Implications, a sort of crucial fuzzy logical connectives, perform considerable actions in both the theory and applications concerning fuzzy sets and systems. Meanwhile, cause of the comprehensive applications in fuzzy systems, fuzzy control, fuzzy reasoning, etc., the research on the construction methods of implications has become an interesting topic at the theoretical level. This paper focuses the extension construction of implications on function space \(P^X\) consisted with any nonempty set X and bounded poset P. To begin with, it proposes a method to extend implications on P to \(P^X\) , which supply a unified construction approach to obtain the commonly used implications (see, e.g., residual implications, S-implications, QL-implications, etc.) on \(P^X\) via a family of known ones on P. Secondly, it presents notion of representable implications on \(P^X\) and shows equivalent characterizations of them. Thirdly, it proves that the provided extension construction method of implications on \(P^X\) maintains almost all of common properties owned by implications on P. As consequence, the situations of implications on \(P^X\) consisting of type-2 fuzzy sets and interval-valued fuzzy sets are contained.
