Quantic Basis of Filter Theory

Ulrich Höhle
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Ulrich Höhle: Bergische Universität, Fachbereich Mathematik

Chapter Chapter 4 in Many Valued Topology and its Applications, 2001, pp 107-143 from Springer

Abstract: Abstract In this chapter we use commutative quantales as lattice–theoretic basis for the development of many valued filter theory. First we recall some basic definitions and facts of the theory of quantales (cf. [91]). A triple Q = (L, ≤, *) is called a commutative quantale iff (L, ≤) is a complete lattice and (L, *) is a commutative semigroup such that * is distributive over arbitrary joins in L. Since for every α ∈ L the map α * _ preserves arbitrary joins in L, it has a right adjoint denoted by α → _. Thus $$ \alpha * \gamma \leqslant \beta \Leftrightarrow \gamma \leqslant \alpha \to \beta $$

Keywords: Galois Connection; Filter Theory; Quantic Basis; Complete Boolean Algebra; Underlying Lattice (search for similar items in EconPapers)
Date: 2001
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4615-1617-0_5

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DOI: 10.1007/978-1-4615-1617-0_5

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