H.A. Priestley. Representation of distributive lattices. Bull. London Math. Soc., 2:186--190, 1970.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....will be a set of events ordered by time. State spaces generalize bc domains (Scott domains or bounded complete algebraic cpo s) Gun92] while their complementary event spaces generalize Winskel s event structures [Win86] This particular duality is one small fragment of Birkhoff Stone duality [Bir33, Sto36, Sto37, Pri70], with the partial distributive lattices alluded to in the section on measurement constituting a much larger fragment. A state space can be thought of as a representation of behavior somewhere in between a formal language 3 We shall see later that the terms dual, complement, and converse form a ....

H.A. Priestley. Representation of distributive lattices. Bull. London Math. Soc., 2:186--190, 1970.

....notion of infinite meets and joins is expressed formally as the Stone topology. In the case of posets and distributive lattices, putting the Stone topology on the left, duality 48, yields what Stone described obscurely [Sto37] but which was much later described more naturally by Priestley [Pri70] as ordered Stone spaces. In the special case where the order is discrete (but still duality 48) we have Stone spaces on the left and Boolean algebras on the right. When we add the Stone topology to the bounded distributive lattices instead we still have the same posets on the left but now we have ....

H.A. Priestley. Representation of distributive lattices. Bull. London Math. Soc., 2:186--190, 1970.

.... distributive lattice, and vice versa, making Fpos dual to FDL (isomorphic to FDL op ) just as for Fchn and Fchn01, shown by Birkhoff in 1933 [Bir33] Stone [Sto37] a little later found one extremal extension of this duality to infinite objects, much later characterized nicely by Priestley [Pri70] as the duality of partially ordered Stone spaces and distributive lattices; the other extremal extension is between posets and profinite distributive lattices, for whose definition, history, and many further extensions see Johnstone [Joh82, Ch.VII] Complement continues to be 2 A . All three of ....

H.A. Priestley. Representation of distributive lattices. Bull. London Math. Soc., 2:186-- 190, 1970.

.... are realizable as Chu spaces [Pra93, Pra95b] as are topological spaces [LS91] These representations can be combined to represent topological relational structures such as topological groups, topological vector spaces (the main autonomous category studied in [Bar79] ordered Stone spaces [Pri70], and so on. More recently (note in preparation) we have shown that every small concrete category C is realizable in ChuK where K is the disjoint union of the underlying sets of the objects of C. 3 Process 3.1 Interaction of events and states This section gives the computational or process ....

H.A. Priestley. Representation of distributive lattices. Bull. London Math. Soc., 2:186-- 190, 1970.

....automaton of size 2 n . This duality can now be spiced up in two essentially orthogonal ways, a combinatorial one due to G. Birkhoff [Bir33] and a topological one due to M. Stone [Sto36] Remarkably, that these ways were orthogonal passed unnoticed until pointed out by H. Priestley in 1970 [Pri70]. Keeping everything finite, Birkhoff duality generalizes the discrete schedules to partially ordered schedules, and generalizes the automata to distributive lattices. On the other hand, keeping the automata Boolean, Stone duality generalizes everything to the infinite case. In order to allow ....

H.A. Priestley. Representation of distributive lattices. Bull. London Math. Soc., 2:186--190, 1970.

....strictly less than x is itself strictly less than x. Stone s original duality put the Stone topology on the set [Sto36] or poset [Sto37] side. In the latter case this view of a poset topologized with a Stone topology was not apparent from Stone s paper and was found much later by Priestley [Pri70]. In the case of the distributive lattice 2 X op the compact elements are exactly those subsets Y of X for which there exists an element y 2 Y such that no proper subset of Y containing y belongs to 2 X op . In terms of states, these are the states containing an event absent from all ....

H.A. Priestley. Representation of distributive lattices. Bull. London Math. Soc., 2:186-- 190, 1970.

....CATEGORY OF CHU SPACES 33 this is a categorical duality. The involution P = P for any Chu space P follows from the definition. The dual of a pdlat can be written directly as (P; L) L ; P ) this is the dual pdlat representation. Here L is the set of all filters of L [Pri70], partially ordered by set inclusion. A filter is an upwards closed subset of the lattice which is closed under all meets. P is the set of all upwards closed subsets of P , and it forms a lattice with the operations of set union and intersection. L embeds in P by sending each element ....

H.A. Priestley. Representation of distributive lattices. Bull. London Math. Soc., 2:186--190, 1970.

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