#257 ⟨a, b | aa=b, bbb=a⟩

Up:Monoid enumeration
Prev:#254 ⟨a, b | aa=b, abb=b⟩
Next:#258 ⟨a, b | aa=b, bbb=b⟩

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.

1aa2a3a4a5
11aa2a3a4a5
aaa2a3a4a5a
a2a2a3a4a5aa2
a3a3a4a5aa2a3
a4a4a5aa2a3a4
a5a5aa2a3a4a5

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. a6 ⇒ a [2]
2. b ⇒ a2 [1]
# ab:aa=b,bbb=a a/b - -
aaaaaa=a
b=aa

Same cardinality

19 unique, 1864 total

Σ#PresentationPropertiesDescriptionφ
622⟨a, b | ab=aa, bb=1⟩Noc 6107
626⟨a, b | aaa=1, abb=1⟩Grp Com 6ℤ61373
7160⟨a, b | ab=aa, bb=b⟩Noc 64
7245⟨a, b | aa=a, bbb=a⟩Noc Com 6ℕ(6 = 3)49
7258⟨a, b | aa=b, bbb=b⟩Noc Com 6ℕ(6 = 2)55
8639⟨a, b | ab=aa, baa=b⟩Noc 613
8644⟨a, b | ab=aa, bbb=a⟩Noc Com 6ℕ(6 = 4)46
8893⟨a, b | aa=a, abbb=b⟩Noc 619
81011⟨a, b | ab=a, baa=bb⟩Noc 613
91427⟨a, b | abba=b, baba=1⟩Grp 666
91581⟨a, b | aaa=aa, abb=b⟩Noc 65
91648⟨a, b | aab=bb, abb=a⟩Noc Com 613
91686⟨a, b | abb=aa, bbb=a⟩Noc Com 6ℕ(6 = 5)49
93075⟨a, b | ab=a, aaaa=bb⟩Noc Com 614
93132⟨a, b | ab=a, bbbb=aa⟩Noc Com 65
93134⟨a, b | ab=a, bbbb=ba⟩Noc 62
93193⟨a, b | ab=a, bbb=aaa⟩Noc Com 69
93199⟨a, b | ab=a, bbb=bba⟩Noc 63
1124145⟨a, b | aa=a, abbbbba=b⟩Noc Com 6

Isomorphic instances

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

61 total

Σ#PresentationMapping
8906⟨a, b | aa=b, aabb=a⟩φ(a) = bbb, φ(b) = b
8908⟨a, b | aa=b, abab=a⟩φ(a) = bbb, φ(b) = b
8910⟨a, b | aa=b, abba=a⟩φ(a) = bbb, φ(b) = b
8914⟨a, b | aa=b, baab=a⟩φ(a) = bbb, φ(b) = b
92002⟨a, b | aaa=b, aaab=a⟩φ(a) = bb, φ(b) = b
92004⟨a, b | aaa=b, aaba=a⟩φ(a) = bb, φ(b) = b
92021⟨a, b | aab=a, aaaa=b⟩φ(a) = bbbb, φ(b) = b
92085⟨a, b | aba=a, aaaa=b⟩φ(a) = bbbb, φ(b) = b
92898⟨a, b | aa=b, aaaab=a⟩φ(a) = bbb, φ(b) = b
92900⟨a, b | aa=b, aaaba=a⟩φ(a) = bbb, φ(b) = b
92904⟨a, b | aa=b, aabaa=a⟩φ(a) = bbb, φ(b) = b
92937⟨a, b | ab=a, aaaaa=b⟩φ(a) = b, φ(b) = bbbbb
108640⟨a, b | aa=b, aaaaaa=a⟩φ(a) = bbb, φ(b) = b
108715⟨a, b | ab=a, aaaaab=b⟩φ(a) = b, φ(b) = bbbbb
108717⟨a, b | ab=a, aaaaba=b⟩φ(a) = b, φ(b) = bbbbb
108721⟨a, b | ab=a, aaabaa=b⟩φ(a) = b, φ(b) = bbbbb
108729⟨a, b | ab=a, aabaaa=b⟩φ(a) = b, φ(b) = bbbbb
108745⟨a, b | ab=a, abaaaa=b⟩φ(a) = b, φ(b) = bbbbb
1114440⟨a, b | aaab=a, abbbb=b⟩φ(a) = bb, φ(b) = b
1114672⟨a, b | aabb=a, aaabb=b⟩φ(a) = bbb, φ(b) = b
1114678⟨a, b | aabb=a, aabba=b⟩φ(a) = bbb, φ(b) = b
1114684⟨a, b | aabb=a, abaab=b⟩φ(a) = bbb, φ(b) = b
1114686⟨a, b | aabb=a, ababa=b⟩φ(a) = bbb, φ(b) = b
1114690⟨a, b | aabb=a, abbaa=b⟩φ(a) = bbb, φ(b) = b
1114736⟨a, b | abab=a, aaabb=b⟩φ(a) = bbb, φ(b) = b
1114740⟨a, b | abab=a, aabab=b⟩φ(a) = bbb, φ(b) = b
1114742⟨a, b | abab=a, aabba=b⟩φ(a) = bbb, φ(b) = b
1114748⟨a, b | abab=a, abaab=b⟩φ(a) = bbb, φ(b) = b
1114750⟨a, b | abab=a, ababa=b⟩φ(a) = bbb, φ(b) = b
1114754⟨a, b | abab=a, abbaa=b⟩φ(a) = bbb, φ(b) = b
1114806⟨a, b | abba=a, aabba=b⟩φ(a) = bbb, φ(b) = b
1114812⟨a, b | abba=a, ababa=b⟩φ(a) = bbb, φ(b) = b
1114847⟨a, b | abba=b, aabbb=a⟩φ(a) = bb, φ(b) = b
1114853⟨a, b | abba=b, ababb=a⟩φ(a) = bb, φ(b) = b
1114855⟨a, b | abba=b, abbab=a⟩φ(a) = bb, φ(b) = b
1114857⟨a, b | abba=b, abbba=a⟩φ(a) = bb, φ(b) = b
1114863⟨a, b | abba=b, baabb=a⟩φ(a) = bb, φ(b) = b
1114865⟨a, b | abba=b, babab=a⟩φ(a) = bb, φ(b) = b
1118760⟨a, b | aaa=b, aaaaaa=a⟩φ(a) = bb, φ(b) = b
1118835⟨a, b | aab=a, aaaaab=b⟩φ(a) = bbbb, φ(b) = b
1118837⟨a, b | aab=a, aaaaba=b⟩φ(a) = bbbb, φ(b) = b
1118841⟨a, b | aab=a, aaabaa=b⟩φ(a) = bbbb, φ(b) = b
1118849⟨a, b | aab=a, aabaaa=b⟩φ(a) = bbbb, φ(b) = b
1118865⟨a, b | aab=a, abaaaa=b⟩φ(a) = bbbb, φ(b) = b
1119093⟨a, b | aba=a, aaaaba=b⟩φ(a) = bbbb, φ(b) = b
1119097⟨a, b | aba=a, aaabaa=b⟩φ(a) = bbbb, φ(b) = b
1124339⟨a, b | ab=a, aaaaabb=b⟩φ(a) = b, φ(b) = bbbbb
1124343⟨a, b | ab=a, aaaabab=b⟩φ(a) = b, φ(b) = bbbbb
1124345⟨a, b | ab=a, aaaabba=b⟩φ(a) = b, φ(b) = bbbbb
1124351⟨a, b | ab=a, aaabaab=b⟩φ(a) = b, φ(b) = bbbbb
1124353⟨a, b | ab=a, aaababa=b⟩φ(a) = b, φ(b) = bbbbb
1124357⟨a, b | ab=a, aaabbaa=b⟩φ(a) = b, φ(b) = bbbbb
1124367⟨a, b | ab=a, aabaaab=b⟩φ(a) = b, φ(b) = bbbbb
1124369⟨a, b | ab=a, aabaaba=b⟩φ(a) = b, φ(b) = bbbbb
1124373⟨a, b | ab=a, aababaa=b⟩φ(a) = b, φ(b) = bbbbb
1124381⟨a, b | ab=a, aabbaaa=b⟩φ(a) = b, φ(b) = bbbbb
1124399⟨a, b | ab=a, abaaaab=b⟩φ(a) = b, φ(b) = bbbbb
1124401⟨a, b | ab=a, abaaaba=b⟩φ(a) = b, φ(b) = bbbbb
1124405⟨a, b | ab=a, abaabaa=b⟩φ(a) = b, φ(b) = bbbbb
1124413⟨a, b | ab=a, ababaaa=b⟩φ(a) = b, φ(b) = bbbbb
1124429⟨a, b | ab=a, abbaaaa=b⟩φ(a) = b, φ(b) = bbbbb