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. Grätzer’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.