Lattice Theory (Order Theory)

ABSTRACT Let P be a finite poset. In this paper a Boolean poset B is constructed such that there is a cover-preserving embedding of P in B. In addition it is proved that the number of maximal chains in P is at most equal to the number of linear extensions of join-irreducibles of P.

La taille d'un treillis de concepts peut augmenter de facon exponentielle avec la taille du contexte. Lorsque le nombre de noeuds devient important, l'´ etude et la generation d'un tel treillis devient impossible. Decomposer le treillis en petit sous-treillis est un moyen de contourner ceprobi eme. Dans la decomposition sous-directe, les petits sous-treillis generes sont des quotients qui ont une interpretation interessante dans le cadre de l'Analyse de Concepts Formels. Dans cet article, nous presentons les etapes pour obtenir une decomposition sous-directe en treillis irreductibles , en partant d'un contexte fini et reduit. Cette decomposition est obtenue en utilisant trois points de vue : les treillis quotients, les relationsfi eches et les sous-contextes compatibles. Cette approche est essentiellement algebrique car elle repose sur la theorie des treillis, sauf pour le dernier point. Nous donnons un algorithme polynomial permettant de generer cette decomposit...

In quantum theory, the modulus-square of the inner product of two normalized Hilbert space elements is to be interpreted as the transition probability between the pure states represented by these elements. A probabilistically motivated and more general definition of this transition probability was introduced in a preceding paper and is extended here to a general type of quantum logics: the orthomodular partially ordered sets. A very general version of the quantum no-cloning theorem, creating promising new opportunities for quantum cryptography, is presented and an interesting relationship between the transition probability and Jordan algebras is highlighted.

We present Nonstandard Analysis by three axioms: the Extension, Transfer and Saturation Principles in the framework of the superstructure of a given infinite set. We also present several applications of this axiomatic approach to point-set topology. Some of the topological topics such as the Hewitt realcompactification and the nonstandard characterization of the sober spaces seem to be new in the literature on nonstandard analysis. Others have already close counterparts but they are presented here with essential simplifications.

In this paper, we study the class of a BH lattices as a common frame to Brouwerian and Heyting lattices and investigate some related properties. Also, we characterize the divisibility condition in the definition of BH lattice and we obtain that the set H of all idempotent elements in a BH lattice L forms a Heyting algebra. We introduce the notion of an IBH lattice and under some specific conditions, we characterize an MV algebra, a bounded Wajsberg hoop, a Boolean Algebra, and a commutative bounded BCK algebra in terms on IBH lattices.

Cahiers de topologie et géométrie différentielle catégoriques, tome 34, n o 3 (1993), p. 167-183 <http © Andrée C. Ehresmann et les auteurs, 1993, tous droits réservés. L'accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l'accord avec les conditions générales d'utilisation (). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques EQUATIONALLY CLOSED SUBFRAMES AND REPRESENTATION OF QUOTIENT SPACES by Ale0161 PULTR and Anna TOZZI CAHIERS DE TOPOLOGIE

We consider the one-dimensional massive Thirring model formulated on the lattice with staggered fermions and an auxiliary compact vector (link) field, which is exactly solvable and shows a phase transition with increasing the chemical potential of fermion number: the crossover at a finite temperature and the first order transition at zero temperature. We complexify its path-integration on Lefschetz thimbles and examine its phase transition by hybrid Monte Carlo simulations on the single dominant thimble. We observe a discrepancy between the numerical and exact results in the crossover region for small inverse coupling β and/or large lattice size L, while they are in good agreement in the lower and higher density regions. We also observe that the discrepancy persists in the continuum limit to keep the temperature finite and it becomes more significant toward the low-temperature limit. This numerical result is consistent with our analytical study of the model and implies that the contributions of subdominant thimbles should be summed up in order to reproduce the first order transition in the low-temperature limit.

We study a resurgence structure of a quantum field theory with a phase transition to uncover relations between resurgence and phase transitions. In particular, we focus on three-dimensional N = 4 supersymmetric quantum electrodynamics (SQED) with multiple hypermultiplets, where a second-order quantum phase transition has been recently proposed in the large-flavor limit. We provide interpretations of the phase transition from the viewpoints of Lefschetz thimbles and resurgence. For this purpose, we study the Lefschetz thimble structure and properties of the large-flavor expansion for the partition function obtained by the supersymmetric localization. We show that the second-order phase transition is understood as a phenomenon where a Stokes and anti-Stokes phenomenon occurs simultaneously. The order of the phase transition is determined by how saddles collide at the critical point. In addition, the phase transition accompanies an infinite number of Stokes phenomena due to the supersymmetry. These features are appropriately mapped to the Borel plane structures as the resurgence theory expects. Given the lessons from the SQED, we provide a more general discussion on the relationship between the resurgence and phase transitions. In particular, we show how the information on the phase transition is decoded from the Borel resummation technique. YITP-21-13, KUNS-2859 * toshiaki.fujimori018(at)gmail.com † masazumi.honda(at)yukawa.kyoto-u.ac.jp ‡ skamata11phys(at)gmail.com § tatsuhiromisumi(at)gmail.com ¶ norisuke.sakai(at)gmail.com ‖ t.yoda(at)gauge.scphys.kyoto-u.ac.jp

We consider the one-dimensional massive Thirring model formulated on the lattice with staggered fermions and an auxiliary compact vector (link) field, which is exactly solvable and shows a phase transition with increasing the chemical potential of fermion number: the crossover at a finite temperature and the first order transition at zero temperature. We complexify its path-integration on Lefschetz thimbles and examine its phase transition by hybrid Monte Carlo simulations on the single dominant thimble. We observe a discrepancy between the numerical and exact results in the crossover region for small inverse coupling β and/or large lattice size L, while they are in good agreement in the lower and higher density regions. We also observe that the discrepancy persists in the continuum limit to keep the temperature finite and it becomes more significant toward the low-temperature limit. This numerical result is consistent with our analytical study of the model and implies that the contributions of subdominant thimbles should be summed up in order to reproduce the first order transition in the low-temperature limit.

We study a resurgence structure of a quantum field theory with a phase transition to uncover relations between resurgence and phase transitions. In particular, we focus on three-dimensional N = 4 supersymmetric quantum electrodynamics (SQED) with multiple hypermultiplets, where a second-order quantum phase transition has been recently proposed in the large-flavor limit. We provide interpretations of the phase transition from the viewpoints of Lefschetz thimbles and resurgence. For this purpose, we study the Lefschetz thimble structure and properties of the large-flavor expansion for the partition function obtained by the supersymmetric localization. We show that the second-order phase transition is understood as a phenomenon where a Stokes and anti-Stokes phenomenon occurs simultaneously. The order of the phase transition is determined by how saddles collide at the critical point. In addition, the phase transition accompanies an infinite number of Stokes phenomena due to the supersymmetry. These features are appropriately mapped to the Borel plane structures as the resurgence theory expects. Given the lessons from the SQED, we provide a more general discussion on the relationship between the resurgence and phase transitions. In particular, we show how the information on the phase transition is decoded from the Borel resummation technique. YITP-21-13, KUNS-2859 * toshiaki.fujimori018(at)gmail.com † masazumi.honda(at)yukawa.kyoto-u.ac.jp ‡ skamata11phys(at)gmail.com § tatsuhiromisumi(at)gmail.com ¶ norisuke.sakai(at)gmail.com ‖ t.yoda(at)gauge.scphys.kyoto-u.ac.jp

We discuss the non-perturbative contributions from real and complex saddle point solutions in the CP 1 quantum mechanics with fermionic degrees of freedom, using the Lefschetz thimble formalism beyond the gaussian approximation. We find bion solutions, which correspond to (complexified) instanton-antiinstanton configurations stabilized in the presence of the fermonic degrees of freedom. By computing the one-loop determinants in the bion backgrounds, we obtain the leading order contributions from both the real and complex bion solutions. To incorporate quasi zero modes which become nearly massless in a weak coupling limit, we regard the bion solutions as well-separated instanton-antiinstanton configurations and calculate a complexified quasi moduli integral based on the Lefschetz thimble formalism. The non-perturbative contributions from the real and complex bions are shown to cancel out in the supersymmetric case and give an (expected) ambiguity in the non-supersymmetric case, which plays a vital role in the resurgent trans-series. For nearly supersymmetric situation, evaluation of the Lefschetz thimble gives results in precise agreement with those of the direct evaluation of the Schrödinger equation. We also perform the same analysis for the sine-Gordon quantum mechanics and point out some important differences showing that the sine-Gordon quantum mechanics does not correctly describe the 1d limit of the CP N -1 field theory of R × S 1 . * Electronic address: toshiaki.fujimori018(at)gmail.com † Electronic address: skamata(at)rikkyo.ac.jp ‡ Electronic address: misumi(at)phys.akita-u.ac.jp § Electronic address: nitta(at)phys-h.keio.ac.jp ¶ Electronic address: norisuke.sakai(at)gmail.com Contents I. Introduction II. CP N -1 Quantum Mechanics from 2d field theory A. 2d CP N -1 non-linear sigma model B. CP N -1 quantum mechanics C. CP 1 quantum mechanics with fermion and supersymmetry III. Bion saddle points and one-loop approximation A. Real bion solution B. Complex bion solution C. One-loop determinant in bion backgrounds D. Normalizable quasi zero modes around bion configurations IV. Bion contributions and quasi moduli integral A. Lefschetz thimble formalism B. Quasi-moduli Integral in sine-Gordon quantum mechanics C. Bion contributions in CP 1 model V. Comments on bion contributions in sine-Gordon model VI.

A. Quantum Mechanics on Riemannian manifolds B. Real multiplet: generating function and Lefschetz thimble C. Chiral multiplet: generating function and Lefschetz thimble III. Resurgence Structure in the Sine-Gordon Quantum Mechanics A. Sine-Gordon quantum mechanics B. Exact results around the SUSY and QES points 1. Expansion around the SUSY point 2. Expansion around the QES points C. Multi-bion solutions and semiclassical contributions 1. Multi-bion solutions 2. Multi-bion contributions and Quasi-moduli integral 3. Comparison with the exact results IV. Resurgence Structure in CP N -1 quantum mechanics A. CP N -1 quantum mechanics B. Near-SUSY exact results C. Bion solutions and semiclassical contributions 1. Embedding CP 1 bion solutions 2. Semi-classical contribution of bion saddle points 3. Multi-bion contributions D. Quasi-exact solvability of CP N -1 quantum mechanics V.

For a non-empty set X, the collection T op(X) of all topologies on X sits inside the Boolean lattice P(P(X)) (when ordered by set-theoretic inclusion) which in turn can be naturally identified with the Stone space 2 P(X) . Via this identification then, T op(X) naturally inherits the subspace topology from 2 P(X) . Extending ideas of Frink (1942), we apply lattice-theoretic methods to establish an equivalence between the topological closures of sublattices of 2 P(X) and their (completely distributive) completions. We exploit this equivalence when searching for countably infinite compact subsets within T op(X) and in crystalizing the Borel complexity of T op(X). We exhibit infinite compact subsets of T op(X) including, in particular, copies of the Stone-Čech and one-point compactifications of discrete spaces.

In this paper, the concepts of C -ideal are defined and explored the various properties C -ideals in posets. The equivalent conditions for an ideal to be a C -ideal is obtained. Further the relations between strongly prime ideals and C -ideals are discussed.

Le champ disciplinaire de notre travail est L’analyse des concepts formels. Dans la gestion
des Bases de données, on est très souvent amenés à regrouper les informations tout en cherchant
à gagner en espace mémoire d’où le terme Généralisation des attributs. Cette Généralisation
sur les attributs n’est pas toujours bénéfique quant à la réduction de la taille du treillis de concepts
d’où l’importance d’introduire la notion de Mesure de similarité qui permet de savoir quant est
ce que deux attributs peuvent être mis en commun sans autant augmenter la taille du treillis de
concept. Cette étude présente une application particulière de la généralisation existentielle sur la
base de données Mushroom.
Mots clés :
Concept formel ; Généralisation existentielle ;  Généralisation universelle ;
Mushroom

Cahiers de topologie et géométrie différentielle catégoriques, tome 35, n o 4 (1994), p. 309-319 <http © Andrée C. Ehresmann et les auteurs, 1994, tous droits réservés. L'accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l'accord avec les conditions générales d'utilisation (). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

In their seminal paper Birkhoff and von Neumann revealed the following dilemma : "... whereas for logicians the orthocomplementation properties of negation were the ones least able to withstand a critical analysis, the study of mechanics points to the distributive identities as the weakest link in the algebra of logic." In this paper we eliminate this dilemma, providing a way for maintaining both. Via the introduction of the "missing" disjunctions in the lattice of properties of a physical system while inheriting the meet as a conjunction we obtain a complete Heyting algebra of propositions on physical properties. In particular there is a bijective correspondence between property lattices and propositional lattices equipped with a so called operational resolution , an operation that exposes the properties on the level of the propositions. If the property lattice goes equipped with an orthocomplementation, then this bijective correspondence can be refined to one with propositional lattices equipped with an operational complementation , as such establishing the claim made above. Formally one rediscovers via physical and logical considerations as such respectively a specification and a refinement of the purely mathematical result by on injective hulls of meet-semilattices. From our representation we can derive a truly intuitionistic functional implication on property lattices, as such confronting claims made in previous writings on the matter. We also make a detailed analysis of disjunctivity vs. distributivity and finitary vs. infinitary conjunctivity, we briefly review the Bruns-Lakser construction and indicate some questions which are left open.

In this paper intuitionistic topological system and its properties have been introduced. Categorical interrelationships among Heyting algebra, Gödel algebra, Esakia space and proposed intuitionistic topological systems have also been studied in details.

In this paper intuitionistic topological system and its properties have been introduced. Categorical interrelationships among Heyting algebra, G\&quot;odel algebra, Esakia space and proposed intuitionistic topological systems have also been studied in details.

We show that the karoubian envelope of the category of correspondances is equivalent to the category of sup-compatible mappings between complete completly distributive lattices. On sait que l’enveloppe karoubienne per met de ramener les notions semi-fonctor ielles à des notions fonctorielles, et en particulier produit des catégories car tésiennes fer mées à partir de catégories faiblement cartésiennes fermées, ce qui est utilisé en théorie des dolnaines pour obtenir des modèles de la logique linéaire. Voir [2], [3]. On sait aussi que la modélisation et la manipulation algébrique de systèmes informatiques fait appel à des correspondances de Galois analysables et génér alisables en termes de transversalité de correspondances difonctionnelles. Voir [9]. Enfin on sait que Raney a établi un lien étroit entre les &quot;tight&quot; correspondances de Galois et les treillis co1plètement distributifs, qui sont des treillis continus particuliers. Voir [5, 6, 7] et [1]. Le lien entre ces diver...

We construct a countable set \( S \) that possesses the essential properties of the real numbers: it is everywhere dense in \( \mathbb{R} \) and Dedekind complete, with no gaps. Each element of \( S \) is part of a unique triplet \( (a_\alpha, x_\alpha, b_\alpha) \), where \( a_\alpha \) and \( b_\alpha \) are rational endpoints, and \( x_\alpha \) is a real number confined between them. By construction, every rational pair \( (a_\alpha, b_\alpha) \) is used, but each is used exactly once, ensuring the countability of \( S \) by prohibiting multiple \( x_\alpha \) values within the same interval. This restriction does not alter \( S \)'s density (it remains everywhere dense in \( \mathbb{R} \)) or its Dedekind completeness (no gaps are introduced, as no numbers are excluded), thereby preserving properties that make \( S \) equivalent to the set of all real numbers.

Through rigorous proofs, we demonstrate that \( S \) encompasses all real numbers by recognizing each within a nested sequence of intervals, corresponding to an element \( x_\alpha \) in \( S \). Furthermore, we establish that any Dedekind cut in \( S \) does not produce gaps, thereby confirming \( S \) as a complete and countable representation of the real line.

In the present paper we construct compact spaces satisfying the countable chain condition and not carrying a strictly positive measure. Our spaces are completely different from already known examples, as it follows from special chain conditions established for them. Spaces satisfying property (*) and not carrying strictly positive family of measures, of given size, are also constructed.

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les classes. Les deux algorithmes ont été implémentés dans un environnement commun afin de comparer leurs performances. Les résultats d'expériences sont rapportés et analysés. ABSTRACT. Many algorithms are proposed for building class hierarchies from the specification of their properties. Among those, the algorithms proposed in ] and [Godin 1995a] preserve the Galois sub-hierarchy of the relationship between the classes and their properties. Furthermore, the algorithms are incremental and can therefore incorporate a new class within an existing hierarchy. The result is a hierarchy that guarantees the maximal factorization of the properties and conformity to the specialization relationship between classes. Both algorithms are implemented in a common environment and their performance is compared. The results of experiments are reported and analyzed. MOTS-CLÉS : sous-hiérarchie de Galois, algorithme, conception de hiérarchie de classes

By a semiring we understand a commutative semiring with nonzero identity. The notions of ω-absolutely (semiprime-) irreducible ideals in a semiring R are introduced and we prove that the prime spectrum Spec(R )o fR is a P-space if and only if every prime ideal of R is ω-absolutely semiprime-irreducible. Also, Spec(R) is an Artin space totally-ordered by inclusion if and only if every (semiprime) ideal of R is ω-absolutely semiprime-irreducible if and only if the family Spec(R) is well-ordered by inclusion. Further, we characterize when every ideal of R is ω-absolutely irreducible as well as when every (semiprime) prime ideal of R is ω-absolutely irreducible.

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Set open topology Topology of uniform convergence on a family of sets Let X and Y be two topological spaces and C(X, Y) be the set of all continuous functions from X into Y. A criterion for the coincidence of the set open and uniform topologies on C(X, Y) is given. Also the comparison of the set open topologies on C(X, Y) for various families α is studied.

Quasi-lattices are introduced in terms of 'join' and 'meet' operations. It is observed that quasi-lattices become lattices when these operations are associative and when these operations satisfy 'modularity' conditions. A fundamental theorem of homomorphism proved in this article states that a quasi-lattice can be mapped onto a lattice when some conditions are satisfied.

Non-classical generalizations of classical modal logic have been developed in the contexts of constructive mathematics and natural language semantics. In this paper, we discuss a general approach to the semantics of non-classical modal logics via algebraic representation theorems. We begin with complete lattices L equipped with an antitone operation ¬ sending 1 to 0, a completely multiplicative operation □, and a completely additive operation ◊. Such lattice expansions can be represented by means of a set X together with binary relations ◁, R, and Q, satisfying some first-order conditions, used to represent (L, ¬), □, and ◊, respectively. Indeed, any lattice L equipped with such a ¬, a multiplicative □, and an additive ◊ embeds into the lattice of propositions of a frame (X, ◁, R, Q). Building on our recent study of fundamental logic, we focus on the case where ¬ is dually self-adjoint (a ≤ ¬b implies b ≤ ¬a) and ◊¬a ≤ ¬□a. In this case, the representations can be constrained so that R = Q, i.e., we need only add a single relation to (X, ◁) to represent both □ and ◊. Using these results, we prove that a system of fundamental modal logic is sound and complete with respect to an elementary class of bi-relational structures (X, ◁, R).

In our previous papers, together with J. Paseka we introduced so-called sectionally pseudocomplemented lattices and posets and illuminated their role in algebraic constructions. We believe that-similar to relatively pseudocomplemented lattices-these structures can serve as an algebraic semantics of certain intuitionistic logics. The aim of the present paper is to define congruences and filters in these structures, derive mutual relationships between them and describe basic properties of congruences in strongly sectionally pseudocomplemented posets. For the description of filters both in sectionally pseudocomplemented lattices and posets, we use the tools introduced by A. Ursini, i.e. ideal terms and the closedness with respect to them. It seems to be of some interest that a similar machinery can be applied also for strongly sectionally pseudocomplemented posets in spite of the fact that the corresponding ideal terms are not everywhere defined.

For an algebraic structure sJ = (A, F, R) or type r and a set S of open formulas of the first order language L(T) we introduce the concept of S-closed subsets of s/. The set 'tfs (.c/) of all E-closed subsets forms a complete lattice. Algebraic structures sj, 8$ of type r are called E-isomorphic if 'rfsf.c/) = ^(Sl). Examples of such S-closed subsets are e.g. subalgebras of an algebra, ideals of a ring, ideals of a lattice, convex subsets of an ordered or quasiordered set etc. We study E-isomorphic algebraic structures in dependence on the properties of E.

Throughout this paper po-group will mean partially ordered abelian group. A subgroup H of a po-group G is an o-ideal if H is a convex, directed subgroup of G. A subgroup M of G is a value of 0 Φ g e G if M is an o-ideal of G that is maximal without g. Let <^€ (g) = {M Q G | M is a value of #} and ^^*(g)= f\^st(g). Two positive elements a y beG are pseudo disjoint (p-dis joint) if α€^C*(6) and &e^^*(α), and G is a pseudo-lattice ordered group (pl-group) if each g e G can be written g = a-b where a and b are ^-disjoint. The main result of § 2 shows that every pl-group G is a Riesz group. That is, G is semiclosed ing ^ 0 implies g ^ 0 for all #eG and all positive integers n), and G satisfies the Riesz interpolation property; if, whenever x u , # TO , y u-,y n are elements of G and #» ^ ?/i for 1 ^ ΐ ^ m, 1 ^ i ^ n, then there is an element 26G such that Xi ^ z S yj.

We first present a Priestley-style dualitiy for the classes of algebras that are the algebraic counterpart of some congruential, finitary and filter-distributive logic with theorems. Then we analyze which properties of the dual spaces correspond to properties that the logic might enjoy, like the deduction theorem or the existence of a disjunction.

The paper is the Mizar encoding of the chapter 0 section 3 of [9] In the paper the following concept are defined: Galois connections, Heyting algebras, and Boolean algebras. the notation and terminology for this paper. 1. PRELIMINARIES Let L 1 , L 2 be non empty 1-sorted structures and let f be a map from L 1 into L 2. Let us observe that f is one-to-one if and only if: (Def. 1) For all elements x, y of L 1 such that f (x) = f (y) holds x = y. Next we state the proposition (1) Let L be a non empty 1-sorted structure and f be a map from L into L. If for every element x of L holds f (x) = x, then f = id L. Let L 1 , L 2 be non empty relational structures and let f be a map from L 1 into L 2. Let us observe that f is monotone if and only if: (Def. 2) For all elements x, y of L 1 such that x ≤ y holds f (x) ≤ f (y). The following four propositions are true: (2) Let L be an antisymmetric transitive relational structure with g.l.b.&#39;s and x, y, z be elements of L. If x ≤ y, then x z ≤ ...

Abs tract Over the last thirty years much progress has been made in the investigation of the lattice of uniformities on a set X. In particular, P elant, Reiterman, Rodl and Simon have published several articles concerning anti-atoms and complements in this lattice. The aim of this dissertation is to begin a similar investigation into the lattice of quasi-uniformi t ies 8 (X) on a set X. It starts off with a summary of results obtained for the lattice of topologies on X, which, having been studied in great detail in the past, is intended as an example as to what may be achieved with 8(X). An exposit ion of the lattice of uniformities is then given. \Ne conclude by commencing an investigation into the lattice of quasi-uniformiti es on X. Where possible, results obtained for the lattice of uniformities are generalized to 8(X) , and some original results for 8 (X) are also presented. First and foremost , I would like to thank my supervisor, Prof. H.-P. Kunzi , for all his help and support over the last year and a half. I have greatly benefitted from his expertise in this area of mathematics, and I appreciate the countless times that he has pointed me in the right direction. In addition to this, I would like to thauk the entire teaching staff in the Department of Mathematics and Applied Mathematics at the University of Cape Town. From as early as the first and second years of my bachelors degree, their encouragment and love for the subject inspired me to do my very best. In my third and fourth years of study their enthusiasm and excellent teaching abilities continued to inspire me, and made an already interresting subject that much more enjoyable. I would also like to thank Assoc. Prof. B. D. Hahn, who t the last moment made available to me a scholarship for my honours degree without which I would not have been here today. Last but not least, I would also like to thank the RF for my Prestigious Scholarhip, which made st dying towards my masters degree possible.

In this article, class of all bounded commutative self distributive modal BE-algebras mBE S is introduced. Finally, we show that the class of all bounded commutative self distributive modal BEalgebras, mBE S , coincides with the class of all topological Heyting algebras.

Subspaces of 2 P(X) Top(X) Model-theoretic conditions for compactness A topology on a nonempty set X specifies a natural subset of P(X). By identifying P(P(X)) with the totally disconnected compact Hausdorff space 2 P(X) , the lattice Top(X) of all topologies on X is a natural subspace therein. We investigate topological properties of Top(X) and give sufficient model-theoretic conditions for a general subspace of 2 P(X) to be compact.

In Huang and Weng (2004), Huang and Weng introduced pooling spaces, and constructed pooling designs from a pooling space. In this paper, we introduce the concept of pooling semilattices and prove that a pooling semilattice is a pooling space, then show how to construct pooling designs from a pooling semilattice. Moreover, we give many examples of pooling semilattices and thus obtain the corresponding pooling designs.

In this paper, we provide an alternative description of the duality result for distributive lattices and coherent locales using ultraposet. In particular, we show that there are fully faithful embeddings from the opposite of the category of distributive lattices into the category of ultraposets with ultrafunctors, and from the category of coherent locales into the category of ultraposets with left ultrafunctors. We also define the notion of zero-dimensional ultraposets, which characterises the essential image of these embeddings.