Abstract
In this paper we deal with systems of sets on the category of all semilattices. Based on the result of Z–frame freely generated by Z–sites studied in [D. Zhao, “Generalization of Frames and Continuous Lattices,” Ph.D. thesis, Cambridge University, 1993], we prove that every semilattice admits a Z–frame envelope. It is a universal approach for any system Z of sets. We define and study a new continuity named Za–continuty on semilattice and prove that in some specific systems of sets (Low, Fin and Idl), every Za–continuous semilattice admits a Z–continuous envelope.
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