aaaaa
101277⟨a, b | aaaaaaaaba=1⟩φ(a) = a, φ(b) = abbbbb
101279⟨a, b | aaaaaaabaa=1⟩φ(a) = a, φ(b) = abbbbb
101283⟨a, b | aaaaaabaaa=1⟩φ(a) = a, φ(b) = abbbbb
101291⟨a, b | aaaaabaaaa=1⟩φ(a) = a, φ(b) = abbbbb
101397⟨a, b | aabaaabaab=1⟩φ(a) = abb, φ(b) = aaaaaaa
101403⟨a, b | aabaabaaab=1⟩φ(a) = abb, φ(b) = aaaaaaa
101404⟨a, b | aabaabaaba=1⟩φ(a) = abb, φ(b) = aaaaaaa
101484⟨a, b | abaaabaaba=1⟩φ(a) = abb, φ(b) = aaaaaaa
112779⟨a, b | aaaaaaaaaba=1⟩φ(a) = a, φ(b) = bbbbb
112781⟨a, b | aaaaaaaabaa=1⟩φ(a) = a, φ(b) = bbbbb
112785⟨a, b | aaaaaaabaaa=1⟩φ(a) = a, φ(b) = bbbbb
112793⟨a, b | aaaaaabaaaa=1⟩φ(a) = a, φ(b) = bbbbb
112809⟨a, b | aaaaabaaaaa=1⟩φ(a) = a, φ(b) = bbbbb
112840⟨a, b | aaaabaaaaab=1⟩φ(a) = b, φ(b) = aaaaaaaaa
112841⟨a, b | aaaabaaaaba=1⟩φ(a) = b, φ(b) = aaaaaaaaa
112900⟨a, b | aaabaaaaaba=1⟩φ(a) = b, φ(b) = aaaaaaaaa
112902⟨a, b | aaabaaaabaa=1⟩φ(a) = b, φ(b) = aaaaaaaaa
112913⟨a, b | aaabaabaaab=1⟩φ(a) = aaa, φ(b) = bbbb
113008⟨a, b | aabaaaaabaa=1⟩φ(a) = b, φ(b) = aaaaaaaaa
113017⟨a, b | aabaaabaaab=1⟩φ(a) = aaa, φ(b) = bbbb
113018⟨a, b | aabaaabaaba=1⟩φ(a) = aaa, φ(b) = bbbb
113030⟨a, b | aabaabaaaba=1⟩φ(a) = aaa, φ(b) = bbbb
113035⟨a, b | aabaababaab=1⟩φ(a) = aaaa, φ(b) = abbbb
113057⟨a, b | aababaabaab=1⟩φ(a) = aaaa, φ(b) = abbbb
113186⟨a, b | abaaabaaaba=1⟩φ(a) = aaa, φ(b) = bbbb
113203⟨a, b | abaabaabaab=1⟩φ(a) = aaaa, φ(b) = abbbb
113204⟨a, b | abaabaababa=1⟩φ(a) = aaaa, φ(b) = abbbb
113208⟨a, b | abaababaaba=1⟩φ(a) = aaaa, φ(b) = abbbb
113209⟨a, b | abaabababab=1⟩φ(a) = abbb, φ(b) = aaaaaa
113234⟨a, b | ababaababab=1⟩φ(a) = abbb, φ(b) = aaaaaa
113240⟨a, b | abababaabab=1⟩φ(a) = abbb, φ(b) = aaaaaa
113242⟨a, b | ababababaab=1⟩φ(a) = abbb, φ(b) = aaaaaa
113243⟨a, b | abababababa=1⟩φ(a) = abbb, φ(b) = aaaaaa