Constructions of quasi-overlap functions and their generalized forms on bounded partially ordered sets

Abstract

Recently, Paiva et al. introduced the concept of quasi-overlap functions on bounded lattices and investigated some vital properties of them. In this paper, we continue consider this research topic and focus on the constructions of quasi-overlap functions along with their generalized forms on bounded partially ordered sets. To be specific, firstly, we generalize the truth values set of quasi-overlap functions from bounded lattices to bounded partially ordered sets and introduce the notions of 0P-quasi-overlap functions, 1P-quasi-overlap functions and 0P,1P-quasi-overlap functions on any bounded partially ordered set P by considering the weaker boundary conditions than the quasi-overlap functions on P. Secondly, we give the constructions of quasi-overlap functions, 0P-quasi-overlap functions, 1P-quasi-overlap functions and 0P,1P-quasi-overlap functions on any bounded partially ordered set P via the so-called Galois s-connections and 0,1-homomorphisms, 1-homomorphisms, 0-homomorphisms and ord-homomorphisms, respectively. In particular, we prove that those constructions contain the methods of extending the known quasi-overlap functions, 0P-quasi-overlap functions, 1P-quasi-overlap functions and 0P,1P-quasi-overlap functions from any bounded partially ordered set P to any other bounded partially ordered sets. Finally, we show that those extensions maintain some basic properties of the known quasi-overlap functions, 0P-quasi-overlap functions, 1P-quasi-overlap functions and 0P,1P-quasi-overlap functions on P, such as, idempotent, Archimedean property and cancellation law.