On some categories of structured sets
Giampiero Chiaselotti, Tommaso Gentile, Federico G. Infusino · European Journal of Mathematics · 2024
Abstract Given an arbitrary set $$\Omega $$ Ω , we consider the collections $$\textrm{SS}\hspace{0.55542pt}(\Omega )$$ SS ( Ω ) , $$\textrm{SR}\hspace{0.55542pt}(\Omega )$$ SR ( Ω ) and $$\textrm{SO}\hspace{0.55542pt}(\Omega )$$ SO ( Ω ) of all the set systems, the binary set relations and the set operators on $$\Omega $$ Ω . We introduce the notion of linking map on $$\Omega $$ Ω as any map whose domain and codomain may be chosen between the above collections. After providing a descriptive overview useful for framing the notion of linking map in a broad non-specialized context, we explain how linking maps occur in a very natural way in two specific results. The first of these results concerns the classic identification between the subfamily $$\textrm{EQ}\hspace{0.55542pt}(\Omega )$$ EQ ( Ω ) of all the equivalence relations on $$\Omega $$ Ω and the subfamily $$\textrm{SP}\hspace{0.55542pt}(\Omega )$$ SP ( Ω ) of all the set partitions on $$\Omega $$ Ω . Starting from it, we introduce a new subfamily $$\textrm{ESO}\hspace{0.55542pt}(\Omega )$$ ESO ( Ω ) of closure operators on $$\Omega $$ Ω and four linking maps whose restrictions to the subfamilies $$\textrm{EQ}\hspace{0.55542pt}(\Omega )$$ EQ ( Ω ) , $$\textrm{SP}\hspace{0.55542pt}(\Omega )$$ SP ( Ω ) and $$\textrm{ESO}\hspace{0.55542pt}(\Omega )$$ ESO ( Ω ) are bijections. The second result concerns the identification between the subfamily $$\textrm{CSO}\hspace{0.55542pt}(\Omega )$$ CSO ( Ω ) of all closure set operators on $$\Omega $$ Ω and the subfamily $$\textrm{CSS}\hspace{0.55542pt}(\Omega )$$ CSS ( Ω ) of all closure set systems on $$\Omega $$ Ω . Starting from it, we introduce a new subfamily $$\textrm{DSR}\hspace{0.55542pt}(\Omega )$$ DSR ( Ω ) of binary set relations and four linking maps whose restrictions to the subfamilies $$\textrm{CSO}\hspace{0.55542pt}(\Omega )$$ CSO ( Ω ) , $$\textrm{CSS}\hspace{0.55542pt}(\Omega )$$ CSS ( Ω )