delooping of the double bicyclic monoid

 algebra
Notation BDBD Objects a single object ∗* Morphisms the elements of the monoid D≔⟨f,g,s:fs=gs=1⟩D \coloneqq \langle f,g,s : fs = gs = 1 \rangle Related BCBC, BNB\IN
Every monoid MM induces a category BMBM with a single object ∗*, morphisms given by the elements of MM, and composition given by the monoid operation. Some of the properties of this category depend on the specific monoid.In this entry, we take the monoid DD freely generated by f,g,sf,g,s subject to the relations fs=gs=1fs = gs = 1, which we call the double bicyclic monoid. It contains two copies of the bicyclic monoid, generated by f,sf,s and by g,sg,s. Every element of DD can be uniquely written as snws^n w for n≥0n \geq 0 and a word ww in f,gf,g.This category provides an example of a core-connected category that does not have reflexive coequalizers.

Satisfied properties

Assigned properties

Deduced properties

Unsatisfied properties

Assigned properties

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

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Special objects

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Special morphisms