#571 ⟨a, b | aaa=a, bbb=a⟩

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Properties

Element profile

Complete 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
  1. b9 ⇒ b3
  2. a ⇒ b3 
# ab:aaa=a,bbb=a b/a
bbbbbbbbb=bbb
a=bbb

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.

Others with 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

Other 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