Quasi-projective posets, lattices, permutations, graphs, digraphs, hypergraphs, point-line geometries

TL;DR

This paper characterizes quasi-projective structures across various mathematical domains, including posets, lattices, graphs, hypergraphs, and geometries, providing a comprehensive understanding of their properties and classifications.

Contribution

It offers the first complete characterizations of quasi-projective posets, lattices, permutations, graphs, hypergraphs, and geometries of arbitrary sizes.

Findings

Characterization of quasi-projective posets and lattices of any size.

Classification of quasi-projective finite permutations, graphs, and digraphs.

Analysis of quasi-projective hypergraphs and point-line geometries.

Abstract

A structure $\cal S$ is quasi-projective if for every structure $\cal T$, for every homomorphism $f : {\cal S} \rightarrow {\cal T}$ and every epimorphism $j: {\cal S}\rightarrow {\cal T}$ there is an endomorphism $\phi$ of $\cal S$ such that $\phi\circ j=f$. In this paper, we characterise the quasi-projective posets and lattices of arbitrary cardinalities, finite permutations, graphs and digraphs of arbitrary cardinalities with loops and without loops, finite hypergraphs, and finite point-line geometries.