How can we refine the understanding of continuous lattices in computer science? This paper explores continuous lattices using maps that preserve all suprema, rather than just directed ones. It shows how that turns out to be *-autonomous and maximal. The study presents the theory with extreme detail. By introducing FS-lattices, the study delves into the complexities of distributivity and algebraicity within these structures. FS-lattices are studied in the presence of distributivity and algebraicity. These investigations reveal a rich tapestry of connections to classical Domain Theory, complete distributivity, and Topology. Ultimately, the paper establishes links to models of Linear Logic, furthering the theoretical foundations of computer science and paving the way for new computational paradigms.
Appearing in Mathematical Structures in Computer Science, this work directly contributes to the journal's exploration of mathematical foundations relevant to computing. By examining linear types and their relationship to continuous lattices, it adds to the journal's discussion of formal structures underpinning computation.