Universal Approach to Z–frame Envelopes of Semilattices

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.