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.

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)

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)

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)

Georgians' acquaintance with Islam was with the first Arab raids. From the first Muslim Arab domination, Georgians started to become Muslims with cultural interaction. Islam spread especially in Eastern Georgia during the time of Muslim Arabs, Seljuks and Mongols. The spread of Islam in Western Georgia started with the Ottomans. The Ottoman Empire's contact with Georgia begins with the conquest of Trabzon by Fatih Sultan Mehmed. When the Ottomans contacted the region, the geography of Georgia was divided into small kingdoms. (...) With the conquest of Trabzon, the Georgian border was reached and in 1479, Acaristan, Borcka and Maçahel surroundings joined the Ottoman lands. On the one hand, Sukhumi was captured from the north. Then, continuing to advance in the region, Yavuz Sultan Selim paid tribute to the Kingdom of Güryel and Imeret (Açıkbaş) in 1508 during his governorship of Trabzon. Thus, the Ottoman borders had reached beyond the Black Sea coast into Western Georgia. As a result of the increasing relations around Adjaria, which came under Ottoman rule, Islamization began to spread rapidly among the people. Contrary to what is claimed, the spread of Islam in the region did not spread from the ruling strata to the lower, but from the popular strata to the ruling strata. This process was surprisingly fast. In the process of Islamization, many mosques, masjids and schools were built in Adjara, especially with the help of the people. The foundations operating in the region and the mosques and masjids serving under them show that, contrary to the claims of Russian and Georgian historians, Islamization took place rapidly at the beginning of the 17th century in Adjara. In this process, many mosques, masjids and schools were built in the region. In this research, based on the numerous Hurufat records belonging to Adjara, Batumi, Gönye, Çürüksu and Maçahel, the Islamization process will be evaluated by making a social reading with the data obtained from the religious structures established in the rural settlements of the Adjara geography and their officials. Evaluations were made on the Islamization process in Adjara by making a social reading. (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)