Lattices

Abstract

In the previous chapter we have seen that the arithmetical properties of elements of formal systems may be described in operational structures. Operations may serve to generate new elements from a given set of basic elements, and thus we may view an operational or an algebraic structure naturally as a syntactic system which generates elements in a formally precise way. The relation of this dynamic conception of such systems and the linguistic notion of a grammar which generates strings as elements of a natural or formal language will be explored in much more detail in Part E. The present chapter is concerned with certain ordering relations between elements of systems or domains of objects and the order-theoretic or ‘topological’ properties of such ordered structures. We will see that the concepts introduced in this chapter provide a universal perspective on set theory and algebra in which important correlations between the two mathematical theories can be insightfully described. Recently linguistic applications of lattices have been made primarily to semantic topics such as plural NPs, mass terms and events, using the ordering relations to structure the domains of an interpretation of a language. The potential usefulness in linguistics of syntactic applications of lattice theory is explored in research on feature systems, for instance. In this chapter we will introduce lattice theory without paying attention to any particular linguistic applications or motivations.