#2019 ⟨a, b | aaa=b, bbbb=b⟩

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Contents

  1. Properties
  2. Commutative structure
  3. Finite structure
  4. Staircase diagram
  5. Cayley table
  6. Right Cayley graph
  7. Rewriting system
  8. Same cardinality
  9. Isomorphic instances

Properties

Commutative structure

Finite structure

An idempotent element x satisfies x2 = x.
The index and period of x is the least m (index) and n (period) such that x(m+n) = xm.

Staircase diagram

Cayley table

Idempotents are shown in bold.

1aa2a3a4a5a6a7a8a9a10a11
11aa2a3a4a5a6a7a8a9a10a11
aaa2a3a4a5a6a7a8a9a10a11a3
a2a2a3a4a5a6a7a8a9a10a11a3a4
a3a3a4a5a6a7a8a9a10a11a3a4a5
a4a4a5a6a7a8a9a10a11a3a4a5a6
a5a5a6a7a8a9a10a11a3a4a5a6a7
a6a6a7a8a9a10a11a3a4a5a6a7a8
a7a7a8a9a10a11a3a4a5a6a7a8a9
a8a8a9a10a11a3a4a5a6a7a8a9a10
a9a9a10a11a3a4a5a6a7a8a9a10a11
a10a10a11a3a4a5a6a7a8a9a10a11a3
a11a11a3a4a5a6a7a8a9a10a11a3a4

Right Cayley graph

Idempotents are shown in bold.

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. a12 ⇒ a3 [2]
2. b ⇒ a3 [1]
# ab:aaa=b,bbbb=b a/b - -
aaaaaaaaaaaa=aaa
b=aaa

Same cardinality

37 unique, 636 total

Σ#PresentationPropertiesDescriptionφ
7124⟨a, b | aab=a, bbb=1⟩Noc 1261
8458⟨a, b | aaaa=1, abbb=1⟩Grp Com 12ℤ12325
8645⟨a, b | ab=aa, bbb=b⟩Noc 121
91646⟨a, b | aab=bb, aba=a⟩Noc 122
91963⟨a, b | aba=b, aaabb=1⟩Grp 1230
91998⟨a, b | aaa=a, bbbb=a⟩Noc Com 12ℕ(12 = 4)6
92018⟨a, b | aaa=b, bbbb=a⟩Noc Com 12ℕ(12 = 1)27
92259⟨a, b | ab=aa, bbb=bb⟩Noc 12
105040⟨a, b | aaa=aa, bbbb=a⟩Noc Com 12ℕ(12 = 8)
105072⟨a, b | aaa=ab, bbbb=a⟩Noc Com 12ℕ(12 = 5)7
105092⟨a, b | aaa=bb, bbbb=a⟩Noc Com 12ℕ(12 = 2)43
105202⟨a, b | aab=bb, abba=a⟩Noc 1241
105330⟨a, b | aaa=ab, bab=bb⟩Noc 121
106597⟨a, b | aaa=b, bbbb=bb⟩Noc Com 12ℕ(12 = 6)4
106660⟨a, b | aab=a, bbbb=ba⟩Noc 121
106661⟨a, b | aab=a, bbbb=bb⟩Noc 121
107105⟨a, b | ab=aa, bbaa=bb⟩Noc 122
1112897⟨a, b | aab=aaa, baaa=b⟩Noc 125
1112910⟨a, b | aab=aaa, bbbb=a⟩Noc Com 12ℕ(12 = 9)2
1115428⟨a, b | aaa=aa, abbbb=b⟩Noc 12
1115996⟨a, b | aaa=ab, aabb=bb⟩Noc 122
1116057⟨a, b | aaa=bb, aabb=ab⟩Noc 124
1116104⟨a, b | aab=aa, abab=bb⟩Noc 12
1116274⟨a, b | aab=bb, aaaa=ab⟩Noc 121
1116275⟨a, b | aab=bb, aaaa=ba⟩Noc 12
1116448⟨a, b | aba=bb, aabb=aa⟩Noc 122
1119773⟨a, b | aba=b, bbbbb=aa⟩Noc Com 12
1119917⟨a, b | aaa=b, bbbb=abb⟩Noc Com 12ℕ(12 = 7)2
1120054⟨a, b | aab=b, aaaa=bba⟩Noc 12
1120686⟨a, b | bb=aa, aaaaab=a⟩Noc Com 1215
1120787⟨a, b | ab=aa, baaaa=bb⟩Noc 127
1121086⟨a, b | bb=aa, aaaa=aba⟩Noc 124
1121092⟨a, b | bb=aa, aaab=aba⟩Noc 123
1121112⟨a, b | bb=aa, abab=aba⟩Noc 12
1124147⟨a, b | aa=a, abbbbbb=b⟩Noc 12
1124991⟨a, b | ab=a, baaaaa=bb⟩Noc 12
1125603⟨a, b | ab=a, bbaaa=bbb⟩Noc 12

Isomorphic instances

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

23 total

Σ#PresentationMapping
105093⟨a, b | aaa=bb, bbbb=b⟩φ(a) = a, φ(b) = aaaaaa
1113111⟨a, b | bbb=aaa, aaaa=b⟩φ(a) = a, φ(b) = aaaa
1113112⟨a, b | bbb=aaa, aaab=a⟩φ(a) = aaaa, φ(b) = a
1113114⟨a, b | bbb=aaa, aaba=a⟩φ(a) = aaaa, φ(b) = a
1113116⟨a, b | bbb=aaa, aabb=a⟩φ(a) = aaaaaaa, φ(b) = a
1113117⟨a, b | bbb=aaa, abab=a⟩φ(a) = aaaaaaa, φ(b) = a
1115512⟨a, b | aaa=bb, aaabb=b⟩φ(a) = a, φ(b) = aaaaaa
1115516⟨a, b | aaa=bb, aabab=b⟩φ(a) = a, φ(b) = aaaaaa
1115518⟨a, b | aaa=bb, aabba=b⟩φ(a) = a, φ(b) = aaaaaa
1115522⟨a, b | aaa=bb, abaab=b⟩φ(a) = a, φ(b) = aaaaaa
1115534⟨a, b | aaa=bb, baaab=b⟩φ(a) = a, φ(b) = aaaaaa
1118775⟨a, b | aaa=b, aaabbb=b⟩φ(a) = a, φ(b) = aaa
1118781⟨a, b | aaa=b, aababb=b⟩φ(a) = a, φ(b) = aaa
1118785⟨a, b | aaa=b, aabbab=b⟩φ(a) = a, φ(b) = aaa
1118787⟨a, b | aaa=b, aabbba=b⟩φ(a) = a, φ(b) = aaa
1118795⟨a, b | aaa=b, abaabb=b⟩φ(a) = a, φ(b) = aaa
1118797⟨a, b | aaa=b, ababab=b⟩φ(a) = a, φ(b) = aaa
1118799⟨a, b | aaa=b, ababba=b⟩φ(a) = a, φ(b) = aaa
1118803⟨a, b | aaa=b, abbaab=b⟩φ(a) = a, φ(b) = aaa
1118815⟨a, b | aaa=b, baaabb=b⟩φ(a) = a, φ(b) = aaa
1118817⟨a, b | aaa=b, baabab=b⟩φ(a) = a, φ(b) = aaa
1119914⟨a, b | aaa=b, bbbb=aaa⟩φ(a) = a, φ(b) = aaa
1124858⟨a, b | aa=b, bbbbbb=ab⟩φ(a) = a, φ(b) = aa