T. Hiraguchi, On the dimension of partially ordered sets, Sci. Reports Kanazawa Univ. 1 (1951) pp. 77-94.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....other words, upward planar source sink graphs are the Hasse diagrams of planar lattices. The first paper shows that in the restricted case, L and R are easily maintained as the graph changes. The second paper shows under which updates the orderings remain maintainable in the general case. Kelly [41] has shown that for general planar graphs, the number of total orders needed to express the transitive closure as their intersection is unbounded. Our approach uses a different characterisation of the transitive closure. We maintain two orders (call them S and T for a moment) with the property ....

David Kelly. On the dimension of partially ordered sets. Discrete Mathematics, 35:135--156, 1981.

....2, see [13] for a description of the proof in [1] The 0 and 1 of the poset guarantee that every pair of elements will have a meet and a join, while uniqueness of meet and join holds because there can be no edge crossings. There are planar posets without 0 and 1 of arbitrary dimension (see [11]) that would not be planar with 0 and 1 element added; see Figure 1 (b) and (c) Figure 1 (a) is not planar, Figure 1 (b) is the planar 6 cycle, and Figure 1 (c) is the 6 cycle with 0 and 1 added, which is the non planar Boolean lattice on 3 elements. s s s s s s ....

D. Kelly (1981) On the dimension of partially ordered sets, Discrete Math. 35, pp. 135-156.

....R . Symbolically, u OE v u L v u R v; where we write OE for the transitive closure. The first paper shows that in the restricted case, L and R are easily maintained as the graph changes. The second paper shows under which updates the orderings remain maintainable in the general case. Kelly [8] has shown that for general planar graphs, the number of total orders needed to express the transitive closure as their intersection is unbounded. The present algorithm subsumes and extends the results from [17, 18] in that it removes the restrictions of both. To this end, we use a different ....

David Kelly, On the dimension of partially ordered sets, Discrete Mathematics 35 (1981), 135--156.

....set of S, i.e. the set of all subsets of S, partially ordered by set inclusion. This class of partial orders arises naturally in the context of protection groups. For example S can be the set of attributes whose subsets determine the compartments in military security policies [1, 2, 8] Komm [7] proved that the dimension of the subset partial ordering on 2 S is j S j. Since this partial order can be represented using j S j bits for each subset of S, the dimension approach is clearly not useful for this case. In our opinion this is the most important case excluded by ntrees. Clearly ....

Komm, H. "On the Dimension of Partially Ordered Sets." Amer. J. Math. 70:507520 (1948).

....The partial order on S is defined by setting (x 1 ; y x1 ) x 2 ; y x2 ) in S if and only if (1) x 1 x 2 in P , or (2) x 1 = x 2 and y x1 y x2 in Q x1 . Both dimension and fractional dimension are well behaved with respect to lexicographic sums. The statement for dimension is due to Hiraguchi [10], and the statement for fractional dimension is given in [1] PROPOSITION 2.3. Let S be the lexicographic sum of D = fY x : x 2 Xg over P = X; P ) Then (1) dim(S) maxfdim(P) max x2X dim(Y x )g. 2) fdim(S) maxffdim(P) max x2X fdim(Y x )g. A poset S is decomposable with respect to ....

....for fractional dimension. Let P = X; P ) be a poset and let S and T be disjoint subsets of X. When M is a linear extension of P , we say S is over T in M , and write S=T in M , when x y in M , for every (x; y) 2 inc(P) with x 2 S and y 2 T . The following elementary result is due to Hiraguchi [10]. 8 FRACTIONAL DIMENSION PROPOSITION 6.2. Let P = X; P ) be a poset and let C ae X be a chain. Then there exist linear extensions M 1 , M 2 of P with C= X Gamma C) in M 1 and (X Gamma C) C in M 2 . Recall that Dilworth s theorem [6] asserts that a poset P of width w can be partitioned into ....

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T. Hiraguchi, On the dimension of partially ordered sets, Sci. Rep. Kanazawa Univ. 1 (1951), 77--94.

....possibly high dimension partial orders arises in military security policies [1, 2, 9] Let S be the set of compartments whose subsets determine the categories and let 2 S be the power set of S, that is the set of all subsets of S. The dimension of the set inclusion partial order on 2 S is jSj [8]. Since this partial order can be represented using jSj bits for each subset of S, the dimension approach is clearly not useful for this case. Although some of the theory we develop (Section 4.1 in particular) is applicable partial orders of arbitrary dimension we are skeptical about whether the ....

Komm, H. "On the dimension of partially ordered sets." Amer. J. Math., Volume 70 (1948), 507-520.

....is actually applicable to partial orders of any dimension while the representation of Section 4.2 applies only to ntrees. Section 5 concludes the paper. 3 NTREES A fundamental result of dimension theory allows us to construct new partial orders from existing ones without increasing the dimension [6]. Let P and Q be partial orders on disjoint sets G and H respectively. Consider some u 2 G. The refinement of u in P into Q is the partial order P on the set (G Gamma fug) H formed by the union of the following sets of ordered pairs. 1. f(x; x 0 ) j (x; x 0 ) 2 P for all x; x 0 2 G ....

....while maintaining the same relationship between the new groups and other previously existing groups which the exploded group had. Refinement is a natural method for incrementally developing more detail in a top down manner. It is particularly important because of the following result. Theorem 2 [6] Let P and Q be partial orders on disjoint sets G and H respectively. Let u 2 G. If P is the refinement of u in P into Q, then dim(P 0 ) maxfdim(P) dim(Q)g. Proof. Let d = dim(P 0 ) and e = maxfdim(P) dim(Q)g. By assumption there are realizers L 1 ; L e and J 1 ; J e for P ....

Hirugachi, T. "On the dimension of partially ordered sets." Sci. Rep. Kanazawa Univ., Volume 1(1951), 77-94.

.... and one sink, we can find two orderings L and R of the nodes in G such that a node v is reachable from another node u in G if and only if v occurs after u in both the orderings [11,58] Unfortunately it seems unlikely that these methods can be extended to all planar digraphs, since Kelly [61] has shown that for general planar digraphs we might need a large number of orderings to represent all pairs reachability information as an intersection of these orderings. In this chapter we give a fully dynamic data structure to maintain the all pairs reachability information in planar digraphs. ....

D. Kelly, "On the Dimension of Partially Ordered Sets," Discrete Mathematics 35 (1981), 135--156.

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T. Hiraguchi, On the dimension of partially ordered sets, Sci. Reports Kanazawa Univ. 1 (1951) pp. 77-94.

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#1955#, 1-20. #8# Komm, H. #On the dimension of partially ordered sets."Amer. J. Math.,Volume

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