#571 ⟨a, b | aaa=a, 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

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.

1bb2b3b4b5b6b7b8
11bb2b3b4b5b6b7b8
bbb2b3b4b5b6b7b8b3
b2b2b3b4b5b6b7b8b3b4
b3b3b4b5b6b7b8b3b4b5
b4b4b5b6b7b8b3b4b5b6
b5b5b6b7b8b3b4b5b6b7
b6b6b7b8b3b4b5b6b7b8
b7b7b8b3b4b5b6b7b8b3
b8b8b3b4b5b6b7b8b3b4

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. b9 ⇒ b3 [4]
2. a ⇒ b3 [2]
# ab:aaa=a,bbb=a b/a - -
bbbbbbbbb=bbb
a=bbb

Same cardinality

26 unique, 894 total

Σ#PresentationPropertiesDescriptionφ
7118⟨a, b | aaa=b, bbb=1⟩Grp Com 9ℤ9562
8568⟨a, b | aaa=a, abb=b⟩Noc 95
8581⟨a, b | aaa=b, bbb=a⟩Noc Com 9ℕ(9 = 1)62
8601⟨a, b | aba=b, bab=a⟩Noc 93
91584⟨a, b | aaa=aa, bbb=a⟩Noc Com 9ℕ(9 = 6)16
91595⟨a, b | aaa=ab, bab=b⟩Noc 99
91598⟨a, b | aaa=ab, bbb=a⟩Noc Com 9ℕ(9 = 4)40
91608⟨a, b | aaa=bb, bbb=a⟩Noc Com 9ℕ(9 = 2)70
91991⟨a, b | aaa=a, abba=b⟩Noc 93
92064⟨a, b | aab=b, abba=a⟩Noc 938
92255⟨a, b | ab=aa, bba=bb⟩Noc 9
104125⟨a, b | aab=aaa, baa=b⟩Noc 92
104130⟨a, b | aab=aaa, bbb=a⟩Noc Com 9ℕ(9 = 7)1
104156⟨a, b | abb=aaa, bbb=a⟩Noc Com 9ℕ(9 = 5)12
105035⟨a, b | aaa=aa, abbb=b⟩Noc 93
105079⟨a, b | aaa=bb, aaba=b⟩Noc Com 98
105124⟨a, b | aab=aa, bbbb=a⟩Noc Com 9ℕ(9 = 8)5
105339⟨a, b | aaa=bb, aab=ab⟩Noc 91
106824⟨a, b | aab=b, bbb=aaa⟩Noc Com 95
107139⟨a, b | bb=aa, aaaa=ab⟩Noc 93
109327⟨a, b | ab=a, baaa=bbb⟩Noc 93
109359⟨a, b | ab=a, bbaa=bbb⟩Noc 92
1119506⟨a, b | aab=b, aaaaa=ba⟩Noc 9
1119507⟨a, b | aab=b, aaaaa=bb⟩Noc Com 9
1120723⟨a, b | ab=aa, aaaaa=bb⟩Noc 915
1125656⟨a, b | ab=a, bbbbb=baa⟩Noc 9

Isomorphic instances

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

48 total

Σ#PresentationMapping
106285⟨a, b | aaa=b, aaabb=b⟩φ(a) = b, φ(b) = bbb
106289⟨a, b | aaa=b, aabab=b⟩φ(a) = b, φ(b) = bbb
106291⟨a, b | aaa=b, aabba=b⟩φ(a) = b, φ(b) = bbb
106295⟨a, b | aaa=b, abaab=b⟩φ(a) = b, φ(b) = bbb
106297⟨a, b | aaa=b, ababa=b⟩φ(a) = b, φ(b) = bbb
106307⟨a, b | aaa=b, baaab=b⟩φ(a) = b, φ(b) = bbb
106802⟨a, b | aaa=a, bbb=aaa⟩φ(a) = bbb, φ(b) = b
106807⟨a, b | aaa=b, bbb=aaa⟩φ(a) = b, φ(b) = bbb
106815⟨a, b | aab=a, bbb=aaa⟩φ(a) = bbbbb, φ(b) = b
108961⟨a, b | aa=b, abbbb=ab⟩φ(a) = b, φ(b) = bb
108962⟨a, b | aa=b, abbbb=ba⟩φ(a) = b, φ(b) = bb
108975⟨a, b | aa=b, babbb=ab⟩φ(a) = b, φ(b) = bb
108976⟨a, b | aa=b, babbb=ba⟩φ(a) = b, φ(b) = bb
108979⟨a, b | aa=b, bbabb=ab⟩φ(a) = b, φ(b) = bb
1112965⟨a, b | abb=aaa, aaaa=b⟩φ(a) = b, φ(b) = bbbb
1112971⟨a, b | abb=aaa, aabb=b⟩φ(a) = b, φ(b) = bbbb
1112975⟨a, b | abb=aaa, abab=b⟩φ(a) = b, φ(b) = bbbb
1112977⟨a, b | abb=aaa, abba=b⟩φ(a) = b, φ(b) = bbbb
1113081⟨a, b | bab=aaa, aaaa=b⟩φ(a) = b, φ(b) = bbbb
1113087⟨a, b | bab=aaa, aabb=b⟩φ(a) = b, φ(b) = bbbb
1113089⟨a, b | bab=aaa, abab=b⟩φ(a) = b, φ(b) = bbbb
1115456⟨a, b | aaa=ab, aabbb=b⟩φ(a) = b, φ(b) = bbbbbbbb
1115464⟨a, b | aaa=ab, ababb=b⟩φ(a) = b, φ(b) = bbbbbbbb
1115468⟨a, b | aaa=ab, abbab=b⟩φ(a) = b, φ(b) = bbbbbbbb
1115470⟨a, b | aaa=ab, abbba=b⟩φ(a) = b, φ(b) = bbbbbbbb
1124752⟨a, b | aa=b, aaabbb=ab⟩φ(a) = b, φ(b) = bb
1124753⟨a, b | aa=b, aaabbb=ba⟩φ(a) = b, φ(b) = bb
1124764⟨a, b | aa=b, aababb=ab⟩φ(a) = b, φ(b) = bb
1124765⟨a, b | aa=b, aababb=ba⟩φ(a) = b, φ(b) = bb
1124771⟨a, b | aa=b, aabbab=ab⟩φ(a) = b, φ(b) = bb
1124772⟨a, b | aa=b, aabbab=ba⟩φ(a) = b, φ(b) = bb
1124775⟨a, b | aa=b, aabbba=ab⟩φ(a) = b, φ(b) = bb
1124776⟨a, b | aa=b, aabbba=ba⟩φ(a) = b, φ(b) = bb
1124790⟨a, b | aa=b, abaabb=ab⟩φ(a) = b, φ(b) = bb
1124791⟨a, b | aa=b, abaabb=ba⟩φ(a) = b, φ(b) = bb
1124794⟨a, b | aa=b, ababab=ab⟩φ(a) = b, φ(b) = bb
1124795⟨a, b | aa=b, ababab=ba⟩φ(a) = b, φ(b) = bb
1124798⟨a, b | aa=b, ababba=ab⟩φ(a) = b, φ(b) = bb
1124799⟨a, b | aa=b, ababba=ba⟩φ(a) = b, φ(b) = bb
1124806⟨a, b | aa=b, abbaab=ab⟩φ(a) = b, φ(b) = bb
1124807⟨a, b | aa=b, abbaab=ba⟩φ(a) = b, φ(b) = bb
1124828⟨a, b | aa=b, baaabb=ab⟩φ(a) = b, φ(b) = bb
1124829⟨a, b | aa=b, baaabb=ba⟩φ(a) = b, φ(b) = bb
1124832⟨a, b | aa=b, baabab=ab⟩φ(a) = b, φ(b) = bb
1124833⟨a, b | aa=b, baabab=ba⟩φ(a) = b, φ(b) = bb
1125356⟨a, b | aa=b, abbbb=aaa⟩φ(a) = b, φ(b) = bb
1125384⟨a, b | aa=b, babbb=aaa⟩φ(a) = b, φ(b) = bb
1125392⟨a, b | aa=b, bbabb=aaa⟩φ(a) = b, φ(b) = bb