Cometic functors and representing order-preserving maps by principal lattice congruences

Abstract

Let $$\mathbf {Lat}^{\mathrm{sd}}_{5}$$ and $$\mathbf {Pos}_{01}^{{\scriptscriptstyle \mathrm{{+}}}}$$ denote the category of selfdual bounded lattices of length 5 with $$\{0,1\}$$ -preserving lattice homomorphisms and that of bounded ordered sets with $$\{0,1\}$$ -preserving isotone maps, respectively. For an object L in $$\mathbf {Lat}^{\mathrm{sd}}_{5}$$ , the ordered set of principal congruences of the lattice L is denoted by $$\mathrm{Princ}(L)$$ . By means of congruence generation, $$\mathrm{Princ}:\mathbf {Lat}^{\mathrm{sd}}_{5}\rightarrow \mathbf {Pos}_{01}^{{\scriptscriptstyle \mathrm{{+}}}}$$ is a functor. We prove that if $$\mathbf {A}$$ is a small subcategory of $$\mathbf {Pos}_{01}^{{\scriptscriptstyle \mathrm{{+}}}}$$ such that every morphism of $$\mathbf {A}$$ is a monomorphism, understood in $$\mathbf {A}$$ , then $$\mathbf {A}$$ is the $$\mathrm{Princ}$$ -image of an appropriate subcategory of $$\mathbf {Lat}^{\mathrm{sd}}_{5}$$ . This result extends G. Gratzer’s earlier theorems where $$\mathbf {A}$$ consisted of one or two objects and at most one non-identity morphism, and the author’s earlier result where all morphisms of $$\mathbf {A}$$ were 0-separating and no hom-set had more the two morphisms. Furthermore, as an auxiliary tool, we derive some families of maps, also known as functions, from injective maps and surjective maps; this can be useful in various fields of mathematics, not only in lattice theory. Namely, for every small concrete category $$\mathbf {A}$$ , we define a functor $${F_{{\scriptscriptstyle \mathrm{{com}}}}}$$ , called cometic functor, from $$\mathbf {A}$$ to the category $$\mathbf Set $$ of sets and a natural transformation $${{\varvec{\pi }}^{{\scriptscriptstyle \mathrm{{com}}}}}$$ , called cometic projection, from $${F_{{\scriptscriptstyle \mathrm{{com}}}}}$$ to the forgetful functor of $$\mathbf {A}$$ into $$\mathbf Set $$ such that the $${F_{{\scriptscriptstyle \mathrm{{com}}}}}$$ -image of every monomorphism of $$\mathbf {A}$$ is an injective map and the components of $${{\varvec{\pi }}^{{\scriptscriptstyle \mathrm{{com}}}}}$$ are surjective maps.