# Zero elements and $z$-ideals in modified pointfree topology

Document Type: Research Paper

Authors

1 Faculty of Mathematics and Computer Sciences‎, ‎Hakim Sabzevari University‎, ‎P.O‎. ‎Box 397‎, ‎Sabzevar‎, ‎Iran.

2 Department of Mathematics‎, ‎Gorgan Branch‎, ‎Islamic Azad University‎, ‎Gorgan‎, ‎Iran.

3 Faculty of Mathematics and Computer Sciences‎, ‎Hakim Sabzevari University‎, ‎P.O‎. ‎Box 397‎, ‎Sabzevar‎, ‎Iran.

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.

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Main Subjects

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### History

• Receive Date: 11 September 2016
• Revise Date: 18 January 2017
• Accept Date: 20 January 2017