This new edition of Introduction to Lattices and Order presents a radical reorganization and updating, though its primary aim is unchanged. The explosive development of theoretical computer science in recent years has, in particular, influenced the book's evolution: a fresh treatment of fixpoints testifies to this and Galois connections now feature prominently. An early presentation of concept analysis gives both a concrete foundation for the subsequent theory of complete lattices and a glimpse of a methodology for data analysis that is (...) of commercial value in social science. Classroom experience has led to numerous pedagogical improvements and many new exercises have been added. As before, exposure to elementary abstract algebra and the notation of set theory are the only prerequisites, making the book suitable for advanced undergraduates and beginning graduate students. It will also be a valuable resource for anyone who meets ordered structures. (shrink)

The techniques of natural duality theory are applied to certain finitely generated varieties of Heyting algebras to obtain optimal dualities for these varieties, and thereby to address algebraic questions about them. In particular, a complete characterisation is given of the endodualisable finite subdirectly irreducible Heyting algebras. The procedures involved rely heavily on Priestley duality for Heyting algebras.

Both syntactic and semantic solutions are given for the entailment problem of duality theory. The test algebra theorem provides both a syntactic solution to the entailment problem in terms of primitive positive formulae and a new derivation of the corresponding result in clone theory, viz. the syntactic description of $\operatorname{Inv(Pol}(R))$ for a given set R of finitary relations on a finite set. The semantic solution to the entailment problem follows from the syntactic one, or can be given in the form (...) of an algorithm. It shows, in the special case of a purely relational type, that duality-theoretic entailment is describable in terms of five constructs, namely trivial relations, intersection, repetition removal, product, and retractive projection. All except the last are concrete, in the sense that they are described by a quantifier-free formula. It is proved that if the finite algebra $\underline{M}$ generates a congruence-distributive variety and all subalgebras of $\underline{M}$ are subdirectly irreducible, then concrete constructs suffice to describe entailment. The concept of entailment appropriate to strong dualities is also introduced, and described in terms of coordinate projections, restriction of domains, and composition of partial functions. (shrink)

Anyone who has ever worked with a variety \ of algebras with a reduct in the variety of bounded distributive lattices will know a restricted Priestley duality when they meet one—but until now there has been no abstract definition. Here we provide one. After deriving some basic properties of a restricted Priestley dual category \ of such a variety, we give a characterisation, in terms of \, of finitely generated discriminator subvarieties of \. As an application of our characterisation, we (...) give a new proof of Sankappanavar’s characterisation of finitely generated discriminator varieties of distributive double p-algebras. A substantial portion of the paper is devoted to the application of our results to Cornish algebras. A Cornish algebra is a bounded distributive lattice equipped with a family of unary operations each of which is either an endomorphism or a dual endomorphism of the bounded lattice. They are a natural generalisation of Ockham algebras, which have been extensively studied. We give an external necessary-and-sufficient condition and an easily applied, completely internal, sufficient condition for a finite set of finite Cornish algebras to share a common ternary discriminator term and so generate a discriminator variety. Our results give a characterisation of discriminator varieties of Ockham algebras as a special case, thereby yielding Davey, Nguyen and Pitkethly’s characterisation of quasi-primal Ockham algebras. (shrink)

The paper investigates completions in the context of finitely generated lattice-based varieties of algebras. In particular the structure of canonical extensions in such a variety $${\mathcal {A}}$$ is explored, and the role of the natural extension in providing a realisation of the canonical extension is discussed. The completions considered are Boolean topological algebras with respect to the interval topology, and consequences of this feature for their structure are revealed. In addition, we call on recent results from duality theory to show (...) that topological and discrete dualities for $${\mathcal {A}}$$ exist in partnership. (shrink)

This paper illustrates how Priestley duality can be used in the transfer of an optimal natural duality from a minimal generating algebra for a quasi-variety to other generating algebras. Detailed calculations are given for the quasi-variety \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{I}\mathbb{S}\mathbb{P}(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{4} )$$ \end{document} of Kleene algebras and the quasi-varieties \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$B$$ \end{document}n of pseudocomplemented distributive lattices (n ≥ 1).

This paper illustrates how Priestley duality can be used in the transfer of an optimal natural duality from a minimal generating algebra for a quasi-variety to other generating algebras. Detailed calculations are given for the quasi-variety \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{I}\mathbb{S}\mathbb{P}(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{4} )$$ \end{document} of Kleene algebras and the quasi-varieties \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$B$$ \end{document}n of pseudocomplemented distributive lattices (n ≥ 1).

Guido de Ruggiero (1888-1948) was perhaps the greatest Italian intellectual historian in the twentieth century. He was a fierce champion of liberalism, an ardent opponent of Fascism, an insightful critic and interpreter of his contemporaries, and a formidable philosopher in his own right. Idealism & Experience: The Philosophy of Guido de Ruggiero comprises eight new critical essays, as well as English translations of five of de Ruggiero's most important shorter writings, which chart the development of his thought between 1914 and (...) 1946. Taken together, these essays reveal the range, richness and philosophical sophistication of de Ruggiero's ideas, enabling Anglophone readers to appreciate their enduring relevance in our own troubled times. (shrink)

Psychoactive drugs are being prescribed to millions of Americans at an increasing rate. In many cases these drugs are necessary in order to overcome debilitating emotional problems. Yet in other instances, these drugs are used to supplant, not supplement, interpersonal therapy. The process of overcoming emotional obstacles by introspection and the attainment of self knowledge is gradually being eroded via the gratuitous use of psychoactive medication in order to rapidly attain a release from the common problems that life inevitably presents (...) us with. In this paper, I argue that Kant’s formula of humanity, which maintains that persons ought never to treat others or themselves soley as a means to an end, proscribes this. Moreover, Kant argues that we have an imperfect duty of self development, and I argue that we fail to adhere to such a duty whenever we seek to evade the process of introspection and self knowledge in favour of the expedient results that drugs may provide us with as we attempt to overcome the emotional hurdles in our lives. (shrink)

Sections R1 to R3 attempt to take the sting out of hostile commentaries. Sections R4 to R5 engage Berlin and Kay and the World Color Survey to correct the record. Section R6 begins the formulation of a new theory of colour as an engineering project with a technological developmental trajectory. It is recommended that the colour space be abandoned.

In this target article the following hypotheses are discussed: (1) Colour is autonomous: a perceptuolinguistic and behavioural universal. (2) It is completely described by three independent attributes: hue, brightness, and saturation: (3) Phenomenologically and psychophysically there are four unique hues: red, green, blue, and yellow; (4) The unique hues are underpinned by two opponent psychophysical and/or neuronal channels: red/green, blue/yellow. The relevant literature is reviewed. We conclude: (i) Psychophysics and neurophysiology fail to set nontrivial constraints on colour categorization. (ii) Linguistic (...) evidence provides no grounds for the universality of basic colour categories. (iii) Neither the opponent hues red/green, blue/yellow nor hue, brightness, and saturation are intrinsic to a universal concept of colour. (iv) Colour is not autonomous. (shrink)

Nigeria is seen as a poverty-stricken society as about 70% of her estimated 138 million population live in abject poverty. She is a nation blessed with mineral resources like crude oil, limestone, iron ore, timber, coal and a host of other commodities. Yet, majority of her citizenry lives in grinding poverty. Nigeria has been rated one of the most corrupt nations in the world. In fact, corruption has constituted an albatross for the progress of the nation. To do theology (to (...) theologize) in Nigeria today and still wink at the dehumanizing condition of the masses will amount to playing the ostrich. Therefore, the Church in Nigeria has an urgent duty to manifest her witness of action, as a prophetic church, to address the problem of poverty in Nigeria. (shrink)

This comprehensive book covers the research, theory, policy and practice context of unusual reproduction using third parties. Olga Van den Akker details the psychological adaptation required to continuing changes in public opinion, advances in technologies and new legislations in surrogate motherhood and discusses their impact at an individual, societal and global level. She describes the competing interests and interactions between legal, organisational, personal, social, psychological and cultural issues in relation to biological and genetic surrogate and commissioning parenthood. This book is (...) intended for professionals, practitioners, academics and students interested in the complexities of unusual reproduction using multidisciplinary perspectives. (shrink)