Introduction to Lattice Theory
In this chapter we introduce the notion of lattices and obtain techniques for obtaining extremal solutions of inequations involving operations over lattices. Our motivation for studying these concepts and techniques stems from our interest in solving a system of inequations involving operations over the lattice of languages. Results presented in this chapter have a “primal” and a “dual” version. We only prove the primal version, as the dual version can be proved analogously.
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- For a general introduction to the subject oflattice theory we refer the readers to Davey-Priestley [DP90]. One of the early results on existence of fixed points of a monotone function is due to Knaster-Tarski [Tar55]. Lassez-Nguyen-Sonenberg [LNS82] provide a nice historical account of this and other fixed point theorems. Our treatment of inequations is influenced by Dijkstra-Scholten [DS90]; in particular, the terms conjugate and converse of a function have been borrowed from there. The existence and computation of extremal solutions of a system of inequations over lattices is reported in Kumar-Garg [KG94b].Google Scholar