A note on topology of Z-continuous posets
نویسنده
چکیده
Z-continuous posets are common generalizations of continuous posets, completely distributive lattices, and unique factorization posets. Though the algebraic properties of Z-continuous posets had been studied by several authors, the topological properties are rather unknown. In this short note an intrinsic topology on a Z-continuous poset is de ned and its properties are explored.
منابع مشابه
Meet-continuity on $L$-directed Complete Posets
In this paper, the definition of meet-continuity on $L$-directedcomplete posets (for short, $L$-dcpos) is introduced. As ageneralization of meet-continuity on crisp dcpos, meet-continuity on$L$-dcpos, based on the generalized Scott topology, ischaracterized. In particular, it is shown that every continuous$L$-dcpo is meet-continuous and $L$-continuous retracts ofmeet-continuous $L$-dcpos are al...
متن کاملCountable composition closedness and integer-valued continuous functions in pointfree topology
For any archimedean$f$-ring $A$ with unit in whichbreak$awedge (1-a)leq 0$ for all $ain A$, the following are shown to be equivalent: 1. $A$ is isomorphic to the $l$-ring ${mathfrak Z}L$ of all integer-valued continuous functions on some frame $L$. 2. $A$ is a homomorphic image of the $l$-ring $C_{Bbb Z}(X)$ of all integer-valued continuous functions, in the usual se...
متن کاملOn the pointfree counterpart of the local definition of classical continuous maps
The familiar classical result that a continuous map from a space $X$ to a space $Y$ can be defined by giving continuous maps $varphi_U: U to Y$ on each member $U$ of an open cover ${mathfrak C}$ of $X$ such that $varphi_Umid U cap V = varphi_V mid U cap V$ for all $U,V in {mathfrak C}$ was recently shown to have an exact analogue in pointfree topology, and the same was done for the familiar cla...
متن کاملZero sets in pointfree topology and strongly $z$-ideals
In this paper a particular case of z-ideals, called strongly z-ideal, is defined by introducing zero sets in pointfree topology. We study strongly z-ideals, their relation with z-ideals and the role of spatiality in this relation. For strongly z-ideals, we analyze prime ideals using the concept of zero sets. Moreover, it is proven that the intersection of all zero sets of a prime ideal of C(L),...
متن کاملZero elements and $z$-ideals in modified pointfree topology
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, t...
متن کامل