#1620 ⟨a, b | aab=aa, bbb=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

The zero element z satisfies zy = yz = z 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.

Staircase diagram

Cayley table

Idempotents are shown in bold.

1bb2b3b4b5b6
11bb2b3b4b5b6
bbb2b3b4b5b6b6
b2b2b3b4b5b6b6b6
b3b3b4b5b6b6b6b6
b4b4b5b6b6b6b6b6
b5b5b6b6b6b6b6b6
b6b6b6b6b6b6b6b6

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. b7 ⇒ b6 [4]
2. a ⇒ b3 [2]
# ab:aab=aa,bbb=a b/a - -
bbbbbbb=bbbbbb
a=bbb

Same cardinality

22 unique, 1188 total

Σ#PresentationPropertiesDescriptionφ
7116⟨a, b | aaa=b, abb=1⟩Grp Com 7ℤ7763
8577⟨a, b | aaa=b, abb=a⟩Noc Com 7ℕ(7 = 1)59
8578⟨a, b | aaa=b, abb=b⟩Noc Com 7ℕ(7 = 3)44
8913⟨a, b | aa=b, abbb=b⟩Noc Com 7ℕ(7 = 2)82
8937⟨a, b | ab=a, baaa=b⟩Noc 766
91593⟨a, b | aaa=ab, baa=b⟩Noc 73
91601⟨a, b | aaa=bb, aab=b⟩Noc Com 75
91632⟨a, b | aab=ab, bbb=a⟩Noc Com 7ℕ(7 = 4)50
92074⟨a, b | aab=b, babb=a⟩Noc Com 720
92231⟨a, b | ab=aa, aaa=bb⟩Noc 73
92271⟨a, b | bb=aa, aaa=ab⟩Noc 73
93197⟨a, b | ab=a, bbb=baa⟩Noc 75
104162⟨a, b | abb=aab, bbb=a⟩Noc Com 7ℕ(7 = 5)42
106251⟨a, b | aaa=a, aabba=b⟩Noc 71
108987⟨a, b | ab=a, aaaaa=bb⟩Noc Com 75
109108⟨a, b | ab=a, bbbbb=aa⟩Noc Com 72
109110⟨a, b | ab=a, bbbbb=ba⟩Noc 71
109263⟨a, b | ab=a, aaaa=bbb⟩Noc Com 74
109375⟨a, b | ab=a, bbba=bbb⟩Noc 73
109376⟨a, b | ab=a, bbbb=aaa⟩Noc Com 73
109382⟨a, b | ab=a, bbbb=bba⟩Noc 71
1114843⟨a, b | abba=b, aabab=a⟩Noc 71

Isomorphic instances

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

35 total

Σ#PresentationMapping
91664⟨a, b | aba=aa, bbb=a⟩φ(a) = bbb, φ(b) = b
108838⟨a, b | ab=a, bbbbbb=a⟩φ(a) = bbbbbb, φ(b) = b
109235⟨a, b | aa=b, abbb=bbb⟩φ(a) = b, φ(b) = bb
109249⟨a, b | aa=b, babb=bbb⟩φ(a) = b, φ(b) = bb
1115610⟨a, b | aab=ab, aaaaa=b⟩φ(a) = b, φ(b) = bbbbb
1115674⟨a, b | aab=ba, aaaaa=b⟩φ(a) = b, φ(b) = bbbbb
1115842⟨a, b | aba=ab, aaaaa=b⟩φ(a) = b, φ(b) = bbbbb
1119310⟨a, b | aaa=b, aaaab=bb⟩φ(a) = b, φ(b) = bbb
1119314⟨a, b | aaa=b, aaaba=bb⟩φ(a) = b, φ(b) = bbb
1119321⟨a, b | aaa=b, aabaa=bb⟩φ(a) = b, φ(b) = bbb
1119857⟨a, b | aaa=b, aaab=abb⟩φ(a) = b, φ(b) = bbb
1119859⟨a, b | aaa=b, aaab=bab⟩φ(a) = b, φ(b) = bbb
1119860⟨a, b | aaa=b, aaab=bba⟩φ(a) = b, φ(b) = bbb
1119865⟨a, b | aaa=b, aaba=abb⟩φ(a) = b, φ(b) = bbb
1119867⟨a, b | aaa=b, aaba=bab⟩φ(a) = b, φ(b) = bbb
1119868⟨a, b | aaa=b, aaba=bba⟩φ(a) = b, φ(b) = bbb
1125113⟨a, b | ab=a, bbbbbb=ab⟩φ(a) = bbbbbb, φ(b) = b
1125289⟨a, b | aa=b, aaabb=bbb⟩φ(a) = b, φ(b) = bb
1125303⟨a, b | aa=b, aabab=bbb⟩φ(a) = b, φ(b) = bb
1125311⟨a, b | aa=b, aabba=bbb⟩φ(a) = b, φ(b) = bb
1125327⟨a, b | aa=b, abaab=bbb⟩φ(a) = b, φ(b) = bb
1125333⟨a, b | aa=b, ababa=bbb⟩φ(a) = b, φ(b) = bb
1125369⟨a, b | aa=b, baaab=bbb⟩φ(a) = b, φ(b) = bb
1125748⟨a, b | aa=b, abbb=aabb⟩φ(a) = b, φ(b) = bb
1125750⟨a, b | aa=b, abbb=abab⟩φ(a) = b, φ(b) = bb
1125751⟨a, b | aa=b, abbb=abba⟩φ(a) = b, φ(b) = bb
1125762⟨a, b | aa=b, baab=abbb⟩φ(a) = b, φ(b) = bb
1125765⟨a, b | aa=b, baba=abbb⟩φ(a) = b, φ(b) = bb
1125769⟨a, b | aa=b, babb=aabb⟩φ(a) = b, φ(b) = bb
1125771⟨a, b | aa=b, babb=abab⟩φ(a) = b, φ(b) = bb
1125772⟨a, b | aa=b, babb=abba⟩φ(a) = b, φ(b) = bb
1125775⟨a, b | aa=b, babb=baab⟩φ(a) = b, φ(b) = bb
1125776⟨a, b | aa=b, babb=baba⟩φ(a) = b, φ(b) = bb
1125778⟨a, b | aa=b, bbaa=abbb⟩φ(a) = b, φ(b) = bb
1125779⟨a, b | aa=b, bbaa=babb⟩φ(a) = b, φ(b) = bb