Zero elements and z-ideals in modified pointfree topology

ýIn this paperý, ýwe define and study the notion of zero elements in topoframes; a topoframe is a pairý ý(Lý,ýτ)ý, ýabbreviated Lτý, ýconsisting of a frame L and aý ýsubframe τ all of whose elements are complemented elements iný ýLý. ýWe show thatý ýthe f-ring R(Lτ)ý, ýthe set of τ-real continuous functions on Lý, ýis uniformly completeý. ýAlsoý, ýthe set of all zero elements in a topoframeý ýis closed under the formation of countable meets and finite joinsý. ýAlsoý, ýwe introduce the notion of z-filters and z-ideals in modified pointfree topologyý ýand make ready some results about themý.