Thesis (Ph. D.)--Lehigh University, 2004. / Includes vita. Includes bibliographical references (leaves 73-74).

Thesis--University of Michigan. / "Reprinted from American journal of mathematics, vol. LXX, no. 3, July, 1948."

It may be said of certain pairs of elements of a set that one element precedes the other. If the collection of all such pairs of elements in a given set exhibits certain properties, the set and the collection of pairs is said to constitute a partially ordered set. The purpose of this paper is to explore some of the properties of partially ordered sets.

Gives the basic definitions and theorems of similar partially ordered sets; studies finite partially ordered sets, including the problem of combinatorial analysis; and includes the ideas of complete, dense, and continuous partially ordered sets, including proofs.

Thesis (Ph.D.)--University of Nebraska-Lincoln, 2008. / Title from title screen (site viewed Dec. 15, 2008). PDF text: vi, 93 p. ; 616 K. UMI publication number: AAT 3333017. Includes bibliographical references. Also available in microfilm and microfiche formats.

Thesis (Ph. D.)--Mathematics, Georgia Institute of Technology, 2008. / Committee Chair: Tetali, Prasad; Committee Member: Duke, Richard; Committee Member: Heitsch, Christine; Committee Member: Randall, Dana; Committee Member: Trotter, William T.

Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 115-119).

The main results of this dissertation are various algorithms related to partially ordered sets. The dissertation basically consists of two parts. The first part treats algorithms that generate ideals of partially ordered sets. The second part concerns the generation of partially ordered sets themselves. First, we present two algorithms for listing ideals of a forest poset. These algorithms generate ideals in a Gray Code manner, that is, consecutive ideals differ by exactly one element. Both algorithms use storage O(n), where n is the number of elements in the poset. The first algorithm traverses, at each phase, the current ideal being listed and runs in time O(nN), where N is the number of ideals of the poset. The second algorithm mimics the first but eliminates the traversal and runs in time O(N). This algorithm has the property that the amount of computation between successive ideals is O(1). Secondly, we give orderly algorithms for constructing acyclic digraphs, acyclic transitive digraphs, finite topologies and finite topologies and finite lattices. For the first time we show that the number of finite lattices on 11, 12, and 13 elements are 37622, 262775, and 2018442, respectively, and the number of finite topologies on 8 and 9 elements are 35979 and 363083, respectively. We also describe orderly algorithms for generating k-colored graphs. We present, in particular, an algorithm for generating connected bicolorable graphs. We also prove some properties of a canonic matrix which might be generally useful for improving the efficiency of orderly algorithms. / Graduate

This thesis is concerned with the development of two generalized set covering models. The first model is formulated for the total set covering problem where cost is minimized subject to the constraint that each customer must be served by at least one facility. The second model is constructed for the partial set covering problem in which customer coverage is maximized subject to a budget constraint. The conventional formulations of both the total set covering and partial set covering problems are shown to be special cases of the two generalized models that arc developed. Appropriate solution strategies arc discussed for each generalized model. A specialized algorithm for a particular case of the partial covering problem is constructed and computational results are presented. / M.S.

The LYM property of a finite standard graded poset is one of the central notions in Sperner theory. It is known that the product of two finite standard graded posets satisfying the LYM properties may not have the LYM property again. In 1974, Harper proved that if two finite standard graded posets satisfying the LYM properties also satisfy rank logarithmic concavities, then their product also satisfies these two properties. However, Harper's proof is rather non-intuitive. Giving a natural proof of Harper's theorem is one of the goals of this thesis. The main new result of this thesis is a characterization of rank-finite standard graded LYM posets that satisfy rank logarithmic concavities. With this characterization theorem, we are able to give a new, natural proof of Harper's theorem. In fact, we prove a strengthened version of Harper's theorem by weakening the finiteness condition to the rank-finiteness condition. We present some interesting applications of the main characterization theorem. We also give a brief history of Sperner theory, and summarize all the ingredients we need for the main theorem and its applications, including a new equivalent condition for the LYM property that is a key for proving our main theorem.