#2019 ⟨a, b | aaa=b, bbbb=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. a12 ⇒ a3
  2. b ⇒ a3 
# ab:aaa=b,bbbb=b a/b
aaaaaaaaaaaa=aaa
b=aaa

Staircase diagram

Cayley table

Idempotents are shown in bold.

1aa2a3a4a5a6a7a8a9a10a11
11aa2a3a4a5a6a7a8a9a10a11
aaa2a3a4a5a6a7a8a9a10a11a3
a2a2a3a4a5a6a7a8a9a10a11a3a4
a3a3a4a5a6a7a8a9a10a11a3a4a5
a4a4a5a6a7a8a9a10a11a3a4a5a6
a5a5a6a7a8a9a10a11a3a4a5a6a7
a6a6a7a8a9a10a11a3a4a5a6a7a8
a7a7a8a9a10a11a3a4a5a6a7a8a9
a8a8a9a10a11a3a4a5a6a7a8a9a10
a9a9a10a11a3a4a5a6a7a8a9a10a11
a10a10a11a3a4a5a6a7a8a9a10a11a3
a11a11a3a4a5a6a7a8a9a10a11a3a4

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

37 unique, 636 total

Σ#PresentationDescriptionRelated
7124⟨a, b | aab=a, bbb=1⟩Finite non-commutative monoid with 12 elements23 iso, 38 anti-iso
8458⟨a, b | aaaa=1, abbb=1⟩Isomorphic to ℤ12325 iso
8645⟨a, b | ab=aa, bbb=b⟩Finite non-commutative monoid with 12 elements1 anti-iso
91646⟨a, b | aab=bb, aba=a⟩Finite non-commutative monoid with 12 elements2 iso
91963⟨a, b | aba=b, aaabb=1⟩Finite non-Abelian group with 12 elements30 iso
91998⟨a, b | aaa=a, bbbb=a⟩Isomorphic to ℕ(12 = 4)6 iso
92018⟨a, b | aaa=b, bbbb=a⟩Isomorphic to ℕ(12 = 1)27 iso
92259⟨a, b | ab=aa, bbb=bb⟩Finite non-commutative monoid with 12 elements
105040⟨a, b | aaa=aa, bbbb=a⟩Isomorphic to ℕ(12 = 8)
105072⟨a, b | aaa=ab, bbbb=a⟩Isomorphic to ℕ(12 = 5)7 iso
105092⟨a, b | aaa=bb, bbbb=a⟩Isomorphic to ℕ(12 = 2)43 iso
105202⟨a, b | aab=bb, abba=a⟩Finite non-commutative monoid with 12 elements9 iso, 32 anti-iso
105330⟨a, b | aaa=ab, bab=bb⟩Finite non-commutative monoid with 12 elements1 iso
106597⟨a, b | aaa=b, bbbb=bb⟩Isomorphic to ℕ(12 = 6)4 iso
106660⟨a, b | aab=a, bbbb=ba⟩Finite non-commutative monoid with 12 elements1 anti-iso
106661⟨a, b | aab=a, bbbb=bb⟩Finite non-commutative monoid with 12 elements1 anti-iso
107105⟨a, b | ab=aa, bbaa=bb⟩Finite non-commutative monoid with 12 elements2 iso
1112897⟨a, b | aab=aaa, baaa=b⟩Finite non-commutative monoid with 12 elements5 iso
1112910⟨a, b | aab=aaa, bbbb=a⟩Isomorphic to ℕ(12 = 9)2 iso
1115428⟨a, b | aaa=aa, abbbb=b⟩Finite non-commutative monoid with 12 elements
1115996⟨a, b | aaa=ab, aabb=bb⟩Finite non-commutative monoid with 12 elements2 iso
1116057⟨a, b | aaa=bb, aabb=ab⟩Finite non-commutative monoid with 12 elements2 iso, 2 anti-iso
1116104⟨a, b | aab=aa, abab=bb⟩Finite non-commutative monoid with 12 elements
1116274⟨a, b | aab=bb, aaaa=ab⟩Finite non-commutative monoid with 12 elements1 iso
1116275⟨a, b | aab=bb, aaaa=ba⟩Finite non-commutative monoid with 12 elements
1116448⟨a, b | aba=bb, aabb=aa⟩Finite non-commutative monoid with 12 elements2 iso
1119773⟨a, b | aba=b, bbbbb=aa⟩Finite commutative monoid with 12 elements
1119917⟨a, b | aaa=b, bbbb=abb⟩Isomorphic to ℕ(12 = 7)2 iso
1120054⟨a, b | aab=b, aaaa=bba⟩Finite non-commutative monoid with 12 elements
1120686⟨a, b | bb=aa, aaaaab=a⟩Finite commutative monoid with 12 elements15 iso
1120787⟨a, b | ab=aa, baaaa=bb⟩Finite non-commutative monoid with 12 elements7 iso
1121086⟨a, b | bb=aa, aaaa=aba⟩Finite non-commutative monoid with 12 elements4 iso
1121092⟨a, b | bb=aa, aaab=aba⟩Finite non-commutative monoid with 12 elements3 iso
1121112⟨a, b | bb=aa, abab=aba⟩Finite non-commutative monoid with 12 elements
1124147⟨a, b | aa=a, abbbbbb=b⟩Finite non-commutative monoid with 12 elements
1124991⟨a, b | ab=a, baaaaa=bb⟩Finite non-commutative monoid with 12 elements
1125603⟨a, b | ab=a, bbaaa=bbb⟩Finite non-commutative monoid with 12 elements

Other isomorphic instances

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

23 total

Σ#PresentationMapping
105093⟨a, b | aaa=bb, bbbb=b⟩φ(a) = a, φ(b) = aaaaaa
1113111⟨a, b | bbb=aaa, aaaa=b⟩φ(a) = a, φ(b) = aaaa
1113112⟨a, b | bbb=aaa, aaab=a⟩φ(a) = aaaa, φ(b) = a
1113114⟨a, b | bbb=aaa, aaba=a⟩φ(a) = aaaa, φ(b) = a
1113116⟨a, b | bbb=aaa, aabb=a⟩φ(a) = aaaaaaa, φ(b) = a
1113117⟨a, b | bbb=aaa, abab=a⟩φ(a) = aaaaaaa, φ(b) = a
1115512⟨a, b | aaa=bb, aaabb=b⟩φ(a) = a, φ(b) = aaaaaa
1115516⟨a, b | aaa=bb, aabab=b⟩φ(a) = a, φ(b) = aaaaaa
1115518⟨a, b | aaa=bb, aabba=b⟩φ(a) = a, φ(b) = aaaaaa
1115522⟨a, b | aaa=bb, abaab=b⟩φ(a) = a, φ(b) = aaaaaa
1115534⟨a, b | aaa=bb, baaab=b⟩φ(a) = a, φ(b) = aaaaaa
1118775⟨a, b | aaa=b, aaabbb=b⟩φ(a) = a, φ(b) = aaa
1118781⟨a, b | aaa=b, aababb=b⟩φ(a) = a, φ(b) = aaa
1118785⟨a, b | aaa=b, aabbab=b⟩φ(a) = a, φ(b) = aaa
1118787⟨a, b | aaa=b, aabbba=b⟩φ(a) = a, φ(b) = aaa
1118795⟨a, b | aaa=b, abaabb=b⟩φ(a) = a, φ(b) = aaa
1118797⟨a, b | aaa=b, ababab=b⟩φ(a) = a, φ(b) = aaa
1118799⟨a, b | aaa=b, ababba=b⟩φ(a) = a, φ(b) = aaa
1118803⟨a, b | aaa=b, abbaab=b⟩φ(a) = a, φ(b) = aaa
1118815⟨a, b | aaa=b, baaabb=b⟩φ(a) = a, φ(b) = aaa
1118817⟨a, b | aaa=b, baabab=b⟩φ(a) = a, φ(b) = aaa
1119914⟨a, b | aaa=b, bbbb=aaa⟩φ(a) = a, φ(b) = aaa
1124858⟨a, b | aa=b, bbbbbb=ab⟩φ(a) = a, φ(b) = aa