#159 ⟨a, b | ab=aa, bb=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. b4 ⇒ b3
  2. a ⇒ b2 
# ab:ab=aa,bb=a b/a
bbbb=bbb
a=bb

Staircase diagram

Cayley table

Idempotents are shown in bold.

1bb2b3
11bb2b3
bbb2b3b3
b2b2b3b3b3
b3b3b3b3b3

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

8 unique, 1591 total

Σ#PresentationDescriptionRelated
57⟨a, b | aa=b, bb=1⟩Isomorphic to ℤ41419 iso
657⟨a, b | aa=a, bb=a⟩Isomorphic to ℕ(4 = 2)37 iso
661⟨a, b | aa=b, bb=a⟩Isomorphic to ℕ(4 = 1)72 iso
7158⟨a, b | ab=aa, ba=b⟩Finite non-commutative monoid with 4 elements8 iso, 6 anti-iso
7242⟨a, b | aa=a, abb=b⟩Finite non-commutative monoid with 4 elements14 iso
7280⟨a, b | ab=a, bb=aa⟩Finite commutative monoid with 4 elements17 iso
92881⟨a, b | aa=a, abbba=b⟩Finite commutative monoid with 4 elements8 iso
105033⟨a, b | aaa=aa, abba=b⟩Finite commutative monoid with 4 elements2 iso

Other isomorphic instances

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

16 total

Σ#PresentationMapping
7273⟨a, b | ab=a, bbb=a⟩φ(a) = bbb, φ(b) = b
8974⟨a, b | aa=b, aaa=bb⟩φ(a) = b, φ(b) = bb
8976⟨a, b | aa=b, aab=ab⟩φ(a) = b, φ(b) = bb
8977⟨a, b | aa=b, aab=ba⟩φ(a) = b, φ(b) = bb
8980⟨a, b | aa=b, aba=ab⟩φ(a) = b, φ(b) = bb
81021⟨a, b | ab=a, bbb=ab⟩φ(a) = bbb, φ(b) = b
92001⟨a, b | aaa=b, aaaa=b⟩φ(a) = b, φ(b) = bbb
93037⟨a, b | aa=b, aaaa=ab⟩φ(a) = b, φ(b) = bb
93154⟨a, b | aa=b, aab=aaa⟩φ(a) = b, φ(b) = bb
93155⟨a, b | aa=b, aba=aaa⟩φ(a) = b, φ(b) = bb
93196⟨a, b | ab=a, bbb=abb⟩φ(a) = bbb, φ(b) = b
109184⟨a, b | aa=b, aaaa=aaa⟩φ(a) = b, φ(b) = bb
109319⟨a, b | ab=a, abbb=bbb⟩φ(a) = bbb, φ(b) = b
1119848⟨a, b | aaa=b, aaaa=aaa⟩φ(a) = b, φ(b) = bbb
1119983⟨a, b | aab=a, abbb=bbb⟩φ(a) = bbb, φ(b) = b
1125531⟨a, b | ab=a, abbbb=bbb⟩φ(a) = bbb, φ(b) = b