Associative operations on the poset of closed intervals of a given lower-bounded poset

Under the inclusion order, our focus lies in exploring associative binary operations on the poset of closed intervals of a given lower-bounded poset. Initially, we propose a method for extending an increasing and associative binary operation on a given lower-bounded poset to the poset of closed intervals of that given lower-bounded poset. It involves the use of an interior operator between that given lower-bounded poset and one of its subposets, alongside another appropriately chosen increasing and associative binary operation on the same lower-bounded poset. Subsequently, we explore conditions under which this extended operation can facilitate the constructions of t-conorms and t-norms on the poset of closed intervals of a given bounded poset. Lastly, we delve into the properties of the constructed t-conorms and t-norms, specifically addressing the Archimedean property and the existence of idempotent and nilpotent elements.