#12324 ⟨a, b | aaab=bb, bbbb=a⟩

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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.

1bb2b3b4b5b6b7b8b9b10b11b12
11bb2b3b4b5b6b7b8b9b10b11b12
bbb2b3b4b5b6b7b8b9b10b11b12b2
b2b2b3b4b5b6b7b8b9b10b11b12b2b3
b3b3b4b5b6b7b8b9b10b11b12b2b3b4
b4b4b5b6b7b8b9b10b11b12b2b3b4b5
b5b5b6b7b8b9b10b11b12b2b3b4b5b6
b6b6b7b8b9b10b11b12b2b3b4b5b6b7
b7b7b8b9b10b11b12b2b3b4b5b6b7b8
b8b8b9b10b11b12b2b3b4b5b6b7b8b9
b9b9b10b11b12b2b3b4b5b6b7b8b9b10
b10b10b11b12b2b3b4b5b6b7b8b9b10b11
b11b11b12b2b3b4b5b6b7b8b9b10b11b12
b12b12b2b3b4b5b6b7b8b9b10b11b12b2

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. b13 ⇒ b2 [3]
2. a ⇒ b4 [2]
# ab:aaab=bb,bbbb=a b/a - -
bbbbbbbbbbbbb=bb
a=bbbb

Same cardinality

26 unique, 289 total

Σ#PresentationPropertiesDescriptionφ
91328⟨a, b | aaaa=b, abbb=1⟩Grp Com 13ℤ13189
92118⟨a, b | aba=b, baab=a⟩Noc 134
104642⟨a, b | aaaa=b, abbb=a⟩Noc Com 13ℕ(13 = 1)10
104643⟨a, b | aaaa=b, abbb=b⟩Noc Com 13ℕ(13 = 4)6
104683⟨a, b | aaab=b, abba=a⟩Noc 1316
105065⟨a, b | aaa=ab, babb=b⟩Noc 137
105334⟨a, b | aaa=ab, bba=bb⟩Noc 13
105336⟨a, b | aaa=ab, bbb=ab⟩Noc 131
105337⟨a, b | aaa=ab, bbb=ba⟩Noc 13
105340⟨a, b | aaa=bb, aab=ba⟩Noc 13
106305⟨a, b | aaa=b, abbbb=b⟩Noc Com 13ℕ(13 = 3)2
107143⟨a, b | bb=aa, aaab=ba⟩Noc 131
1112240⟨a, b | aaab=aa, bbbb=a⟩Noc Com 13ℕ(13 = 8)1
1112268⟨a, b | aaab=ab, bbbb=a⟩Noc Com 13ℕ(13 = 5)3
1114647⟨a, b | aaba=b, babbb=a⟩Noc Com 132
1115520⟨a, b | aaa=bb, aabbb=b⟩Noc Com 132
1116012⟨a, b | aaa=ab, abbb=bb⟩Noc 13
1116069⟨a, b | aaa=bb, abbb=ba⟩Noc 131
1116205⟨a, b | aab=ab, bbbb=aa⟩Noc 132
1116459⟨a, b | aba=bb, abbb=aa⟩Noc 131
1116506⟨a, b | aab=ab, bbb=aaa⟩Noc 131
1116515⟨a, b | aab=ba, bbb=aaa⟩Noc 13
1118958⟨a, b | aab=a, bbbbbb=a⟩Noc Com 13ℕ(13 = 6)4
1120046⟨a, b | aab=a, bbbb=bba⟩Noc 13
1120927⟨a, b | ab=aa, aaaa=bbb⟩Noc 137
1120991⟨a, b | ab=aa, baaa=bbb⟩Noc 133

Isomorphic instances

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

14 total

Σ#PresentationMapping
1112428⟨a, b | aaba=bb, bbbb=a⟩φ(a) = bbbb, φ(b) = b
1115519⟨a, b | aaa=bb, aabbb=a⟩φ(a) = bbbbbbbb, φ(b) = b
1115525⟨a, b | aaa=bb, ababb=a⟩φ(a) = bbbbbbbb, φ(b) = b
1115527⟨a, b | aaa=bb, abbab=a⟩φ(a) = bbbbbbbb, φ(b) = b
1115529⟨a, b | aaa=bb, abbba=a⟩φ(a) = bbbbbbbb, φ(b) = b
1115535⟨a, b | aaa=bb, baabb=a⟩φ(a) = bbbbbbbb, φ(b) = b
1115537⟨a, b | aaa=bb, babab=a⟩φ(a) = bbbbbbbb, φ(b) = b
1119352⟨a, b | aaa=b, abbbb=aa⟩φ(a) = b, φ(b) = bbb
1119366⟨a, b | aaa=b, babbb=aa⟩φ(a) = b, φ(b) = bbb
1119370⟨a, b | aaa=b, bbabb=aa⟩φ(a) = b, φ(b) = bbb
1124291⟨a, b | aa=b, abbbbbb=b⟩φ(a) = b, φ(b) = bb
1124319⟨a, b | aa=b, babbbbb=b⟩φ(a) = b, φ(b) = bb
1124327⟨a, b | aa=b, bbabbbb=b⟩φ(a) = b, φ(b) = bb
1124329⟨a, b | aa=b, bbbabbb=b⟩φ(a) = b, φ(b) = bb