#5079 ⟨a, b | aaa=bb, aaba=b⟩

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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. ba ⇒ ab
  2. a3 ⇒ b2
  3. b3 ⇒ b 
# ab:aaa=bb,aaba=b ab
ba=ab
aaa=bb
bbb=b

Staircase diagram

Cayley table

Idempotents are shown in bold.

1aba2abb2a2bab2a2b2
11aba2abb2a2bab2a2b2
aaa2abb2a2bab2ba2b2b2
bbabb2a2bab2ba2b2aba2b
a2a2b2a2bab2ba2b2abb2ab2
ababa2bab2ba2b2abb2a2bb
b2b2ab2ba2b2abb2a2bab2a2b2
a2ba2bba2b2abb2a2bab2bab
ab2ab2a2b2abb2a2bab2ba2b2b2
a2b2a2b2b2a2bab2ba2b2abb2ab2

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

26 unique, 934 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
8571⟨a, b | aaa=a, bbb=a⟩Isomorphic to ℕ(9 = 3)48 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
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.

8 total

Σ#PresentationMapping
1112967⟨a, b | abb=aaa, aaab=b⟩φ(a) = a, φ(b) = ab
1112969⟨a, b | abb=aaa, aaba=b⟩φ(a) = a, φ(b) = ab
1112973⟨a, b | abb=aaa, abaa=b⟩φ(a) = a, φ(b) = ab
1112979⟨a, b | abb=aaa, abbb=b⟩φ(a) = a, φ(b) = ab
1113083⟨a, b | bab=aaa, aaab=b⟩φ(a) = a, φ(b) = ab
1113085⟨a, b | bab=aaa, aaba=b⟩φ(a) = a, φ(b) = ab
1113093⟨a, b | bab=aaa, abbb=b⟩φ(a) = a, φ(b) = ab
1113097⟨a, b | bab=aaa, babb=b⟩φ(a) = a, φ(b) = ab