On Morgado and Sette’s Implicative Hyperlattices as Models of da Costa Logic Cω

José Morgado introduced in 1962 a novel notion of hyperlattices, which he called reticuloides. In his master’s thesis submitted in 1971 (under the supervision of Newton da Costa), Antonio M. Sette introduced a new class of implicative hyperlattices (here called SIHLs) based on Morgado’s hyperlattices. He also extended SIHLs by adding a unary hyperoperator, thus defining a class of hyperalgebras (denoted SHCω) corresponding to da Costa algebras for Cω, thereby providing suitable semantics for the logic Cω. In this paper, after providing a (hyper)lattice-theoretic characterization of Sette’s implicative hyperlattices and proving some basic results on SIHLs, we introduce a class of swap structures—special hyperalgebras over the signature of Cω that arise naturally from implicative lattices. We prove that these swap structures are indeed SHCω. Finally, we demonstrate that the class SHCω, as well as the aforementioned swap structures, characterizes the logic Cω.