#752 ⟨a, b | aaa=1, babb=a⟩

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Quick links

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

Properties

Elements

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 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. b9 ⇒ 1 [11]
2. ba ⇒ ab7 [9]
3. a3 ⇒ 1 [1]
# ab:aaa=1,babb=a b/a
bbbbbbbbb=1
ba=abbbbbbb
aaa=1

Same cardinality

6 unique, 34 total

Σ#PresentationDescriptionRelated
8450⟨a, b | bab=aa, bbb=1⟩Finite non-commutative monoid with 27 elements27 iso
91695⟨a, b | bab=aa, bbb=b⟩Finite non-commutative monoid with 27 elements1 iso
105300⟨a, b | aaa=aa, aba=bb⟩Finite non-commutative monoid with 27 elements
1115514⟨a, b | aaa=bb, aabaa=b⟩Finite non-commutative monoid with 27 elements
1118739⟨a, b | aaa=a, abbbbb=b⟩Finite non-commutative monoid with 27 elements
1120314⟨a, b | aba=b, bbbb=aaa⟩Finite non-commutative monoid with 27 elements

Isomorphic instances

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

59 total

Σ#PresentationMapping
92290⟨a, b | aaa=1, aababb=1⟩φ(a) = a, φ(b) = b
92292⟨a, b | aaa=1, aabbab=1⟩φ(a) = aa, φ(b) = b
92297⟨a, b | aaa=1, abaabb=1⟩φ(a) = aa, 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
107793⟨a, b | aaa=1, abbab=aa⟩φ(a) = aa, φ(b) = b
107807⟨a, b | aaa=1, baabb=aa⟩φ(a) = aa, φ(b) = b
1113811⟨a, b | aaaa=a, aababb=1⟩φ(a) = a, φ(b) = b
1113813⟨a, b | aaaa=a, aabbab=1⟩φ(a) = aa, φ(b) = b
1113818⟨a, b | aaaa=a, abaabb=1⟩φ(a) = aa, 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
1113950⟨a, b | aaab=b, aabbab=1⟩φ(a) = aa, φ(b) = b
1113956⟨a, b | aaab=b, abaabb=1⟩φ(a) = aa, 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
1113963⟨a, b | aaab=b, abbaba=1⟩φ(a) = aa, φ(b) = b
1113974⟨a, b | aaab=b, baabab=1⟩φ(a) = a, φ(b) = b
1113975⟨a, b | aaab=b, baabba=1⟩φ(a) = aa, φ(b) = b
1113978⟨a, b | aaab=b, babaab=1⟩φ(a) = aa, 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
1113989⟨a, b | aaab=b, bbabaa=1⟩φ(a) = aa, φ(b) = b
1114072⟨a, b | aaba=b, aaabbb=1⟩φ(a) = aabb, φ(b) = b
1114076⟨a, b | aaba=b, aababb=1⟩φ(a) = b, φ(b) = aa
1114078⟨a, b | aaba=b, aabbab=1⟩φ(a) = ab, φ(b) = b
1114079⟨a, b | aaba=b, aabbba=1⟩φ(a) = aabb, φ(b) = b
1114084⟨a, b | aaba=b, abaabb=1⟩φ(a) = ab, φ(b) = b
1114086⟨a, b | aaba=b, ababab=1⟩φ(a) = aabb, φ(b) = b
1114087⟨a, b | aaba=b, ababba=1⟩φ(a) = b, φ(b) = aa
1114090⟨a, b | aaba=b, abbaab=1⟩φ(a) = b, φ(b) = aa
1114091⟨a, b | aaba=b, abbaba=1⟩φ(a) = ab, φ(b) = b
1114093⟨a, b | aaba=b, abbbaa=1⟩φ(a) = aabb, φ(b) = b
1114100⟨a, b | aaba=b, baaabb=1⟩φ(a) = aabb, φ(b) = b
1114102⟨a, b | aaba=b, baabab=1⟩φ(a) = b, φ(b) = aa
1114103⟨a, b | aaba=b, baabba=1⟩φ(a) = ab, φ(b) = b
1114106⟨a, b | aaba=b, babaab=1⟩φ(a) = ab, φ(b) = b
1114107⟨a, b | aaba=b, bababa=1⟩φ(a) = aabb, φ(b) = b
1114109⟨a, b | aaba=b, babbaa=1⟩φ(a) = b, φ(b) = aa
1114114⟨a, b | aaba=b, bbaaab=1⟩φ(a) = aabb, φ(b) = b
1114115⟨a, b | aaba=b, bbaaba=1⟩φ(a) = b, φ(b) = aa
1114117⟨a, b | aaba=b, bbabaa=1⟩φ(a) = ab, φ(b) = b
1114121⟨a, b | aaba=b, bbbaaa=1⟩φ(a) = aabb, φ(b) = b
1121670⟨a, b | aaa=1, aaababb=a⟩φ(a) = a, φ(b) = b
1121674⟨a, b | aaa=1, aaabbab=a⟩φ(a) = aa, φ(b) = b
1121684⟨a, b | aaa=1, aabaabb=a⟩φ(a) = aa, 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
1121696⟨a, b | aaa=1, aabbaba=a⟩φ(a) = aa, φ(b) = b
1121714⟨a, b | aaa=1, abaabab=a⟩φ(a) = a, φ(b) = b
1121716⟨a, b | aaa=1, abaabba=a⟩φ(a) = aa, φ(b) = b
1121720⟨a, b | aaa=1, ababaab=a⟩φ(a) = aa, φ(b) = b
1121754⟨a, b | aaa=1, baaaabb=a⟩φ(a) = a, φ(b) = b
1121756⟨a, b | aaa=1, baaabab=a⟩φ(a) = aa, φ(b) = b
1123304⟨a, b | aaa=1, babb=aaaa⟩φ(a) = a, φ(b) = b