Limit properties in the metric semi-linear space of picture fuzzy numbers

The picture fuzzy set (PFS) just appeared in 2014 and was introduced by Cuong, which is a generalization of intuitionistic fuzzy sets (Atanassov in Fuzzy Sets Syst 20(1):87-96, 1986) and fuzzy sets (Zadeh Inf Control 8(3):338-353, 1965). The picture fuzzy number (PFN) is an ordered value triple, including a membership degree, a neutral-membership degree, a non-membership degree, of a PFS. The PFN is a useful tool to study the problems that have uncertain information in real life. In this paper, the main aim is to develop basic foundations that can become tools for future research related to PFN and picture fuzzy calculus. We first establish a semi-linear space for PFNs by providing two new definitions of two basic operations, addition and scalar multiplication, such that the set of PFNs together with these two operations can form a semi-linear space. Moreover, we also provide some important properties and concepts such as metrics, order relations between two PFNs, geometric difference, multiplication of two PFNs. Next, we introduce picture fuzzy functions with a real domain that is also known as picture fuzzy functions with time-varying values, called geometric picture fuzzy function (GPFFs). In this framework, we give definitions about the limit of GPFFs and sequences of PFN. The important limit properties are also presented in detail. Finally, we prove that the metric semi-linear space of PFNs is complete, which is an important property in the classical mathematical analysis.