Universal specialization semilattices

Abstract

We devise exact conditions under which a contact semilattice can be semilattice embedded into a Boolean algebra with an overlap contact relation, equivalently, into a distributive lattice with additive contact relation.Contact algebras are Boolean algebras endowed with a further contact binary relation. Contact algebras play an important role in region-based theory of space [1]. They might be seen as a point-free algebraization of proximities, a useful generalization of the notion of topology [2,7]. See the quoted sources for more references and details.Ivanova [4] provides arguments suggesting the naturalness of contact joinsemilattices. In a nutshell, if a is a region of space, its complement is possibly not well-defined, since it is dependent on the universe in which we consider a. Moreover, if one considers large but limited regions of space, both complementation and meet might turn out to be inappropriate.Parallel arguments in favor of the use of the join operation only-without meet or complementation-are presented in [5]. The arguments are motivated by the fact that if f → is the image function from P(X) to P(Y ) associated to some function f : X → Y , then f → preserves unions but not necessarily intersections or complements.At a very elementary level, the use of the join operation alone is also suggested by the fact that it is the only operation appearing in the axioms for a contact relation; see Definition 1 below. As another advantage, generally, representation theorems for semilattices do not need the axiom of choice.A standard definition of contact in a semilattice guarantees that the semilattice can be (semilattice-) embedded into a complete lattice, which can be chosen to have trivial contact [6]. On the other hand, the standard definition is too weak to guarantee that a contact semilattice can be (semilattice-) embedded into a distributive lattice. Here we find necessary and sufficient conditions under which the above embeddability can be obtained. In this way we get an equivalent axiomatization for contact join-semilattices in the sense of [4].Definition 1. In the present note semilattices shall always be intended in the sense of join semilattices with a minimum element 0. The existence of 0 is assumed only in order to simplify some statements; see Remark 6 in [6].