Fuzzy implications and coimplications on the poset of closed intervals

In this paper, under the inclusion order, we focus on the investigations of fuzzy implications and coimplications on the poset of closed intervals of a given bounded poset. We first propose a method for constructing a fuzzy implication on the poset of closed intervals of a given bounded poset through the use of a fuzzy implication and a fuzzy coimplication on that given bounded poset. This process of constructing fuzzy implications is an extension of the fuzzy implication on that given bounded poset. We then explore some properties of this extended fuzzy implication and discuss about which properties of fuzzy implications are preserved in this extension. Moreover, we would like to apply the method of extending fuzzy implications for fuzzy coimplications. Although we present two methods for constructing fuzzy coimplications on the poset of closed intervals of a given bounded poset, these two methods are not the extension of the fuzzy coimplication on that given bounded poset. We also provide some illustrative examples.