#571 ⟨a, b | aaa=a, bbb=a⟩

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Quick links

  1. Properties
  2. Elements
  3. Staircase diagram
  4. Cayley table
  5. Right Cayley graph
  6. Rewriting system
  7. Same cardinality
  8. Isomorphic instances

Properties

Elements

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 to reduce:
Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Reduction strategy:
Path to normal form: 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#PresentationDescriptionRelated
7118⟨a, b | aaa=b, bbb=1⟩Isomorphic to ℤ9562 iso
8568⟨a, b | aaa=a, abb=b⟩Finite non-commutative monoid with 9 elements5 iso
8581⟨a, b | aaa=b, bbb=a⟩Isomorphic to ℕ(9 = 1)62 iso
8601⟨a, b | aba=b, bab=a⟩Finite non-commutative monoid with 9 elements3 iso
91584⟨a, b | aaa=aa, bbb=a⟩Isomorphic to ℕ(9 = 6)16 iso
91595⟨a, b | aaa=ab, bab=b⟩Finite non-commutative monoid with 9 elements4 iso, 5 anti-iso
91598⟨a, b | aaa=ab, bbb=a⟩Isomorphic to ℕ(9 = 4)40 iso
91608⟨a, b | aaa=bb, bbb=a⟩Isomorphic to ℕ(9 = 2)70 iso
91991⟨a, b | aaa=a, abba=b⟩Finite non-commutative monoid with 9 elements3 iso
92064⟨a, b | aab=b, abba=a⟩Finite non-commutative monoid with 9 elements14 iso, 24 anti-iso
92255⟨a, b | ab=aa, bba=bb⟩Finite non-commutative monoid with 9 elements
104125⟨a, b | aab=aaa, baa=b⟩Finite non-commutative monoid with 9 elements1 iso, 1 anti-iso
104130⟨a, b | aab=aaa, bbb=a⟩Isomorphic to ℕ(9 = 7)1 iso
104156⟨a, b | abb=aaa, bbb=a⟩Isomorphic to ℕ(9 = 5)12 iso
105035⟨a, b | aaa=aa, abbb=b⟩Finite non-commutative monoid with 9 elements3 iso
105079⟨a, b | aaa=bb, aaba=b⟩Finite commutative monoid with 9 elements8 iso
105124⟨a, b | aab=aa, bbbb=a⟩Isomorphic to ℕ(9 = 8)5 iso
105339⟨a, b | aaa=bb, aab=ab⟩Finite non-commutative monoid with 9 elements1 iso
106824⟨a, b | aab=b, bbb=aaa⟩Finite commutative monoid with 9 elements5 iso
107139⟨a, b | bb=aa, aaaa=ab⟩Finite non-commutative monoid with 9 elements3 iso
109327⟨a, b | ab=a, baaa=bbb⟩Finite non-commutative monoid with 9 elements3 iso
109359⟨a, b | ab=a, bbaa=bbb⟩Finite non-commutative monoid with 9 elements2 iso
1119506⟨a, b | aab=b, aaaaa=ba⟩Finite non-commutative monoid with 9 elements
1119507⟨a, b | aab=b, aaaaa=bb⟩Finite commutative monoid with 9 elements
1120723⟨a, b | ab=aa, aaaaa=bb⟩Finite non-commutative monoid with 9 elements15 iso
1125656⟨a, b | ab=a, bbbbb=baa⟩Finite non-commutative monoid with 9 elements

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