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... It has been used in the characterizations of spectral spaces and T 0 spaces that are determined by their open set lattices. With the development of domain theory, another two properties also emerged as the very useful and important properties for non-Hausdorff topology theory: d-space and well-filtered space (see [1,[11][12][13][14][15][16][17][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][41][42][43][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60]). In the past few years, some remarkable progresses have been achieved in understanding such structures. ...

... [m3L; v1.297] P.5 (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) X. Xu, D. Zhao / Topology and its Applications ...

... In [14], it is shown that for the complete Boolean algebra B of all regular open subsets of the reals, the Scott space ΣB is not a topological join-semilattice (and hence the Scott topology σ(B × B) is properly larger than the product topology σ(B) × σ(B)). It is natural to wonder whether ΣB is sober. ...

In the past few years, the research on sober spaces and well-filtered spaces has got some breakthrough progress. In this paper, we shall present a brief summarising survey on some of such development. Furthermore, we shall pose and illustrate some open problems on well-filtered spaces and sober spaces.

... In domain theory, various kinds of convergence classes were studied (see e.g. [1][2][3][4]9,13,15,16]). By different convergence structures, not only many notions of continuity are characterized, but also they are closely related to order and topology. ...

... In [4], Erné characterized the liminf topology via directed sets. Similarly, we can give a characterization of the O(Q)-topology via filter bases. ...

In this paper, the concepts of gs2-convergence and quasi-liminf convergence of filters in posets are introduced. The main results are: (1) For an order consistent topology τ on a poset P, P is τ-quasicontinuous iff the gs2-convergence coincides with τ-convergence; (2) For an order consistent topology τ on a poset P, the quasi-liminf convergence coincides with τ∨ω(P)-convergence if P is τ-quasicontinuous; (3) For an order consistent topology τ on a poset P, P is τ-continuous iff the s2-convergence coincides with τ-convergence iff it is meet τ-continuous and the quasi-liminf convergence coincides with τ∨ω(P)-convergence.

... Then (1) ⇒ (2). If (X, τ ) is sober, then (1) and (2) (1) If the Scott topology σ(L) is a continuous lattice, then σ(L) is sober. ...

... Hence by Proposition 3.7 W is a meet continuous lattice such that σ(W ) is continuous but not hypercontinuous (cf. [1]). ...

In this note it is proved that for a quasicontinuous lattice L, the lower topology and the Scott topology are duals for each other; and if L is a complete lattice such that is continuous but not hypercontinuous (equivalently, L is not quasicontinuous), then is not the dual of and hence they are not duals for each other.

... In addition there are two topologies which are initially defined in terms of conver- gence: (v) the order topology, 8,(X), as defined, for example, in [16] (note also [13, p. 16]), and (vi) the lim-inf topology, c(X), as defined in [16, Section 111.31. The relationship between these various topologies is shown in Fig. 1; for proofs see [13] and [34]. It is immediate from this diagram that if the interval topology and the order topology are equal then any intermediate topology coincides with both. ...

... For a more complete discussion of the evolution of these results, we refer the reader to [12] [13] [33] [34]. ...

The purpose of this expository note is to draw together and to interrelate a variety of characterisations and examples of spectral sets (alias representable posets or profinite posets) from ring theory, lattice theory and domain theory.

... Another motivating aspect of our study is that various classes of spaces with strong local connectedness properties may be regarded as natural generalizations of semilattices with compatible topologies such that the unary, binary or infinitary semilattice operations become continuous (cf. [19]), or of topological semilattices with local or global bases of subsemilattices. ...

... However, it does not follow that the binary meet operation ∧ : X 2 → X has to be continuous: any complete Boolean lattice is meet-continuous, whence its Scott topology is a coframe; but the binary meet need not be continuous, as the example of the regular open subsets of the reals shows (see [19]). Theorem 2, Proposition 2 and Corollary 2 generalize known characterizations of meet-continuous posets and semilattices (see [25,] and [39]). ...

Various local connectedness and compactness properties of topological spaces are characterized by higher degrees of distributivity for their lattices of open (or closed) sets, and conversely. For example, those topological spaces for which not only the lattice of open sets but also that of closed sets is a frame, are described by the existence of web neighborhood bases, where webs are certain specific path-connected sets. Such spaces are called web spaces. The even better linked wide web spaces are characterized by F-distributivity of their topologies, and the worldwide web spaces (or C-spaces) by complete distributivity of their topologies. Similarly, strongly locally connected spaces and locally hypercompact spaces are characterized by suitable infinite distributive laws. The web space concepts are also viewed as natural extensions of spaces that are semilattices with respect to the specialization order and have continuous (unary, binary or infinitary) semilattice operations.

... Then Σ L is a retract of Σ C Σ L ðÞ and hence Σ C Σ L ðÞ is non-sober. In [56], Erné showed that for the complete Boolean algebra B of all regular open subsets of the reals, the Scott space ΣB is not a topological join-semilattice (and hence the Scott topology σ B Â B ðÞ is strictly finer than the product topology σ B ðÞ Â σ B ðÞ ). It is natural to wonder whether ΣB is sober. ...

In most topology books, the Hausdorff separation property is assumed from the very start and contain very little information on non-Hausdorff spaces. In classical mathematics, most topological spaces are indeed Hausdorff. But non-Hausdorff spaces are important already in algebraic geometry, and crucial in fields such as domain theory. Indeed, in connection with order, non-Hausdorff spaces, especially $T_0$ spaces, play a more significant role than Hausdorff spaces. Sobriety is probably the most important and useful property of non-Hausdorff topological spaces. It has been used in the characterizations of spectral spaces and $T_0$ spaces that are determined by their open set lattices. With the development of domain theory, another two properties also emerged as the very useful and important properties for non-Hausdorff topology theory: $d$-space and well-filtered space. In the past few years, some remarkable progresses have been achieved in understanding such structures. In this chapter, we shall make a brief survey on some of these progresses and list a few related problems.

... This happens for all continuous lattices (see Corollary 2.1 or [14], [15]) but not for all join-and meetcontinuous complete lattices. In [13] it is shown that for bounded semilattices S, the square space Σ(S 2 ) is a topological semilattice iff ΣS is such and Σ(S 2 ) coincides with (ΣS) 2 , and examples of atomless complete (hence join-and meet-continuous) Boolean algebras are given for which the latter coincidence fails. Now, we say a space Y is a CL-space (continuous lattice space) if (C1) Y is order complete; (C2) join operations of arbitrary arity are continuous; (C3) meet operations of arbitrary arity are box continuous. ...

... In [14], it is shown that for the complete Boolean algebra B of all regular open subsets of the reals line, the Scott space ΣB is not a topological join-semilattice (and hence the Scott topology σ(B × B) is properly larger than the product topology σ(B) × σ(B)). It is natural to wonder whether ΣB is sober. ...

In the past few years, the research on sober spaces and well-filtered spaces has got some breakthrough progress. In this paper, we shall present a brief summarising survey on some of such development. Furthermore, we shall pose and illustrate some open problems on well-filtered spaces and sober spaces.

... In [21, III-5], one finds a whole collection of various equivalent characterizations of continuous domains that are compact in their Lawson topology. For a detailed study of joint and separate continuity of operations in posets and lattices, see [18]. ...

We present old and new characterizations of core spaces, alias worldwide web spaces, originally defined by the existence of supercompact neighborhood bases. The patch spaces of core spaces, obtained by joining the original topology with a second topology having the dual specialization order, are the so-called sector spaces, which have good convexity and separation properties and determine the original space. The category of core spaces is shown to be concretely isomorphic to the category of fan spaces; these are certain quasi-ordered spaces having neighborhood bases of so-called fans, obtained by deleting a finite number of principal filters from a principal filter. This approach has useful consequences for domain theory. In fact, endowed with the Scott topology, the continuous domains are nothing but the sober core spaces, and endowed with the Lawson topology, they are the corresponding fan spaces. We generalize the characterization of continuous lattices as meet-continuous lattices with T$_2$ Lawson topology and extend the Fundamental Theorem of Compact Semilattices to non-complete structures. Finally, we investigate cardinal invariants like density and weight of the involved objects.

... As a consequence, liminfs and limsups of filterbases are formed in a way which, coordinatewise, is highly special: the associated nets are eventually constant. The topological structure in ordered topological spaces in general need not interact well with liminfs and limsups (indeed, order-convergence need not correspond to convergence with respect to any topology); see for example [13,II.1 and III.3] and also [7] and [14,Section 2]. In particular, order-convergence and topological convergence do not coincide on a complete lattice which fails to be meet-and join-continuous. ...

... The join of the upper and the lower topology is the interval topology ı P , while the join of the Scott topology and the lower topology is the Lawson topology λ P (cf. [15], [19]). ...

We study several choice principles for systems of finite character and prove their equivalence to the Prime Ideal Theorem in ZF set theory without Axiom of Choice, among them the Intersection Lemma (stating that if S is a system of finite character then so is the system of all collections of finite subsets of S meeting a common member of S), the Finite Cutset Lemma (a finitary version of the Teichmuller-Tukey Lemma), and various compactness theorems. Several implications between these statements re- main valid in ZF even if the underlying set is fixed. Some fundamental algebraic and order-theoretical facts like the Artin-Schreier Theorem on the orderability of real fields, the Erdos-De Bruijn Theorem on the colorability of infinite graphs, and Dilworth's The- orem on chain-decompositions for posets of finite width, are easy consequences of the Intersection Lemma or of the Finite Cutset Lemma.

... The most flexible versions of semicontinuity for lattice-valued functions on a topological space seem to be the continuities with respect to the upper topology ν(L) and the lower topology ν˜(L) (cf. [25,32]) and [8] for notation). As in [25] , we shall assume that the range space is a completely distributive lattice L with a countable join-dense subset consisting of non-supercompact elements. ...

Problems of inserting lattice-valued functions are investigated. We provide an analogue of the clas-sical insertion theorem of Lane [Proc. Amer. Math. Soc. 49 (1975) 90–94] for L-valued functions where L is a -separable completely distributive lattice (i.e. L admits a countable join-dense subset which is free of completely join-irreducible elements). As a corollary we get an L-version of the Katětov–Tong insertion theorem due to Liu and Luo [Topology Appl. 45 (1992) 173–188] (our proof is different and much simpler). We show that -separable completely distributive lattices are closed under the formation of countable products. In particular, the Hilbert cube is a -separable completely distributive lattice and some join-dense subset is shown to be both order and topologically isomor-phic to the hedgehog J (ω) with appropriately defined topology. This done, we deduce an insertion theorem for J (ω)-valued functions which is independent of that of Blair and Swardson [Indian J. Math. 29 (1987) 229–250]. Also, we provide an iff criterion for inserting a pair of semicontinuous function which yields, among others, a characterization of hereditarily normal spaces.

... As a consequence, liminfs and limsups of filterbases are formed in a way which, coordinatewise, is highly special: the associated nets are eventually constant. The topological structure in ordered topological spaces in general need not interact well with liminfs and limsups (indeed , order-convergence need not correspond to convergence with respect to any topology); see for example [13, II.1 and III.3] and also [7] and [14, Section 2]. In particular, order-convergence and topological convergence do not coincide on a complete lattice which fails to be meet-and join-continuous. ...

This paper investigates completions in the context of finitely generated lattice-based varieties of algebras. It is shown that, for such a variety A, the order-theoretic conditions of density and compactness which characterise the canonical extension of (the lattice reduct of) any A ∈ A have truly topological interpreta-tions. In addition, a particular realisation is presented of the canonical extension of A; this has the structure of a topological algebra n A (A) whose underlying algebra belongs to A. Furthermore, each of the operations of n A (A) coincides with both the σ-extension and the π-extension of the corresponding operation on A, with which a canonical extension is customarily equipped. Thus, in partic-ular, the variety A is canonical, and all its operations are smooth. The methods employed rely solely on elementary order-theoretic and topological arguments, and by-pass the subtle theory of canonical extensions that has been developed for lattice-based algebras in general.

... In order to ensure that a subposet M of an algebraic or continuous lattice L is an algebraic or continuous lattice, too, it suffices to require that M be closed under arbitrary meets and under directed joins (up-closed); although the compact elements of M may differ from those in L, they are just the closures of the compact elements in L (see [15,). A map f : L → M between complete lattices is called (Scott) continuous if it preserves directed joins, i.e., f ( D) = f [D] whenever D is directed (see [14] or [15]). Notice that every continuous map f is isotone, i.e., x ≤ y implies f (x) ≤ f (y). ...

Our aim is to investigate groups and their weak congruence lattices in the abstract setting of lattices L with (local) closure operators C in the categorical sense, where L is regarded as a small category and C is a family of closure maps on the principal ideals of L. A useful tool for structural investigations of such “lattices with closure” is the so-called characteristic triangle, a
certain substructure of the square L
2. For example, a purely order-theoretical investigation of the characteristic triangle shows that the Dedekind groups (alias
Hamiltonian groups) are precisely those with modular weak congruence lattices; similar results are obtained for other classes
of algebras.
2000 Mathematics Subject ClassificationPrimary 08A30-secondary 08C05-06B99
Key words and phrasesAlgebraic lattice-characteristic triangle-continuous closure-Dedekind group-diagram-normal subgroup-weak congruence lattice

... However, for isotone functions, the following fact has been established in [9, Proposition 2.21: In the subsequent considerations, continuity of join and meet operations will play a central role. As in [3,4,9] If L is o-continuous and the order convergence of nets is topological on L then we say L is order-topological for nets (the corresponding filter-theoretical notion has been discussed in [3]; for complete lattices, these notions coincide). By 2.2 and its dual, an arbitrary lattice is order-topological for nets iff it is a topological lattice with a topology r such that r-convergence of nets agrees with order convergence (whence r must be the order topology). ...

By a (fine) scale on a lattice L, we mean a (strictly) isotone real-valued function μ on L. The coarsest topology on L such that μ composed with the unary lattice operations becomes continuous is denoted by τμ. The following convergence structures on L are compared with each other: 1.(i) order convergence,2.(ii) convergence in the order topology,3.(iii) τμ-convergence,4.(iv) ρμ-convergence, where ρμ(x, y) = μ(x ∨ y) − μ(x ∧ y).
We show that for any scale μ on an arbitrary complete lattice, order convergence agrees with ρμ-convergence and with τμ-convergence iff μ is a fine continuous scale such that join and meet operations of arbitrary arity are continuous with respect to ρμ-convergence. Furthermore, for any fine continuous scale μ on a bi-algebraic lattice, order convergence agrees with τμ-convergence. From these and related results, we derive various applications to the theory of measures and valuations on orthomodular lattices. For example, if μ is a fine scale on a complete orthomodular lattice then order convergence agrees with τμ-convergence iff μ is continuous and L is algebraic (or atomic and meet-continuous).

... If a subset X of a lattice or poset L has a join (least upper bound), this will be denoted by V X; dually, A X denotes the meet (greatest lower accordance with [6] [13] [14] [19], we define order convergence of follows. For any filtercbase) 9 on L, put bound) of X. ...

A lattice is order-topological iff its order convergence is topological and makes the lattice operations continuous. We show that the following properties are equivalent for any complete orthomodular lattice L: 1.(i) L is order-topological,2.(ii) L is continuous (in the sense of Scott),3.(iii) L is algebraic,4.(iv) L is compactly atomistic,5.(v) L is a totally order-disconnected topological lattice in the order topology.A special class of complete order-topological orthomodular lattices, namely the compact topological orthomodular lattices, are characterized by various algebraic conditions, for example, by the existence of a join-dense subset of so-called hypercompact elements.

... Indeed, it is known that a map between ordered sets preserves up-directed joins and down-directed meets if and only if it is continuous with respect to any topology between the interval topology and the order topology (cf. M. Erné and H. Gatzke [19]). For completeness we include a direct proof based on the preceding lemmas. ...

This paper presents a unified account of a number of dual category equivalences of relevance to the theory of canonical extensions
of distributive lattices. Each of the categories involved is generated by an object having a two-element underlying set; additional
structure may be algebraic (lattice or complete lattice operations) or relational (order) and, in either case, topology may
or may not be included. Among the dualities considered is that due to B.Banaschewski between the categories of Boolean topological
bounded distributive lattices and the category of ordered sets. By combining these dualities we obtain new insights into canonical
extensions of distributive lattices.

... In an arbitrary lattice there is a host of well-known convergence structures in between interval topology and order convergence. Ern4 [2] Using i.a. Lemma 9 and the well-known fact that any chain J is complete (conditionally complete), if and only if t(J) is compact (all bounded ultrafilters on J are t(J)-convergent), we obtain the following two theorems. ...

On ordered sets (posets, lattices) we regard topologies (or, more general convergence
structures) which on any maximal chain of the ordered set induce its own interval topology. This construction generalizes several well-known intrinsic structures, and still contains enough to produce interesting results on for instance compactness and connectedness.
The maximal chain compatibility between topology (convergence structure) and order is preserved by formation of arbitrary products, at least in case the involved order structures are conditionally complete lattices.

We study general notions of convergence and continuity in arbitrary spaces or ordered sets, extending considerably topological concepts in domain theory like those of Scott convergence, alias lower (lim-inf) convergence, and Scott topology. It turns out that the convergence-theoretical properties of being localized, a limit relation, pretopological, or topological, respectively, all correspond to important properties of the underlying ordered sets that reduce to (meet) continuity and similar properties in the classical situation. Basic tools are the cut closure operators and diverse order-theoretical or topological variants of them. We characterize the generalized Scott convergence spaces abstractly as so-called core determined convergence spaces. This unifying concept provides simplifications and new insights into various areas of order theory, topology and theoretical computer science. In particular, some intimate connections between convergence properties, meet preservation by certain closure operations, and the continuity of meet operations are established.

It is known that for a nonempty topological space X and a nonsingleton complete lattice Y endowed with the Scott topology, the partially ordered set [X, Y] of all continuous functions from X into Y is a continuous lattice if and only if both Y and the open set lattice OX are continuous lattices. This result extends to certain classes of Z-distributive lattices, where Z is a subset system replacing the system D of all directed subsets (for which the D-distributive complete lattices are just the continuous ones). In particular, it is shown that if [X, Y] is a complete lattice then it is supercontinuous (i. e. completely distributive) iff both Y and OX are supercontinuous. Moreover, the Scott topology on Y is the only one making that equivalence true for all spaces X with completely distributive topology. On the way to these results, we find necessary and sufficient conditions for [X, Y] to be complete, and some new, purely topological characterizations of continuous lattices by continuity conditions on their (infinitary) lattice operations.

A standard extension (resp. standard completion) is a function Z assigning to each poset P a (closure) system ZP of subsets such that x ⋚ y iff x belongs to every Z ε ZP with y ε Z. A poset P is Z -complete if each Z ε 2P has a join in P. A map f: P → P′ is Z—continuous if f [Z′] ε ZP for all Z′ ε ZP′, and a Z—morphism if, in addition, for all Z ε ZP there is a least Z′ ε ZP′ with f[Z] ⊆ Z′. The standard extension Z is compositive if every map f: P → P′ with {x ε P: f(x) ⋚ y′} ε ZP for all y′ ε P′ is Z -continuous. We show that any compositive standard extension Z is the object part of a reflector from IPZ, the category of posets and Z -morphisms, to IRZ, the category of Z -complete posets and residuated maps. In case of a standard completion Z, every Z -continuous map is a Z -morphism, and IR2 is simply the category of complete lattices and join—preserving maps. Defining in a suitable way so-called Z -embeddings and morphisms between them, we obtain for arbitrary standard extensions Z an adjunction between IPZ and the category of Z -embeddings. Many related adjunctions, equivalences and dualities are studied and compared with each other. Suitable specializations of the function 2 provide a broad spectrum of old and new applications.

The compact–open topology for function spaces is usually attributed to R. H. Fox in 1945 [16]; and indeed, there is no earlier publication to attribute it to. But it is clear from Fox's paper that the idea of the compact–open topology, and its notable success in locally compact spaces, were already familiar. The topology of course goes back to Riemann; and to generalize to locally compact spaces needs only a definition or two. The actual contributions of Fox were (1) to formulate the partial result, and the problem of extending it, clearly and categorically; (2) to show that in separable metric spaces there is no extension beyond locally compact spaces; (3) to anticipate, partially and somewhat awkwardly, the idea of changing the category so as to save the functorial equation. (Scholarly reservations: Fox attributes the question to Hurewicz, and doubtless Hurewicz had a share in (1). As for (2), when Fox's paper was published R. Arens was completing a dissertation which gave a more general result [1] – though worse formulated.)(Received March 07 1986)

By a recent observation of Monjardet and Wille, a finite distributive lattice is generated by its doubly irreducible elements iff the poset of all join-irreducible elements has a distributive MacNeille completion. This fact is generalized in several directions, by dropping the finiteness condition and considering various types of bigeneration via arbitrary meets and certain distinguished joins. This leads to a deeper investigation of so-called L-generators resp. C-subbases, translating well-known notions of topology to order theory. A strong relationship is established between bigeneration by (minimal) L-generators and so-called principal separation, which is defined in order-theoretical terms but may be regarded as a strong topological separation axiom. For suitable L, the complete lattices with a smallest join-dense L-subbasis consisting of L-primes are the L-completions of principally separated posets.

A standard completion γ assigns a closure system to each partially ordered set in such a way that the point closures are precisely
the (order-theoretical) principal ideals. If S is a partially ordered semigroup such that all left and all right translations
are γ-continuous (i.e., Y∈γS implies {x∈S:y·x∈Y}∈γS and {x∈S:x·y∈Y}∈γS for all y∈S), then S is called a γ-semigroup. If S
is a γ-semigroup, then the completion γS is a complete residuated semigroup, and the canonical principal ideal embedding of
S in γS is a semigroup homomorphism. We investigate the universal properties of γ-semigroup completions and find that under
rather weak conditions on γ, the category of complete residuated semigroups is a reflective subcategory of the category of
γ-semigroups. Our results apply, for example, to the Dedekind-MacNeille completion by cuts, but also to certain join-completions
associated with so-called “subset systems”. Related facts are derived for conditional completions.

With every subset selection for posets, there is associated a certain ideal completion . As shown by Erné, such completions help to extend classical results on domains and similar structures in the absence of
the required joins. Some results about –predistributive or –precontinuous posets and –continuous functions are summarized and supplemented. In particular, several central results on function spaces in domain
theory are extended to the setting of productive closed subset selections. The category FSBP, in which objects are finitely separated and upper bounded posets and arrows are continuous functions between them, is shown to be cartesian closed.

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