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

José Morgado introduced in 1962 an original and interesting notion of hyperlattices, that he called reticuloides. In his Master’s thesis defended in 1971 (and supervised by Newton da Costa), Antonio M. Sette proposed a novel notion of implicative hyperlattices (here called SIHLs) based on Morgado’s hyperlattices. He also extended SIHLs by adding an unary hyperoperator, obtaining a class of hyperalgebras (here called SHCωs) which correspond to da Costa algebras for Cω, being so a suitable semantics for the logic Cω. In this paper, after characterizing Sette’s implicative hyperlattices in (hyper)lattice-theoretic terms, and proving some basic results on SIHLs, we introduce a class of swap structures, a special class of hyperalgebras over the signature of Cω naturally induced by implicative lattices. It is proven that these swap structures are indeed SHCωs. Finally, it is proven that the class of SHCωs, as well as the above mentioned class of swap structures, characterize the logic Cω.