#1620 ⟨a, b | aab=aa, bbb=a⟩

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Quick links

  1. Properties
  2. Elements
  3. Staircase diagram
  4. Cayley table
  5. Right Cayley graph
  6. Rewriting system
  7. Same cardinality
  8. Isomorphic instances

Properties

Elements

The zero element z satisfies zy = yz = z for all y.
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.

Staircase diagram

Cayley table

Idempotents are shown in bold.

1bb2b3b4b5b6
11bb2b3b4b5b6
bbb2b3b4b5b6b6
b2b2b3b4b5b6b6b6
b3b3b4b5b6b6b6b6
b4b4b5b6b6b6b6b6
b5b5b6b6b6b6b6b6
b6b6b6b6b6b6b6b6

Right Cayley graph

Idempotents are shown in bold.

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
#RuleProof
1. b7 ⇒ b6 [4]
2. a ⇒ b3 [2]
# ab:aab=aa,bbb=a b/a
bbbbbbb=bbbbbb
a=bbb

Same cardinality

22 unique, 1188 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
8937⟨a, b | ab=a, baaa=b⟩Finite non-commutative monoid with 7 elements54 iso, 12 anti-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
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

Isomorphic instances

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

35 total

Σ#PresentationMapping
91664⟨a, b | aba=aa, bbb=a⟩φ(a) = bbb, φ(b) = b
108838⟨a, b | ab=a, bbbbbb=a⟩φ(a) = bbbbbb, φ(b) = b
109235⟨a, b | aa=b, abbb=bbb⟩φ(a) = b, φ(b) = bb
109249⟨a, b | aa=b, babb=bbb⟩φ(a) = b, φ(b) = bb
1115610⟨a, b | aab=ab, aaaaa=b⟩φ(a) = b, φ(b) = bbbbb
1115674⟨a, b | aab=ba, aaaaa=b⟩φ(a) = b, φ(b) = bbbbb
1115842⟨a, b | aba=ab, aaaaa=b⟩φ(a) = b, φ(b) = bbbbb
1119310⟨a, b | aaa=b, aaaab=bb⟩φ(a) = b, φ(b) = bbb
1119314⟨a, b | aaa=b, aaaba=bb⟩φ(a) = b, φ(b) = bbb
1119321⟨a, b | aaa=b, aabaa=bb⟩φ(a) = b, φ(b) = bbb
1119857⟨a, b | aaa=b, aaab=abb⟩φ(a) = b, φ(b) = bbb
1119859⟨a, b | aaa=b, aaab=bab⟩φ(a) = b, φ(b) = bbb
1119860⟨a, b | aaa=b, aaab=bba⟩φ(a) = b, φ(b) = bbb
1119865⟨a, b | aaa=b, aaba=abb⟩φ(a) = b, φ(b) = bbb
1119867⟨a, b | aaa=b, aaba=bab⟩φ(a) = b, φ(b) = bbb
1119868⟨a, b | aaa=b, aaba=bba⟩φ(a) = b, φ(b) = bbb
1125113⟨a, b | ab=a, bbbbbb=ab⟩φ(a) = bbbbbb, φ(b) = b
1125289⟨a, b | aa=b, aaabb=bbb⟩φ(a) = b, φ(b) = bb
1125303⟨a, b | aa=b, aabab=bbb⟩φ(a) = b, φ(b) = bb
1125311⟨a, b | aa=b, aabba=bbb⟩φ(a) = b, φ(b) = bb
1125327⟨a, b | aa=b, abaab=bbb⟩φ(a) = b, φ(b) = bb
1125333⟨a, b | aa=b, ababa=bbb⟩φ(a) = b, φ(b) = bb
1125369⟨a, b | aa=b, baaab=bbb⟩φ(a) = b, φ(b) = bb
1125748⟨a, b | aa=b, abbb=aabb⟩φ(a) = b, φ(b) = bb
1125750⟨a, b | aa=b, abbb=abab⟩φ(a) = b, φ(b) = bb
1125751⟨a, b | aa=b, abbb=abba⟩φ(a) = b, φ(b) = bb
1125762⟨a, b | aa=b, baab=abbb⟩φ(a) = b, φ(b) = bb
1125765⟨a, b | aa=b, baba=abbb⟩φ(a) = b, φ(b) = bb
1125769⟨a, b | aa=b, babb=aabb⟩φ(a) = b, φ(b) = bb
1125771⟨a, b | aa=b, babb=abab⟩φ(a) = b, φ(b) = bb
1125772⟨a, b | aa=b, babb=abba⟩φ(a) = b, φ(b) = bb
1125775⟨a, b | aa=b, babb=baab⟩φ(a) = b, φ(b) = bb
1125776⟨a, b | aa=b, babb=baba⟩φ(a) = b, φ(b) = bb
1125778⟨a, b | aa=b, bbaa=abbb⟩φ(a) = b, φ(b) = bb
1125779⟨a, b | aa=b, bbaa=babb⟩φ(a) = b, φ(b) = bb