TY - JOUR ID - 1090 TI - Zero elements and $z$-ideals in modified pointfree topology JO - Bulletin of the Iranian Mathematical Society JA - BIMS LA - en SN - 1017-060X AU - Estaji, A.A. AU - Karimi Feizabad, A. AU - Zarghani, M. AD - Faculty of Mathematics and Computer Sciences‎, ‎Hakim Sabzevari University‎, ‎P.O‎. ‎Box 397‎, ‎Sabzevar‎, ‎Iran. AD - Department of Mathematics‎, ‎Gorgan Branch‎, ‎Islamic Azad University‎, ‎Gorgan‎, ‎Iran. AD - Faculty of Mathematics and Computer Sciences‎, ‎Hakim Sabzevari University‎, ‎P.O‎. ‎Box 397‎, ‎Sabzevar‎, ‎Iran. Y1 - 2017 PY - 2017 VL - 43 IS - 7 SP - 2205 EP - 2226 KW - Topoframe‎ KW - ‎zero element‎ KW - ‎$z$-filter‎ KW - ‎$z$-ideal‎ KW - ‎prime ideal DO - N2 - ‎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.   UR - http://bims.iranjournals.ir/article_1090.html L1 - http://bims.iranjournals.ir/article_1090_17ec79f62279ac897d5fe7ebab2a0ff0.pdf ER -