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In this paper, we solve two problems concerning the ideally conjunctive join-semilattices. First, we show that L / R 1 (Id L) | L is ideally conjunctive for all join-semilattices L. Then we characterize those ideally conjunctive join-semilattices L such that coz a is compact for all a ∈ L. Moreover, we give the definition of conjunctive posets and prove that the category of ideally conjunctive join-semilattices and join homomorphisms is reflective in the category of conjunctive posets and weakly ideal-continuous maps. As a corollary, we obtain the free ideally conjunctive join-semilattices over conjunctive posets. [ABSTRACT FROM AUTHOR]

Monads prove to be useful mathematical tools in theoretical computer science, notably in denoting different effects of programming languages. In this paper, we investigate a type of monads which arise naturally from Keimel and Lawson's $\mathbf{K}$ -ification. A subcategory of $\mathbf{TOP}_{\mathbf{0}}$ is called of type $\mathrm{K}^{*}$ if it consists of monotone convergence spaces and is of type $\mathrm K$ in the sense of Keimel and Lawson. Each such category induces a canonical monad $\mathcal K$ on the category $\mathbf{DCPO}$ of dcpos and Scott-continuous maps, which is called the order- $\mathbf{K}$ -ification monad in this paper. First, for each category of type $\mathrm{K}^{*}$ , we characterize the algebras of the corresponding monad $\mathcal K$ as k -complete posets and algebraic homomorphisms as k -continuous maps, from which we obtain that the order- $\mathbf{K}$ -ification monad gives the free k -complete poset construction over the category $\mathbf{POS}_{\mathbf{d}}$ of posets and Scott-continuous maps. In addition, we show that all k -complete posets and Scott-continuous maps form a Cartesian closed category. Moreover, we consider the strongness of the order- K -ification monad and conclude with the fact that each order- K -ification monad is always commutative. [ABSTRACT FROM AUTHOR]

This paper introduces weak logical algebraic domains and a special class of quantum B-algebras, namely, cofinal quantum B-algebras. It proves that there is a dual equivalence between the category of cofinal quantum B-algebras and that of weak logical algebraic domains. In particular, the category of residuated posets is shown to be dual equivalent to that of logical algebraic domains. This correspondence generalizes the classical duality of posets and algebraic domains. [ABSTRACT FROM AUTHOR]

The study of the sobriety of Scott spaces has got a relatively long history in domain theory. Lawson and Hoffmann independently proved that the Scott space of every continuous directed complete poset (usually called domain) is sober. Johnstone constructed the first directed complete poset whose Scott space is non-sober. Soon after, Isbell gave a complete lattice with a non-sober Scott space. Based on Isbell's example, Xu, Xi, and Zhao showed that there is even a complete Heyting algebra whose Scott space is non-sober. Achim Jung then asked whether every countable complete lattice has a sober Scott space. The main aim of this paper is to answer Jung's problem by constructing a countable complete lattice whose Scott space is non-sober. This lattice is then modified to obtain a countable distributive complete lattice with a non-sober Scott space. In addition, we prove that the topology of the product space $\Sigma P\times \Sigma Q$ coincides with the Scott topology of the product poset $P\times Q$ if the set Id (P) and Id (Q) of all incremental ideals of posets P and Q are both countable. Based on this, it is deduced that a directed complete poset P has a sober Scott space, if Id (P) is countable and $\Sigma P$ is coherent and well filtered. In particular, every complete lattice L with Id (L) countable has a sober Scott space. [ABSTRACT FROM AUTHOR]

This paper focuses on the study of H-closedness and absolute H-closedness of the posets. First, we propose a counterexample to indicate that an absolutely H-closed topological semilattice may not be c-complete, which gives a negative answer to an open question proposed by Banakh and Bardyla. However, in the case of continuous semilattice with the Lawson topology, we prove that the absolutely H-closed topological semilattice implies c-completeness. Second, we obtain a characterization for quasicontinuous lattices utilizing the topological embedding mapping. Finally, enlightened by the definitions of H-closedness for Hausdorff spaces and absolute H-closedness for Hausdorff topological semilattices, we introduce the concepts of H-closedness and absolute H-closedness for posets with the Lawson topology. [ABSTRACT FROM AUTHOR]

In this paper, we define a new class of posets which are complemented and ideal-distributive, we call these posets strong Boolean. This definition is a generalization of Boolean lattices on posets, and is different from Boolean posets. We give a topology on the set of all prime Frink ideals in order to obtain the Stone's topological representation for strong Boolean posets. A discussion of a duality between the categories of strong Boolean posets and BP-spaces is also presented. [ABSTRACT FROM AUTHOR]

Recently, Rusu and Ciobanu established that for a continuous domain L , a subset B of L is a basis if and only if B is dense with respect to the d -topology, called the density topology, on L. In situations where directed completeness fails, Erné has proposed in 1991 an alternative definition of continuity called s 2-continuity which remedied the lack of stability of continuity under the classical Dedekind–MacNeille completion. In this paper, we show how the 'Rusu–Ciobanu' type of characterization can be formulated and established over the class of s 2-continuous posets with appropriate modifications. Although we obtain more properties of essential topologies and density topologies on s 2-continuous posets, respectively. [ABSTRACT FROM AUTHOR]

We introduce a new concept of continuity of posets, called θ-continuity. Topological characterizations of θ-continuous posets are put forward. We also present two types of dcpo-completion of posets which are Dθ-completion and Ds2-completion. Connections between these notions of continuity and dcpo-completions of posets are investigated. The main results are (1) a poset P is θ-continuous iff its θ-topology lattice is completely distributive iff it is a quasi θ-continuous and meet θ-continuous poset iff its Dθ-completion is a domain; (2) the Dθ-completion of a poset B is isomorphic to a domain L iff B is a θ-embedded basis of L; (3) if a poset P is θ-continuous, then the Dθ-completion Dθ(P) is isomorphic to the round ideal completion RI(P, ≪θ). [ABSTRACT FROM AUTHOR]

We study a basic problem: in what posets is the order-convergence topological? We introduce the notion of 𝓡∗-doubly continuous posets, which extends the notion of doubly continuous posets, and then prove that the order-convergence in a poset is topological if and only if the poset is 𝓡∗-doubly continuous. This is the main result which can be regarded as a complete characterization of posets for the order-convergence being topological. [ABSTRACT FROM AUTHOR]

Let P denote a cartesian closed full subcategory of the category POSET of posets and Scott continuous functions. We define C-P to be the full subcategory of objects from P whose D -completion is isomorphic to an object from C , where C is a subcategory of the category CONT of domains. The category C-P is always a subcategory of the category CONTP of continuous posets and Scott continuous functions. We prove that if C is a cartesian closed full subcategory of F-L , U-L , F-RB or U-RB , then the category C-P is also cartesian closed. It is known that the category CDCPO of consistent directed complete posets and Scott continuous functions is cartesian closed. In particular, we have the following cartesian closed categories: F-L-CDCPO , U-L-CDCPO , F-RB-CDCPO , U-RB-CDCPO , F-aL-CDCPO , U-aL-CDCPO , F-B-CDCPO , U-B-CDCPO , etc. If the categories FS and RB coincide, then it leads directly to the most potential of this uniform way of finding new cartesian closed categories of continuous posets: for every cartesian closed full subcategory C of CONT , the category C-P is also cartesian closed. [ABSTRACT FROM AUTHOR]

By introducing new approximation relations on posets, the notion of S ⁎ -doubly continuous posets is proposed and its relationship with doubly continuous posets is presented. Aslo the B-topology on posets is defined and the special properties of the B-topology on S ⁎ -doubly continuous posets are studied. Then, a sufficient and necessary condition for the Birkhoff's order-convergence in a poset being topological is obtained. [ABSTRACT FROM AUTHOR]

In this paper, as a special case of fuzzy domains, algebraic fuzzy directed-complete posets (shortly, algebraic fuzzy dcpos) are discussed. Throughout the algebraic fuzzy dcpos, some equivalent characterizations of fuzzy domains are given. Then, the construction of new algebraic fuzzy dcpos (resp., fuzzy domains) from known ones by means of forming products, taking some special subsets, and taking images under maps with appropriate operations is investigated. Finally, a new kind of fuzzy continuous section retraction pairs is proposed. Moreover, it is proved that every fuzzy domain is a fuzzy continuous retraction of an algebraic fuzzy dcpo, and the fuzzy continuous retraction of a fuzzy domain is also a fuzzy domain. [ABSTRACT FROM AUTHOR]

In this paper, we establish some connections between partial metric spaces and domain theory, and give a characterization of stable partially metrizable d-spaces. First, the concept of S-partial metrics on posets is introduced. Then it is demonstrated that the open ball topology is coarser than the Scott topology in an S-partial metric space, and that the partially metrizable d -space is exactly the DS-partial metric space. In addition, some conditions are provided to ensure that the open-ball topology and the Scott topology coincide in an S-partial metric space. Moreover, we prove that, for an S-partial metric space (X , ≤ , p) , (X , O p (X)) is sober iff the set of all limit points of every self-convergent sequence is directed. Finally, we propose the notion of DS-valuation space and obtain the result that there is a bijection between stable DS-partial metric spaces and DS-valuation spaces, thereby giving the characterization of stable partially metrizable d-spaces. [ABSTRACT FROM AUTHOR]

Many extensions of rough sets have been trying to seek appropriate granular structures, such as neighborhood systems, disjoint intervals and coverings. However, few of them consider data-driven approaches to generating posets-structured coverings based on granules of irregular shapes and variable sizes. By generalizing norm granules (intervals, δ -neighborhoods and k -nearest neighbors), the present study proposes a tree-structured model whose information granules are obtained through an "onion-peeling" strategy, CrossSift. Two comparative experiments are conducted in this paper. One shows that granules generated by CrossSift are able to achieve a higher dependency degree with fewer numbers than equal width/frequency intervals, δ -neighborhoods and k -nearest neighbors. The other shows the trees output by CrossSift outperform SVC, KNN, AdaBoost, Cart, LDA in the average rank of classification accuracy. The proposed method bridges a gap between rough sets and perceptrons, and is expected to contribute to dimensionality reduction, computer vision and geometry. [ABSTRACT FROM AUTHOR]

It is well known in [3] that the liminf convergence is topological in a domain. We supply an example to illustrate that a dcpo in which the liminf convergence is topological may not be continuous in this paper. Naturally, there is a problem raised as follows: in what posets is the liminf convergence topological? First, we show that if the liminf convergence is topological in a poset, then the poset is exact. Second, we prove that a poset L is continuous if and only if L is meet continuous and the liminf convergence is topological in L and the Lawson topology equals the liminf topology. Finally, we introduce the concept of weak* continuous and obtain the result that the liminf convergence is topological in a poset L if and only if L is weak* continuous. This result answers the above problem. [ABSTRACT FROM AUTHOR]

It remains an open problem as to whether a core-compact dcpo is sober. We give a sufficient condition for a core-compact dcpo to be sober. We introduce and study a new kind of posets—join continuous posets. Meanwhile, we investigate the relationships among sober, well-filtered and weak well-filtered topological properties. Based on the above results, we give some characterizations for a poset to be Lawson compact. Moreover, we have also solved the open problems posed by Xu and Zhao. Finally, we have considerably simplified the counterexample given by Isbell based on the work of Goubault in his blog. [ABSTRACT FROM AUTHOR]