#2064 ⟨a, b | aab=b, abba=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. b5 ⇒ b
  2. ab ⇒ b3
  3. b4a ⇒ a
  4. a2 ⇒ b2a 
# ab:aab=b,abba=a b/a
bbbbb=b
ab=bbb
bbbba=a
aa=bba

Cayley table

Idempotents are shown in bold.

1abbab2b2ab3b3ab4
11abbab2b2ab3b3ab4
aab2ab3b3ab4abbab2
bbbab2b2ab3b3ab4ab
babab3ab4abbab2b2ab3
b2b2b2ab3b3ab4abbab2
b2ab2aabbab2b2ab3b3ab4
b3b3b3ab4abbab2b2ab3
b3ab3abab2b2ab3b3ab4ab
b4b4abbab2b2ab3b3ab4

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

26 unique, 904 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
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.

14 total

Σ#PresentationMapping
92072⟨a, b | aab=b, baba=a⟩φ(a) = a, φ(b) = b
92076⟨a, b | aab=b, bbaa=a⟩φ(a) = a, φ(b) = b
106449⟨a, b | aba=a, aaaab=b⟩φ(a) = b, φ(b) = ba
1114540⟨a, b | aaba=a, aaaab=b⟩φ(a) = b, φ(b) = a
1118972⟨a, b | aab=b, aaabba=a⟩φ(a) = a, φ(b) = b
1118980⟨a, b | aab=b, aababa=a⟩φ(a) = a, φ(b) = b
1118984⟨a, b | aab=b, aabbaa=a⟩φ(a) = a, φ(b) = b
1118996⟨a, b | aab=b, abaaba=a⟩φ(a) = a, φ(b) = b
1119000⟨a, b | aab=b, ababaa=a⟩φ(a) = a, φ(b) = b
1119008⟨a, b | aab=b, abbaaa=a⟩φ(a) = a, φ(b) = b
1119028⟨a, b | aab=b, baaaba=a⟩φ(a) = a, φ(b) = b
1119032⟨a, b | aab=b, baabaa=a⟩φ(a) = a, φ(b) = b
1119040⟨a, b | aab=b, babaaa=a⟩φ(a) = a, φ(b) = b
1119056⟨a, b | aab=b, bbaaaa=a⟩φ(a) = a, φ(b) = b

Other anti-isomorphic instances

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

24 total

Σ#PresentationMapping
92969⟨a, b | ab=a, baaaa=b⟩φ(a) = b, φ(b) = bba
106351⟨a, b | aab=a, baaaa=b⟩φ(a) = b, φ(b) = ba
108779⟨a, b | ab=a, baaaab=b⟩φ(a) = b, φ(b) = bba
108781⟨a, b | ab=a, baaaba=b⟩φ(a) = b, φ(b) = bba
108785⟨a, b | ab=a, baabaa=b⟩φ(a) = b, φ(b) = bba
108793⟨a, b | ab=a, babaaa=b⟩φ(a) = b, φ(b) = bba
108809⟨a, b | ab=a, bbaaaa=b⟩φ(a) = b, φ(b) = bba
1114442⟨a, b | aaab=a, baaaa=b⟩φ(a) = b, φ(b) = a
1114570⟨a, b | aaba=a, baaaa=b⟩φ(a) = b, φ(b) = a
1124467⟨a, b | ab=a, baaaabb=b⟩φ(a) = b, φ(b) = bba
1124471⟨a, b | ab=a, baaabab=b⟩φ(a) = b, φ(b) = bba
1124473⟨a, b | ab=a, baaabba=b⟩φ(a) = b, φ(b) = bba
1124479⟨a, b | ab=a, baabaab=b⟩φ(a) = b, φ(b) = bba
1124481⟨a, b | ab=a, baababa=b⟩φ(a) = b, φ(b) = bba
1124485⟨a, b | ab=a, baabbaa=b⟩φ(a) = b, φ(b) = bba
1124495⟨a, b | ab=a, babaaab=b⟩φ(a) = b, φ(b) = bba
1124497⟨a, b | ab=a, babaaba=b⟩φ(a) = b, φ(b) = bba
1124501⟨a, b | ab=a, bababaa=b⟩φ(a) = b, φ(b) = bba
1124509⟨a, b | ab=a, babbaaa=b⟩φ(a) = b, φ(b) = bba
1124527⟨a, b | ab=a, bbaaaab=b⟩φ(a) = b, φ(b) = bba
1124529⟨a, b | ab=a, bbaaaba=b⟩φ(a) = b, φ(b) = bba
1124533⟨a, b | ab=a, bbaabaa=b⟩φ(a) = b, φ(b) = bba
1124541⟨a, b | ab=a, bbabaaa=b⟩φ(a) = b, φ(b) = bba
1124557⟨a, b | ab=a, bbbaaaa=b⟩φ(a) = b, φ(b) = bba