@article { author = {Estaji, A.A. and Karimi Feizabad, A. and Zarghani, M.}, title = {Zero elements and $z$-ideals in modified pointfree topology}, journal = {Bulletin of the Iranian Mathematical Society}, volume = {43}, number = {7}, pages = {2205-2226}, year = {2017}, publisher = {Iranian Mathematical Society (IMS)}, issn = {1017-060X}, eissn = {1735-8515}, doi = {}, abstract = {‎In this paper‎, ‎we define and study the notion of zero elements in topoframes; a topoframe is a pair‎ ‎$(L‎, ‎\tau)$‎, ‎abbreviated $L_{ \tau}$‎, ‎consisting of a frame $L$ and a‎ ‎subframe $ \tau $ all of whose elements are complemented elements in‎ ‎$L$‎. ‎We show that‎ ‎the $f$-ring $ \mathcal{R}(L_\tau)$‎, ‎the set of $\tau$-real continuous functions on $L$‎, ‎is uniformly complete‎. ‎Also‎, ‎the set of all zero elements in a topoframe‎ ‎is closed under the formation of countable meets and finite joins‎. ‎Also‎, ‎we introduce the notion of $z$-filters and $z$-ideals in modified pointfree topology‎ ‎and make ready some results about them‎.  }, keywords = {Topoframe‎,‎zero element‎,‎$z$-filter‎,‎$z$-ideal‎,‎prime ideal}, url = {http://bims.iranjournals.ir/article_1090.html}, eprint = {http://bims.iranjournals.ir/article_1090_17ec79f62279ac897d5fe7ebab2a0ff0.pdf} }