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Free and cofree acts of dcpo-monoids on directed complete posets. (English) Zbl 1344.06013

Let \(S\) be a directed complete pomonoid (dcpo). A dcpo which has a bottom element is said to be a cpo. A dcpo \(S\) is said to be good if for every directed subset \(D\) of \(S\), the directed join \(\bigvee^dD\in D\). Let \(\mathbf{Dcpo}\)-\(S\) (\(\mathbf{Cpo}\)-\(S\)) be the category of all directed complete posets (with bottom element) equipped with a compatible right action of a dcpo-monoid (cpo-monoid, respectively) \(S\).
Free objects and a number of specific cases in \(\mathbf{Dcpo}\)-\(S\) and \(\mathbf{Cpo}\)-\(S\) are described. For example, if \(S\) is a good dcpo-monoid, then for a given \(S\)-poset \(A\), the free \(S\)-dcpo on \(A\) is the dcpo \(\mathrm{Id}(A)\) (the collection of all ideals of \(A\)) with the action \(\lambda\colon\mathrm{Id}(A)\times S\to\mathrm{Id}(A)\), given by \((I,s)\mapsto{\downarrow\!(Is)}\). Cofree objects in \(\mathbf{Dcpo}\)-\(S\) are described and proved that if \(P\) is a nontrivial poset with nonidentity order, which is also a dcpo, then the cofree dcpo over \(P\) does not exist. The results allow to decide existence of left and right adjoints of forgetful functors between the relevant categories \(\mathbf{Cpo}\)-\(S\), \(\mathbf{Dcpo}\)-\(S\), \(\mathbf{Pos}\)-\(S\), \(\mathbf{Act}\)-\(S\) and \(\mathbf{Cpo}\), \(\mathbf{Dcpo}\), \(\mathbf{Pos}\), \(\mathbf{Set}\).

MSC:

06F05 Ordered semigroups and monoids
20M50 Connections of semigroups with homological algebra and category theory
20M30 Representation of semigroups; actions of semigroups on sets
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
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