#1470 ⟨a, b | aaa=bb, abab=1⟩

Up:Monoid enumeration
Prev:#1449 ⟨a, b | aaa=aa, bbbb=1⟩
Next:#1472 ⟨a, b | aaa=bb, abbb=1⟩

Contents

  1. Properties
  2. Finite structure
  3. Right Cayley graph
  4. Left Cayley graph
  5. Rewriting system
  6. Same cardinality
  7. Isomorphic instances

Properties

Finite structure

Elements in the center commute with all other elements.
The order of x is the least n such that xn = 1.

Right Cayley graph

Left Cayley graph

Rewriting system

Format:
Word:
Enter a word above to compute its normal form. Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Strategy:
Result: 1 
1
#RuleProof
1. a15 ⇒ 1 [7]
2. ab ⇒ ba11 [6]
3. b2 ⇒ a3 [1]
# ab:aaa=bb,abab=1 a/b - -
aaaaaaaaaaaaaaa=1
ab=baaaaaaaaaaa
bb=aaa

Same cardinality

8 unique, 50 total

Σ#PresentationPropertiesφ
91907⟨a, b | aab=a, bbbbb=1⟩Noc 3015
92443⟨a, b | aaa=1, abbbb=b⟩Noc 3019
106560⟨a, b | aaa=a, bbbb=ab⟩Noc 30
107013⟨a, b | ab=aa, bbbbb=b⟩Noc 30
1111058⟨a, b | abba=bbb, baba=1⟩Grp 308
1113115⟨a, b | bbb=aaa, aaba=b⟩Noc 30
1115979⟨a, b | aaa=aa, bbbb=ab⟩Noc 30
1120847⟨a, b | ab=aa, bbbbb=bb⟩Noc 30

Isomorphic instances

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

73 total

Σ#PresentationMapping
93241⟨a, b | aa=1, ababbbb=1⟩φ(a) = ba, φ(b) = a
93249⟨a, b | aa=1, abbbbab=1⟩φ(a) = ba, φ(b) = a
93262⟨a, b | aa=1, bababbb=1⟩φ(a) = ba, φ(b) = a
93268⟨a, b | aa=1, bbababb=1⟩φ(a) = ba, φ(b) = a
93400⟨a, b | aa=1, babbbb=a⟩φ(a) = ba, φ(b) = a
109458⟨a, b | aa=1, abababbb=1⟩φ(a) = ba, φ(b) = b
109462⟨a, b | aa=1, ababbbab=1⟩φ(a) = ba, φ(b) = b
109477⟨a, b | aa=1, abbbabab=1⟩φ(a) = ba, φ(b) = b
109502⟨a, b | aa=1, babababb=1⟩φ(a) = ba, φ(b) = b
109780⟨a, b | aa=1, bababbb=a⟩φ(a) = ba, φ(b) = b
109784⟨a, b | aa=1, babbbab=a⟩φ(a) = ba, φ(b) = b
1110641⟨a, b | aaaa=abb, abab=1⟩φ(a) = a, φ(b) = b
1110646⟨a, b | aaaa=abb, baba=1⟩φ(a) = a, φ(b) = b
1110771⟨a, b | aaab=bbb, abab=1⟩φ(a) = a, φ(b) = b
1110776⟨a, b | aaab=bbb, baba=1⟩φ(a) = a, φ(b) = b
1112847⟨a, b | bab=aaa, aabbb=1⟩φ(a) = baa, φ(b) = a
1112852⟨a, b | bab=aaa, abbba=1⟩φ(a) = baa, φ(b) = a
1112855⟨a, b | bab=aaa, baabb=1⟩φ(a) = baa, φ(b) = a
1115387⟨a, b | aba=bb, abbabb=1⟩φ(a) = baaaa, φ(b) = a
1115396⟨a, b | aba=bb, babbab=1⟩φ(a) = baaaa, φ(b) = a
1117485⟨a, b | abab=1, aaaaba=b⟩φ(a) = a, φ(b) = b
1117509⟨a, b | abab=1, abaaaa=b⟩φ(a) = a, φ(b) = b
1118019⟨a, b | abab=1, aaaba=bb⟩φ(a) = b, φ(b) = a
1118039⟨a, b | abab=1, abaaa=bb⟩φ(a) = b, φ(b) = a
1118185⟨a, b | aaa=a, ababbbb=1⟩φ(a) = ba, φ(b) = a
1118193⟨a, b | aaa=a, abbbbab=1⟩φ(a) = ba, φ(b) = a
1118206⟨a, b | aaa=a, bababbb=1⟩φ(a) = ba, φ(b) = a
1118212⟨a, b | aaa=a, bbababb=1⟩φ(a) = ba, φ(b) = a
1118463⟨a, b | aab=b, ababbbb=1⟩φ(a) = ba, φ(b) = a
1118477⟨a, b | aab=b, abbbbab=1⟩φ(a) = ba, φ(b) = a
1118503⟨a, b | aab=b, bababbb=1⟩φ(a) = ba, φ(b) = a
1118510⟨a, b | aab=b, babbbba=1⟩φ(a) = ba, φ(b) = a
1118523⟨a, b | aab=b, bbababb=1⟩φ(a) = ba, φ(b) = a
1118533⟨a, b | aab=b, bbbabab=1⟩φ(a) = ba, φ(b) = a
1118538⟨a, b | aab=b, bbbbaba=1⟩φ(a) = ba, φ(b) = a
1125958⟨a, b | aa=1, aaababbbb=1⟩φ(a) = ba, φ(b) = a
1125971⟨a, b | aa=1, aaabbbbab=1⟩φ(a) = ba, φ(b) = a
1125994⟨a, b | aa=1, aabababbb=1⟩φ(a) = ba, φ(b) = a
1126000⟨a, b | aa=1, aababbbba=1⟩φ(a) = ba, φ(b) = a
1126010⟨a, b | aa=1, aabbababb=1⟩φ(a) = ba, φ(b) = a
1126018⟨a, b | aa=1, aabbbabab=1⟩φ(a) = ba, φ(b) = a
1126022⟨a, b | aa=1, aabbbbaba=1⟩φ(a) = ba, φ(b) = a
1126039⟨a, b | aa=1, abaaabbbb=1⟩φ(a) = ba, φ(b) = a
1126049⟨a, b | aa=1, abaabbbab=1⟩φ(a) = ba, φ(b) = a
1126056⟨a, b | aa=1, ababaabbb=1⟩φ(a) = ba, φ(b) = a
1126061⟨a, b | aa=1, abababbba=1⟩φ(a) = ba, φ(b) = a
1126064⟨a, b | aa=1, ababbaabb=1⟩φ(a) = ba, φ(b) = a
1126068⟨a, b | aa=1, ababbbaab=1⟩φ(a) = ba, φ(b) = a
1126081⟨a, b | aa=1, abbaabbab=1⟩φ(a) = ba, φ(b) = a
1126087⟨a, b | aa=1, abbababba=1⟩φ(a) = ba, φ(b) = a
1126096⟨a, b | aa=1, abbbaabab=1⟩φ(a) = ba, φ(b) = a
1126103⟨a, b | aa=1, abbbbaaab=1⟩φ(a) = ba, φ(b) = a
1126123⟨a, b | aa=1, baaababbb=1⟩φ(a) = ba, φ(b) = a
1126132⟨a, b | aa=1, baabababb=1⟩φ(a) = ba, φ(b) = a
1126136⟨a, b | aa=1, baabbabab=1⟩φ(a) = ba, φ(b) = a
1126144⟨a, b | aa=1, babaaabbb=1⟩φ(a) = ba, φ(b) = a
1126148⟨a, b | aa=1, bababaabb=1⟩φ(a) = ba, φ(b) = a
1126166⟨a, b | aa=1, bbaaababb=1⟩φ(a) = ba, φ(b) = a
1126528⟨a, b | aa=1, aababbbb=a⟩φ(a) = ba, φ(b) = a
1126550⟨a, b | aa=1, aabbbbab=a⟩φ(a) = ba, φ(b) = a
1126588⟨a, b | aa=1, abababbb=a⟩φ(a) = ba, φ(b) = a
1126598⟨a, b | aa=1, ababbbba=a⟩φ(a) = ba, φ(b) = a
1126614⟨a, b | aa=1, abbababb=a⟩φ(a) = ba, φ(b) = a
1126626⟨a, b | aa=1, abbbabab=a⟩φ(a) = ba, φ(b) = a
1126654⟨a, b | aa=1, baaabbbb=a⟩φ(a) = ba, φ(b) = a
1126666⟨a, b | aa=1, baabbbab=a⟩φ(a) = ba, φ(b) = a
1126674⟨a, b | aa=1, babaabbb=a⟩φ(a) = ba, φ(b) = a
1126682⟨a, b | aa=1, babbaabb=a⟩φ(a) = ba, φ(b) = a
1127126⟨a, b | aa=1, ababbbb=aa⟩φ(a) = ba, φ(b) = a
1127157⟨a, b | aa=1, abbbbab=aa⟩φ(a) = ba, φ(b) = a
1127205⟨a, b | aa=1, bababbb=aa⟩φ(a) = ba, φ(b) = a
1127227⟨a, b | aa=1, bbababb=aa⟩φ(a) = ba, φ(b) = a
1127740⟨a, b | aa=1, babbbb=aaa⟩φ(a) = ba, φ(b) = a