#124 ⟨a, b | aab=a, bbb=1⟩

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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. a4 ⇒ a
  2. ab ⇒ a3
  3. b3 ⇒ 1 
# ab:aab=a,bbb=1 a/b
aaaa=a
ab=aaa
bbb=1

Cayley table

Idempotents are shown in bold.

1aba2bab2a3ba2b2aba3b2a2b2a3
11aba2bab2a3ba2b2aba3b2a2b2a3
aaa2a3a3aa2aa2a3a3aa2
bbbab2ba2b2a1ba3b2a2ab2a3a2a3
a2a2a3aaa2a3a2a3aaa2a3
bababa2ba3ba3baba2baba2ba3ba3baba2
b2b2b2a1b2a2abb2a3a2baa3ba2ba3
a3a3aa2a2a3aa3aa2a2a3a
ba2ba2ba3bababa2ba3ba2ba3bababa2ba3
b2ab2ab2a2b2a3b2a3b2ab2a2b2ab2a2b2a3b2a3b2ab2a2
ba3ba3baba2ba2ba3baba3baba2ba2ba3ba
b2a2b2a2b2a3b2ab2ab2a2b2a3b2a2b2a3b2ab2ab2a2b2a3
b2a3b2a3b2ab2a2b2a2b2a3b2ab2a3b2ab2a2b2a2b2a3b2a

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

37 unique, 598 total

Σ#PresentationDescriptionRelated
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
92019⟨a, b | aaa=b, bbbb=b⟩Isomorphic to ℕ(12 = 3)23 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
7150⟨a, b | ab=aa, bbb=1⟩φ(a) = a, φ(b) = bb
8440⟨a, b | aba=ab, bbb=1⟩φ(a) = ba, φ(b) = b
8446⟨a, b | abb=aa, bbb=1⟩φ(a) = a, φ(b) = b
91295⟨a, b | abb=aab, bbb=1⟩φ(a) = a, φ(b) = bb
91299⟨a, b | abb=aba, bbb=1⟩φ(a) = a, φ(b) = bb
92580⟨a, b | aaa=1, abba=ab⟩φ(a) = b, φ(b) = a
107493⟨a, b | aaa=1, aaabba=b⟩φ(a) = b, φ(b) = a
107762⟨a, b | aaa=1, aaaba=bb⟩φ(a) = bb, φ(b) = a
107765⟨a, b | aaa=1, aaabb=ba⟩φ(a) = bb, φ(b) = a
107787⟨a, b | aaa=1, ababa=ab⟩φ(a) = bb, φ(b) = a
108036⟨a, b | aaa=1, aaab=bba⟩φ(a) = b, φ(b) = a
108056⟨a, b | aaa=1, abab=aba⟩φ(a) = b, φ(b) = ba
1121669⟨a, b | aaa=1, aaababa=b⟩φ(a) = bb, φ(b) = a
1121673⟨a, b | aaa=1, aaabbaa=b⟩φ(a) = bb, φ(b) = a
1122210⟨a, b | aaa=1, aaabaa=bb⟩φ(a) = b, φ(b) = a
1122213⟨a, b | aaa=1, aaabab=ba⟩φ(a) = b, φ(b) = ba
1122255⟨a, b | aaa=1, abaaba=ab⟩φ(a) = b, φ(b) = ba
1122739⟨a, b | aaa=1, aaaba=bab⟩φ(a) = b, φ(b) = ba
1122746⟨a, b | aaa=1, aaabb=baa⟩φ(a) = b, φ(b) = a
1122765⟨a, b | aaa=1, aabba=aab⟩φ(a) = b, φ(b) = a
1122782⟨a, b | aaa=1, abaab=aba⟩φ(a) = b, φ(b) = a
1123269⟨a, b | aaa=1, aabb=aaba⟩φ(a) = bb, φ(b) = a
1123277⟨a, b | aaa=1, abab=abaa⟩φ(a) = bb, φ(b) = a

Other anti-isomorphic instances

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

38 total

Σ#PresentationMapping
8743⟨a, b | aaa=1, aabb=b⟩φ(a) = b, φ(b) = aa
8745⟨a, b | aaa=1, abab=b⟩φ(a) = b, φ(b) = aa
91308⟨a, b | bab=aab, bbb=1⟩φ(a) = aa, φ(b) = b
92427⟨a, b | aaa=1, aabab=b⟩φ(a) = b, φ(b) = ba
92433⟨a, b | aaa=1, abaab=b⟩φ(a) = b, φ(b) = ba
92572⟨a, b | aaa=1, aabb=ab⟩φ(a) = b, φ(b) = a
92587⟨a, b | aaa=1, baab=ab⟩φ(a) = b, φ(b) = a
107487⟨a, b | aaa=1, aaaabb=b⟩φ(a) = b, φ(b) = a
107497⟨a, b | aaa=1, aabaab=b⟩φ(a) = b, φ(b) = a
107511⟨a, b | aaa=1, abaaab=b⟩φ(a) = b, φ(b) = a
107758⟨a, b | aaa=1, aaaab=bb⟩φ(a) = b, φ(b) = aa
107764⟨a, b | aaa=1, aaabb=ab⟩φ(a) = b, φ(b) = aa
107771⟨a, b | aaa=1, aabab=ab⟩φ(a) = b, φ(b) = aa
107776⟨a, b | aaa=1, aabba=ba⟩φ(a) = b, φ(b) = aa
107805⟨a, b | aaa=1, baaab=ab⟩φ(a) = b, φ(b) = aa
108033⟨a, b | aaa=1, aaab=abb⟩φ(a) = b, φ(b) = a
108044⟨a, b | aaa=1, aaba=bba⟩φ(a) = b, φ(b) = a
108055⟨a, b | aaa=1, abab=aab⟩φ(a) = b, φ(b) = ba
108077⟨a, b | aaa=1, baab=aab⟩φ(a) = b, φ(b) = ba
1121655⟨a, b | aaa=1, aaaaabb=b⟩φ(a) = b, φ(b) = aa
1121659⟨a, b | aaa=1, aaaabab=b⟩φ(a) = b, φ(b) = aa
1121681⟨a, b | aaa=1, aabaaab=b⟩φ(a) = b, φ(b) = aa
1121709⟨a, b | aaa=1, abaaaab=b⟩φ(a) = b, φ(b) = aa
1122198⟨a, b | aaa=1, aaaaab=bb⟩φ(a) = b, φ(b) = a
1122212⟨a, b | aaa=1, aaabab=ab⟩φ(a) = b, φ(b) = ba
1122224⟨a, b | aaa=1, aabaab=ab⟩φ(a) = b, φ(b) = ba
1122229⟨a, b | aaa=1, aababa=ba⟩φ(a) = b, φ(b) = ba
1122293⟨a, b | aaa=1, baaaab=ab⟩φ(a) = b, φ(b) = ba
1122731⟨a, b | aaa=1, aaaab=bab⟩φ(a) = b, φ(b) = ba
1122743⟨a, b | aaa=1, aaabb=aab⟩φ(a) = b, φ(b) = a
1122766⟨a, b | aaa=1, aabba=aba⟩φ(a) = b, φ(b) = a
1122781⟨a, b | aaa=1, abaab=aab⟩φ(a) = b, φ(b) = a
1122825⟨a, b | aaa=1, baaab=aab⟩φ(a) = b, φ(b) = a
1123268⟨a, b | aaa=1, aabb=aaab⟩φ(a) = b, φ(b) = aa
1123274⟨a, b | aaa=1, abab=aaab⟩φ(a) = b, φ(b) = aa
1123280⟨a, b | aaa=1, abba=aaba⟩φ(a) = b, φ(b) = aa
1123291⟨a, b | aaa=1, baaa=aabb⟩φ(a) = b, φ(b) = aa
1123292⟨a, b | aaa=1, baaa=abab⟩φ(a) = b, φ(b) = aa