### A continuous lattice L with DID(L) incomplete.

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$Z$-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 defined and its properties are explored.

Let $(L,\le )$, be an algebraic lattice. It is well-known that $(L,\le )$ with its topological structure is topologically scattered if and only if $(L,\le )$ is ordered scattered with respect to its algebraic structure. In this note we prove that, if $L$ is a distributive algebraic lattice in which every element is the infimum of finitely many primes, then $L$ has Krull-dimension if and only if $L$ has derived dimension. We also prove the same result for $errorL$, the set of all prime elements of $L$. Hence the dimensions on the lattice...

We prove the following theorem: Given a⊆ω and $1\le \alpha <{\omega}_{1}^{CK}$, if for some $\eta <{\aleph}_{1}$ and all u ∈ WO of length η, a is ${\Sigma}_{\alpha}^{0}\left(u\right)$, then a is ${\Sigma}_{\alpha}^{0}$.We use this result to give a new, forcing-free, proof of Leo Harrington’s theorem: ${\Sigma}_{1}^{1}$-Turing-determinacy implies the existence of ${0}^{}$.

A constructively valid counterpart to Bourbaki’s Fixpoint Lemma for chain-complete partially ordered sets is presented to obtain a condition for one closure system in a complete lattice $L$ to be stable under another closure operator of $L$. This is then used to deal with coproducts and other aspects of frames.

Let $A$ be a uniformly closed and locally m-convex $\Phi $-algebra. We obtain internal conditions on $A$ stated in terms of its closed ideals for $A$ to be isomorphic and homeomorphic to ${C}_{k}\left(X\right)$, the $\Phi $-algebra of all the real continuous functions on a normal topological space $X$ endowed with the compact convergence topology.

Posets with property DINT which are compact pospaces with respect to the interval topologies are characterized.

In this paper, some generating methods for principal topology are introduced by means of some logical operators such as uninorms and triangular norms and their properties are investigated. Defining a pre-order obtained from the closure operator, the properties of the pre-order are studied.