An inverse Fra\"iss\'e limit for finite posets and duality for posets and lattices

TL;DR

This paper constructs inverse limits of sequences of finite posets and lattices using Fra"iss"e theory, revealing dualities and structural properties of these inverse limits within category theory.

Contribution

It introduces a novel inverse Fra"iss"e limit framework for finite posets and lattices, establishing duality relations and analyzing their inverse limits.

Findings

Constructed a Fra"iss"e sequence in the category of finite posets with quotient maps.

Established properties of the inverse limit of these sequences.

Explored duality between posets and lattices via inverse limits.

Abstract

We consider a category of all finite partial orderings with quotient maps as arrows and construct a Fra\"iss\'e sequence in this category. Then we use commonly known relations between partial orders and lattices to construct a sequence of lattices associated with it. Each of these two sequences has a limit object -- an inverse limit, which is an object of our interest as well. In the first chapter there are some preliminaries considering partial orders, lattices, topology, inverse limits, category theory and Fra\"iss\'e theory, which are used later. In the second chapter there are our results considering a Fra\"iss\'e sequence in category of finite posets with quotient maps and properties of inverse limit of this sequence. In the third chapter we investigate connections between posets and order ideals corresponding to them, getting an inductive sequence made of these ideals; then we study properties of the inverse limit of this sequence.