Lattices

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Lattices, Page 1 of 1

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After the essentials of the theory of transfinite numbers, investigated in the previous chapter, we now present a few topics from another specialization of the concept of order: lattice theory. Our aim is to provide lattice-theoretical tools broadly used in various fields of mathematics and in particular some latticetheoretical background necessary to universal algebra.

Our way from posets to lattices passes through the intermediate level of semilattices; in §1 the relationship between the order-theoretical and the algebraic definition of these concepts is described in the categorical language. Several characterizations of the important subclasses of modular, distributive and Boolean lattices are given in §2. Complete lattices are dealt with in §3, with emphasis on complete Boolean algebras, which appear in several mathematical contexts; compactly generated lattices are also introduced. The next section is devoted to closure operators, an order-theoretical tool which is applied in almost all branches of mathematics to construct the substructures of a set endowed with a certain structure. Galois connections, intimately related to closure operators, are dealt with in the last section.

There is a huge literature on lattices and their applications. For lattices in general we refer the reader e.g. to Birkho↵ [1967], Blyth [2005], Crawley and Dilworth [1973], Davey and Priestley [1990], Dubreil-Jacotin, Lesieur and Croisot [1953], Gr¨atzer [1978a], Roman [2008], Padmanabhan and Rudeanu [2008], Sz´asz [1963]. For Boolean algebras see e.g. Dwinger [1961], Halmos [1963], Monk and Bonnet [1989], Rudeanu [1973], [2001], Sikorski [1964], Vladimirov [1969]. For Boolean algebras in other fields see e.g. Halmos [1950] for measure theory, N¨obeling [1954], Ponasse and Carrega [1979], Vaidyanathaswamy [1960] for topology, and Kappos [1969], Onicescu and Cuculescu [1976] for probability theory. Last but not least, a branch of algebra which owes much to lattice theory is universal algebra. The reader is referred e.g. to Burris and Sankappanavar [1981], Gr¨atzer [1978b], and Pierce [1968].