#158 ⟨a, b | ab=aa, ba=b⟩

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Contents

  1. Properties
  2. Finite structure
  3. Cayley table
  4. Right Cayley graph
  5. Left Cayley graph
  6. Rewriting system
  7. Same cardinality
  8. Isomorphic instances
  9. Anti-isomorphic instances

Properties

Finite structure

Elements in the center commute with all other elements.
A left zero element x satisfies xy = x for all y.
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.

Cayley table

Idempotents are shown in bold.

1abab
11abab
aaababab
bbbbb
ababababab

Right Cayley graph

Idempotents are shown in bold.

Left 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. b2 ⇒ b [3]
2. ba ⇒ b [2]
3. a2 ⇒ ab [1]
# ab:ab=aa,ba=b ba - -
bb=b
ba=b
aa=ab

Same cardinality

8 unique, 1593 total

Σ#PresentationPropertiesDescriptionφ
57⟨a, b | aa=b, bb=1⟩Grp Com 4ℤ41419
657⟨a, b | aa=a, bb=a⟩Noc Com 4ℕ(4 = 2)37
661⟨a, b | aa=b, bb=a⟩Noc Com 4ℕ(4 = 1)72
7159⟨a, b | ab=aa, bb=a⟩Noc Com 4ℕ(4 = 3)16
7242⟨a, b | aa=a, abb=b⟩Noc 414
7280⟨a, b | ab=a, bb=aa⟩Noc Com 417
92881⟨a, b | aa=a, abbba=b⟩Noc Com 48
105033⟨a, b | aaa=aa, abba=b⟩Noc Com 42

Isomorphic instances

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

8 total

Σ#PresentationMapping
81015⟨a, b | ab=a, bab=bb⟩φ(a) = b, φ(b) = a
93119⟨a, b | ab=a, babb=bb⟩φ(a) = b, φ(b) = a
109079⟨a, b | ab=a, babbb=bb⟩φ(a) = b, φ(b) = a
1119455⟨a, b | aab=a, baabb=bb⟩φ(a) = b, φ(b) = a
1119463⟨a, b | aab=a, babab=bb⟩φ(a) = b, φ(b) = a
1119467⟨a, b | aab=a, babba=bb⟩φ(a) = b, φ(b) = a
1119471⟨a, b | aab=a, babbb=bb⟩φ(a) = b, φ(b) = a
1125051⟨a, b | ab=a, babbbb=bb⟩φ(a) = b, φ(b) = a

Anti-isomorphic instances

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

6 total

Σ#PresentationMapping
91659⟨a, b | aba=aa, abb=b⟩φ(a) = a, φ(b) = b
105235⟨a, b | aba=aa, abbb=b⟩φ(a) = a, φ(b) = b
1112327⟨a, b | aaba=aa, aabb=b⟩φ(a) = a, φ(b) = b
1112331⟨a, b | aaba=aa, abab=b⟩φ(a) = a, φ(b) = b
1112335⟨a, b | aaba=aa, abbb=b⟩φ(a) = a, φ(b) = b
1115828⟨a, b | aba=aa, abbbb=b⟩φ(a) = a, φ(b) = b