#57 ⟨a, b | aa=a, bb=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.

1bb2b3
11bb2b3
bbb2b3b2
b2b2b3b2b3
b3b3b2b3b2

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. b4 ⇒ b2 [4]
2. a ⇒ b2 [2]
# ab:aa=a,bb=a b/a - -
bbbb=bb
a=bb

Same cardinality

8 unique, 1570 total

Σ#PresentationPropertiesDescriptionφ
57⟨a, b | aa=b, bb=1⟩Grp Com 4ℤ41419
661⟨a, b | aa=b, bb=a⟩Noc Com 4ℕ(4 = 1)72
7158⟨a, b | ab=aa, ba=b⟩Noc 414
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.

37 total

Σ#PresentationMapping
7250⟨a, b | aa=b, aab=b⟩φ(a) = b, φ(b) = bb
7252⟨a, b | aa=b, aba=b⟩φ(a) = b, φ(b) = bb
7276⟨a, b | aa=a, bb=aa⟩φ(a) = bb, φ(b) = b
7278⟨a, b | aa=b, bb=aa⟩φ(a) = b, φ(b) = bb
8631⟨a, b | ab=aa, aaa=b⟩φ(a) = b, φ(b) = bbb
8633⟨a, b | ab=aa, aab=b⟩φ(a) = b, φ(b) = bbb
8635⟨a, b | ab=aa, aba=b⟩φ(a) = b, φ(b) = bbb
8637⟨a, b | ab=aa, abb=b⟩φ(a) = b, φ(b) = bbb
8901⟨a, b | aa=b, aaaa=b⟩φ(a) = b, φ(b) = bb
8954⟨a, b | aa=a, aaa=bb⟩φ(a) = bb, φ(b) = b
8975⟨a, b | aa=b, aab=aa⟩φ(a) = b, φ(b) = bb
8979⟨a, b | aa=b, aba=aa⟩φ(a) = b, φ(b) = bb
91611⟨a, b | aab=aa, aba=b⟩φ(a) = b, φ(b) = bb
93002⟨a, b | aa=a, aaaa=bb⟩φ(a) = bb, φ(b) = b
93036⟨a, b | aa=b, aaaa=aa⟩φ(a) = b, φ(b) = bb
105101⟨a, b | aab=aa, aabb=b⟩φ(a) = b, φ(b) = bb
105105⟨a, b | aab=aa, abab=b⟩φ(a) = b, φ(b) = bb
105107⟨a, b | aab=aa, abba=b⟩φ(a) = b, φ(b) = bb
105233⟨a, b | aba=aa, abba=b⟩φ(a) = b, φ(b) = bb
106562⟨a, b | aaa=b, aaaa=aa⟩φ(a) = b, φ(b) = bbb
106613⟨a, b | aab=a, aabb=bb⟩φ(a) = bbb, φ(b) = b
106621⟨a, b | aab=a, abab=bb⟩φ(a) = bbb, φ(b) = b
106625⟨a, b | aab=a, abba=bb⟩φ(a) = bbb, φ(b) = b
106666⟨a, b | aab=b, aaab=aa⟩φ(a) = b, φ(b) = bbb
106765⟨a, b | aba=b, aaab=aa⟩φ(a) = b, φ(b) = bbb
106769⟨a, b | aba=b, aaba=aa⟩φ(a) = b, φ(b) = bbb
108842⟨a, b | aa=a, aaaaa=bb⟩φ(a) = bb, φ(b) = b
1115560⟨a, b | aab=aa, aabbb=b⟩φ(a) = b, φ(b) = bb
1115568⟨a, b | aab=aa, ababb=b⟩φ(a) = b, φ(b) = bb
1115572⟨a, b | aab=aa, abbab=b⟩φ(a) = b, φ(b) = bb
1115574⟨a, b | aab=aa, abbba=b⟩φ(a) = b, φ(b) = bb
1115826⟨a, b | aba=aa, abbba=b⟩φ(a) = b, φ(b) = bb
1119508⟨a, b | aab=b, aaaab=aa⟩φ(a) = b, φ(b) = bb
1119707⟨a, b | aba=b, aaaab=aa⟩φ(a) = b, φ(b) = bb
1119711⟨a, b | aba=b, aaaba=aa⟩φ(a) = b, φ(b) = bb
1119719⟨a, b | aba=b, aabaa=aa⟩φ(a) = b, φ(b) = bb
1124590⟨a, b | aa=a, aaaaaa=bb⟩φ(a) = bb, φ(b) = b