We present a new topological representation and Stone-type duality for general lattices. The dual objects of lattices are triples (X, Þ, Y), where X, Y are the filter and ideal spaces of the lattice, endowed with a natural topology, and Þ is a relation from X to Y. PreliminariesThis paper is largely motivated by the need to develop the second author's Gaggle Theory (Dunn [4]), aiming at providing a systematic procedure for the semantic analysis of various logical calculi via suitable representation results of their associated Lindenbaum algebras. As various calculi drop the distribution axiom (of meets over joins) the project seemed to stumble on a suitable representation result for general lattices. We rectify this situation with the present report (see also Hartonas [6] and Hartonas [7]). We are of course aware that Urquhart [13] has already proposed a topological representation for general lattices. In fact Allwein and the second author (see [1]) exploited Urquhart's representation theorem to provide Kripke semantics for Linear Logic. Allwein and the first author (see [2]) extended Urquhart's topological representation to a full duality result including a duality for congruences and sublattices. Independently, Hartung [8] extended Urquhart's representation to a full duality working within R. Wille's framework of concept lattices [14].We follow a different course here mainly for the reason that Urquhart's representation necessarily leads to a 3-valued semantic. We establish a new representation result, abstracting the essentials from both Urquhart's representation of general lattices and from Goldblatt's topological representation of ortholattices [5]. We prove functoriality of the representation and extend to a full Stone-type duality Presented by W. Blok.

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This paper is an investigation into the logic of information ow. The basic perspective is that logic ows in virtue of constraints (as in 7]), and that constraints classify channels connecting particulars (as in 8]). In this paper we explore some logics intended to model reasoning in the case of idealized information ow, that is, where the constraints involved are exceptionless. We look at this as a step toward the far more challenging task of understanding the logic of imperfect information ow, that is where the constraints admit of exceptional connections. This paper continues and ampli es work presented by the same authors in 10].

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