#937 ⟨a, b | ab=a, baaa=b⟩

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Properties

Elements

Elements in the center commute with all other elements.
An idempotent element x satisfies x2 = x.
The index and period of x is the least m (index) and n (period) such that x(m+n) = xm.

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. ab ⇒ a
  2. b2 ⇒ b
  3. a4 ⇒ a
  4. ba3 ⇒ b 
# ab:ab=a,baaa=b ab
ab=a
bb=b
aaaa=a
baaa=b

Cayley table

Idempotents are shown in bold.

1aba2baa3ba2
11aba2baa3ba2
aaa2aa3a2aa3
bbbabba2babba2
a2a2a3a2aa3a2a
bababa2babba2bab
a3a3aa3a2aa3a2
ba2ba2bba2babba2ba

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

22 unique, 1157 total

Σ#PresentationDescriptionRelated
7116⟨a, b | aaa=b, abb=1⟩Isomorphic to ℤ7763 iso
8577⟨a, b | aaa=b, abb=a⟩Isomorphic to ℕ(7 = 1)59 iso
8578⟨a, b | aaa=b, abb=b⟩Isomorphic to ℕ(7 = 3)44 iso
8913⟨a, b | aa=b, abbb=b⟩Isomorphic to ℕ(7 = 2)82 iso
91593⟨a, b | aaa=ab, baa=b⟩Finite non-commutative monoid with 7 elements1 iso, 2 anti-iso
91601⟨a, b | aaa=bb, aab=b⟩Finite commutative monoid with 7 elements5 iso
91620⟨a, b | aab=aa, bbb=a⟩Isomorphic to ℕ(7 = 6)35 iso
91632⟨a, b | aab=ab, bbb=a⟩Isomorphic to ℕ(7 = 4)50 iso
92074⟨a, b | aab=b, babb=a⟩Finite commutative monoid with 7 elements20 iso
92231⟨a, b | ab=aa, aaa=bb⟩Finite non-commutative monoid with 7 elements3 iso
92271⟨a, b | bb=aa, aaa=ab⟩Finite non-commutative monoid with 7 elements1 iso, 2 anti-iso
93197⟨a, b | ab=a, bbb=baa⟩Finite non-commutative monoid with 7 elements5 iso
104162⟨a, b | abb=aab, bbb=a⟩Isomorphic to ℕ(7 = 5)42 iso
106251⟨a, b | aaa=a, aabba=b⟩Finite non-commutative monoid with 7 elements1 iso
108987⟨a, b | ab=a, aaaaa=bb⟩Finite commutative monoid with 7 elements5 iso
109108⟨a, b | ab=a, bbbbb=aa⟩Finite commutative monoid with 7 elements2 iso
109110⟨a, b | ab=a, bbbbb=ba⟩Finite non-commutative monoid with 7 elements1 iso
109263⟨a, b | ab=a, aaaa=bbb⟩Finite commutative monoid with 7 elements4 iso
109375⟨a, b | ab=a, bbba=bbb⟩Finite non-commutative monoid with 7 elements3 iso
109376⟨a, b | ab=a, bbbb=aaa⟩Finite commutative monoid with 7 elements3 iso
109382⟨a, b | ab=a, bbbb=bba⟩Finite non-commutative monoid with 7 elements1 iso
1114843⟨a, b | abba=b, aabab=a⟩Finite non-commutative monoid with 7 elements1 iso

Other isomorphic instances

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

54 total

Σ#PresentationMapping
92037⟨a, b | aab=a, baaa=b⟩φ(a) = baa, φ(b) = a
92971⟨a, b | ab=a, baaab=b⟩φ(a) = a, φ(b) = b
92973⟨a, b | ab=a, baaba=b⟩φ(a) = a, φ(b) = b
92977⟨a, b | ab=a, babaa=b⟩φ(a) = a, φ(b) = b
92985⟨a, b | ab=a, bbaaa=b⟩φ(a) = a, φ(b) = b
104662⟨a, b | aaab=a, baaa=b⟩φ(a) = ba, φ(b) = a
104668⟨a, b | aaab=a, babb=b⟩φ(a) = ba, φ(b) = a
104672⟨a, b | aaab=a, bbab=b⟩φ(a) = ba, φ(b) = a
104674⟨a, b | aaab=a, bbba=b⟩φ(a) = ba, φ(b) = a
104736⟨a, b | aabb=a, baab=b⟩φ(a) = ba, φ(b) = a
104738⟨a, b | aabb=a, baba=b⟩φ(a) = ba, φ(b) = a
104740⟨a, b | aabb=a, bbaa=b⟩φ(a) = ba, φ(b) = a
104744⟨a, b | abab=a, baab=b⟩φ(a) = ba, φ(b) = a
104746⟨a, b | abab=a, baba=b⟩φ(a) = ba, φ(b) = a
108783⟨a, b | ab=a, baaabb=b⟩φ(a) = a, φ(b) = b
108787⟨a, b | ab=a, baabab=b⟩φ(a) = a, φ(b) = b
108789⟨a, b | ab=a, baabba=b⟩φ(a) = a, φ(b) = b
108795⟨a, b | ab=a, babaab=b⟩φ(a) = a, φ(b) = b
108797⟨a, b | ab=a, bababa=b⟩φ(a) = a, φ(b) = b
108801⟨a, b | ab=a, babbaa=b⟩φ(a) = a, φ(b) = b
108811⟨a, b | ab=a, bbaaab=b⟩φ(a) = a, φ(b) = b
108813⟨a, b | ab=a, bbaaba=b⟩φ(a) = a, φ(b) = b
108817⟨a, b | ab=a, bbabaa=b⟩φ(a) = a, φ(b) = b
108825⟨a, b | ab=a, bbbaaa=b⟩φ(a) = a, φ(b) = b
1118899⟨a, b | aab=a, baaaab=b⟩φ(a) = baa, φ(b) = a
1118901⟨a, b | aab=a, baaaba=b⟩φ(a) = baa, φ(b) = a
1118905⟨a, b | aab=a, baabaa=b⟩φ(a) = baa, φ(b) = a
1118913⟨a, b | aab=a, babaaa=b⟩φ(a) = baa, φ(b) = a
1118927⟨a, b | aab=a, babbbb=b⟩φ(a) = baa, φ(b) = a
1118929⟨a, b | aab=a, bbaaaa=b⟩φ(a) = baa, φ(b) = a
1118943⟨a, b | aab=a, bbabbb=b⟩φ(a) = baa, φ(b) = a
1118951⟨a, b | aab=a, bbbabb=b⟩φ(a) = baa, φ(b) = a
1118955⟨a, b | aab=a, bbbbab=b⟩φ(a) = baa, φ(b) = a
1118957⟨a, b | aab=a, bbbbba=b⟩φ(a) = baa, φ(b) = a
1124475⟨a, b | ab=a, baaabbb=b⟩φ(a) = a, φ(b) = b
1124483⟨a, b | ab=a, baababb=b⟩φ(a) = a, φ(b) = b
1124487⟨a, b | ab=a, baabbab=b⟩φ(a) = a, φ(b) = b
1124489⟨a, b | ab=a, baabbba=b⟩φ(a) = a, φ(b) = b
1124499⟨a, b | ab=a, babaabb=b⟩φ(a) = a, φ(b) = b
1124503⟨a, b | ab=a, bababab=b⟩φ(a) = a, φ(b) = b
1124505⟨a, b | ab=a, bababba=b⟩φ(a) = a, φ(b) = b
1124511⟨a, b | ab=a, babbaab=b⟩φ(a) = a, φ(b) = b
1124513⟨a, b | ab=a, babbaba=b⟩φ(a) = a, φ(b) = b
1124517⟨a, b | ab=a, babbbaa=b⟩φ(a) = a, φ(b) = b
1124531⟨a, b | ab=a, bbaaabb=b⟩φ(a) = a, φ(b) = b
1124535⟨a, b | ab=a, bbaabab=b⟩φ(a) = a, φ(b) = b
1124537⟨a, b | ab=a, bbaabba=b⟩φ(a) = a, φ(b) = b
1124543⟨a, b | ab=a, bbabaab=b⟩φ(a) = a, φ(b) = b
1124545⟨a, b | ab=a, bbababa=b⟩φ(a) = a, φ(b) = b
1124549⟨a, b | ab=a, bbabbaa=b⟩φ(a) = a, φ(b) = b
1124559⟨a, b | ab=a, bbbaaab=b⟩φ(a) = a, φ(b) = b
1124561⟨a, b | ab=a, bbbaaba=b⟩φ(a) = a, φ(b) = b
1124565⟨a, b | ab=a, bbbabaa=b⟩φ(a) = a, φ(b) = b
1124573⟨a, b | ab=a, bbbbaaa=b⟩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.

12 total

Σ#PresentationMapping
92087⟨a, b | aba=a, aaab=b⟩φ(a) = baa, φ(b) = a
104675⟨a, b | aaab=b, aaba=a⟩φ(a) = ba, φ(b) = a
104679⟨a, b | aaab=b, abaa=a⟩φ(a) = ba, φ(b) = a
1114477⟨a, b | aaab=b, aaaba=a⟩φ(a) = a, φ(b) = b
1114481⟨a, b | aaab=b, aabaa=a⟩φ(a) = a, φ(b) = b
1114489⟨a, b | aaab=b, abaaa=a⟩φ(a) = a, φ(b) = b
1114505⟨a, b | aaab=b, baaaa=a⟩φ(a) = a, φ(b) = b
1119095⟨a, b | aba=a, aaaabb=b⟩φ(a) = baa, φ(b) = a
1119099⟨a, b | aba=a, aaabab=b⟩φ(a) = baa, φ(b) = a
1119105⟨a, b | aba=a, aabaab=b⟩φ(a) = baa, φ(b) = a
1119119⟨a, b | aba=a, abaaab=b⟩φ(a) = baa, φ(b) = a
1119139⟨a, b | aba=a, abbbbb=b⟩φ(a) = baa, φ(b) = a