Let O be the poset of open sets of a d-dimensional manifold M, and let [O.sub.k] be the full subposet of O whose objects are disjoint unions of at most k components, each diffeomorphic to [R.sup.d].
The primary example of interest here occurs when M is a smooth codimension zero submanifold of [R.sup.d], and [B.sub.k](M) contains all open sets which are disjoint unions of at most k open balls (in the euclidean metric sense).
Using methods for decomposing regular [omega]-languages into disjoint unions
of parts of simple structure we derive two sufficient conditions under which [omega]-languages with a closure definable by a finite automaton have the same Hausdorff measure as this closure.
Batle On edge-magic labelings of certain disjoint unions
of graphs Australas.
Muntaner-Batle, "On edge-magic labelings of certain disjoint unions
of graphs," The Australasian Journal of Combinatorics, vol.
Let c, called the magnitude set of [gamma], be the set containing the disjoint magnitudes in [[phi].sub.*]., all possible disjoint unions
of these magnitudes, all possible disjoint unions
of disjoint unions
of the same magnitudes, and so on.
Since every vertex has at least k closed neighbors in S, it is the case that S = [[union].sup.r+1.sub.i=k] [S.sub.1] and S' = [[union].sup.r.sub.i=k] [S'.sub.i] are disjoint unions
. Therefore, [absolute value of S] = [[summation].sup.r+1.sub.i=k] [absolute value of S'] and [absolute value of S'] = [[summation].sup.r.sub.i=k] [absolute value of S'].
Muntaner-Batle, On edge-magic labelings of certain disjoint unions
of graphs, Austral.
There exists a notion which is halfway between that of simple and that of arbitrary disjoint unions
of simple sets.
The product of a matrix by a vector or by a matrix is obtained by the classical formulas in sums of products forms, sums being interpreted as disjoint unions
and products as cartesian products, themselves obtained by grafting at a bud following Eq.
The graph G' is constructed from the disjoint union
of [S.sub.1], [S.sub.2], ..., [S.sub.h] and F by joining every vertex of F to every vertex of [I.sub.1] [union] [I.sub.2] [union] ...
Two k-graphs ([G.sub.1], [l.sub.1]) and ([G.sub.2], [l.sub.2]) can be glued together producing a k-graph (G, l) = ([G.sub.1], [l.sub.1]) [[??].sub.k] ([G.sub.2], [l.sub.2]) by taking the disjoint union
of [G.sub.1] and [G.sub.2] and [l.sub.1] and [l.sub.2] and identifying elements with the same label.