Among these are state minimizations, Boolean algebra, and switching algebra. In minimization, three approaches are normally used that are based on equivalence relations. Partial order relations are used today in constructions of Boolean algebra. In this paper we survey this important algebra from its...
Lattice theory evolved as part of algebra in the nineteenth century through the work of Boole, Peirce and Schrder, and in the first half of the twentieth century through the work of Dedekind, Birkhoff, Ore, von Neumann, Mac Lane, Wilcox, Dilworth, and others. In Semimodular Lattices, Manfr...
Prerequisites are kept to a minimum, but an introductory course in abstract algebra is highly recommended, since many of the examples are drawn from this area.\nThe book has an excellent choice of topics, including a chapter on well ordering and ordinal numbers, which is not usually found in...
boolean latticescrisp setsσ-algebralinear spacesrough setsquantum orthomodular latticesThe basic notions of posets and lattices are introduced with a wide presentation of examples. In particular, distributive and modular lattices are treated. With respect to the distributive behavior, we discuss Boolean ...
Let pL; θq be a sup-quasiordered set and let K : L{Eθ ÞÑ L be an arbitrary choice function. If for each two elements x, y P L: x _ y " Kpsupprxs, rysqq " Kprxs _ rysq, then the algebra pL; _q is a weakly idempotent semilattice, which we call upper weakly ...
The first two results are applications of Nies' theorem on the non-arithmeticity of the 1st order theory of the lattice of r.e. ideals on any effectively dense r.e. Boolean algebra. The theorem on degrees of interpretability relies on an adaptation of techniques leading to Nies' theorem....
It is an extension of Boolean algebra with an additional relation C called contact. There are some problems related to the motivation of the operation of Boolean complementation. Because of this operation is dropped and the language of distributive lattices is extended by considering as non-...
One of the most important practical applications and also one of the oldest applications of modern algebra, especially lattice theory, is the use of Boolean algebras in modeling and simplifying switching or relay circuits. This application will be descri
The book surveys and analyzes Garrett Birkhoff's concept of semimodularity and the various related concepts in lattice theory, and it presents theoretical results as well as applications in discrete mathematics group theory and universal algebra. The author also deals with lattices that are 'close' ...
Furthermore, we consider additional axioms for contact and the representation of those structures in topological spaces with richer structure. 展开 关键词: Theoretical or Mathematical/ Boolean algebra set theory topology/ distributive contact lattice topological representation Boolean algebra abstract algebra ...