#752 ⟨a, b | aaa=1, babb=a⟩

Up:Monoid enumeration
Prev:#751 ⟨a, b | aaa=1, baab=b⟩
Next:#753 ⟨a, b | aaa=1, babb=b⟩

Contents

  1. Properties
  2. Finite structure
  3. Right Cayley graph
  4. Left Cayley graph
  5. Rewriting system
  6. Same cardinality
  7. Isomorphic instances
  8. Anti-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. b9 ⇒ 1 [8]
2. ab ⇒ b4a [6]
3. a3 ⇒ 1 [1]
# ab:aaa=1,babb=a reversed:b/a - -
bbbbbbbbb=1
ab=bbbba
aaa=1

Same cardinality

6 unique, 34 total

Σ#PresentationPropertiesφ
8450⟨a, b | bab=aa, bbb=1⟩Noc 2727
91695⟨a, b | bab=aa, bbb=b⟩Noc 271
105300⟨a, b | aaa=aa, aba=bb⟩Noc 27
1115514⟨a, b | aaa=bb, aabaa=b⟩Noc 27
1118739⟨a, b | aaa=a, abbbbb=b⟩Noc 27
1120314⟨a, b | aba=b, bbbb=aaa⟩Noc 27

Isomorphic instances

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

35 total

Σ#PresentationMapping
92290⟨a, b | aaa=1, aababb=1⟩φ(a) = a, φ(b) = b
92299⟨a, b | aaa=1, ababba=1⟩φ(a) = a, φ(b) = b
92301⟨a, b | aaa=1, abbaab=1⟩φ(a) = a, φ(b) = b
92308⟨a, b | aaa=1, baabab=1⟩φ(a) = a, φ(b) = b
107789⟨a, b | aaa=1, ababb=aa⟩φ(a) = a, φ(b) = b
1113811⟨a, b | aaaa=a, aababb=1⟩φ(a) = a, φ(b) = b
1113820⟨a, b | aaaa=a, ababba=1⟩φ(a) = a, φ(b) = b
1113822⟨a, b | aaaa=a, abbaab=1⟩φ(a) = a, φ(b) = b
1113829⟨a, b | aaaa=a, baabab=1⟩φ(a) = a, φ(b) = b
1113948⟨a, b | aaab=b, aababb=1⟩φ(a) = a, φ(b) = b
1113959⟨a, b | aaab=b, ababba=1⟩φ(a) = a, φ(b) = b
1113962⟨a, b | aaab=b, abbaab=1⟩φ(a) = a, φ(b) = b
1113974⟨a, b | aaab=b, baabab=1⟩φ(a) = a, φ(b) = b
1113981⟨a, b | aaab=b, babbaa=1⟩φ(a) = a, φ(b) = b
1113987⟨a, b | aaab=b, bbaaba=1⟩φ(a) = a, φ(b) = b
1114072⟨a, b | aaba=b, aaabbb=1⟩φ(a) = bbaa, φ(b) = b
1114078⟨a, b | aaba=b, aabbab=1⟩φ(a) = ba, φ(b) = b
1114079⟨a, b | aaba=b, aabbba=1⟩φ(a) = bbaa, φ(b) = b
1114084⟨a, b | aaba=b, abaabb=1⟩φ(a) = ba, φ(b) = b
1114086⟨a, b | aaba=b, ababab=1⟩φ(a) = bbaa, φ(b) = b
1114091⟨a, b | aaba=b, abbaba=1⟩φ(a) = ba, φ(b) = b
1114093⟨a, b | aaba=b, abbbaa=1⟩φ(a) = bbaa, φ(b) = b
1114100⟨a, b | aaba=b, baaabb=1⟩φ(a) = bbaa, φ(b) = b
1114103⟨a, b | aaba=b, baabba=1⟩φ(a) = ba, φ(b) = b
1114106⟨a, b | aaba=b, babaab=1⟩φ(a) = ba, φ(b) = b
1114107⟨a, b | aaba=b, bababa=1⟩φ(a) = bbaa, φ(b) = b
1114114⟨a, b | aaba=b, bbaaab=1⟩φ(a) = bbaa, φ(b) = b
1114117⟨a, b | aaba=b, bbabaa=1⟩φ(a) = ba, φ(b) = b
1114121⟨a, b | aaba=b, bbbaaa=1⟩φ(a) = bbaa, φ(b) = b
1121670⟨a, b | aaa=1, aaababb=a⟩φ(a) = a, φ(b) = b
1121690⟨a, b | aaa=1, aababba=a⟩φ(a) = a, φ(b) = b
1121694⟨a, b | aaa=1, aabbaab=a⟩φ(a) = a, φ(b) = b
1121714⟨a, b | aaa=1, abaabab=a⟩φ(a) = a, φ(b) = b
1121754⟨a, b | aaa=1, baaaabb=a⟩φ(a) = a, φ(b) = b
1123304⟨a, b | aaa=1, babb=aaaa⟩φ(a) = a, φ(b) = b

Anti-isomorphic instances

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

24 total

Σ#PresentationMapping
92292⟨a, b | aaa=1, aabbab=1⟩φ(a) = a, φ(b) = b
92297⟨a, b | aaa=1, abaabb=1⟩φ(a) = a, φ(b) = b
107793⟨a, b | aaa=1, abbab=aa⟩φ(a) = a, φ(b) = b
107807⟨a, b | aaa=1, baabb=aa⟩φ(a) = a, φ(b) = b
1113813⟨a, b | aaaa=a, aabbab=1⟩φ(a) = a, φ(b) = b
1113818⟨a, b | aaaa=a, abaabb=1⟩φ(a) = a, φ(b) = b
1113950⟨a, b | aaab=b, aabbab=1⟩φ(a) = a, φ(b) = b
1113956⟨a, b | aaab=b, abaabb=1⟩φ(a) = a, φ(b) = b
1113963⟨a, b | aaab=b, abbaba=1⟩φ(a) = a, φ(b) = b
1113975⟨a, b | aaab=b, baabba=1⟩φ(a) = a, φ(b) = b
1113978⟨a, b | aaab=b, babaab=1⟩φ(a) = a, φ(b) = b
1113989⟨a, b | aaab=b, bbabaa=1⟩φ(a) = a, φ(b) = b
1114076⟨a, b | aaba=b, aababb=1⟩φ(a) = b, φ(b) = a
1114087⟨a, b | aaba=b, ababba=1⟩φ(a) = b, φ(b) = a
1114090⟨a, b | aaba=b, abbaab=1⟩φ(a) = b, φ(b) = a
1114102⟨a, b | aaba=b, baabab=1⟩φ(a) = b, φ(b) = a
1114109⟨a, b | aaba=b, babbaa=1⟩φ(a) = b, φ(b) = a
1114115⟨a, b | aaba=b, bbaaba=1⟩φ(a) = b, φ(b) = a
1121674⟨a, b | aaa=1, aaabbab=a⟩φ(a) = a, φ(b) = b
1121684⟨a, b | aaa=1, aabaabb=a⟩φ(a) = a, φ(b) = b
1121696⟨a, b | aaa=1, aabbaba=a⟩φ(a) = a, φ(b) = b
1121716⟨a, b | aaa=1, abaabba=a⟩φ(a) = a, φ(b) = b
1121720⟨a, b | aaa=1, ababaab=a⟩φ(a) = a, φ(b) = b
1121756⟨a, b | aaa=1, baaabab=a⟩φ(a) = a, φ(b) = b