Basic Posets

The following sections are included:

• Contents

• Preface

Partially ordered sets (posets) have a long history beginning with the first recognition of ordering in the integers. In the early nineteenth century, properties of the ordering of the subsets of a set were investigated by De Morgan, while in the late nineteenth century partial ordering by divisibility was investigated by Dedekind. Although Hausdorff did not originate the idea of a partially ordered set, the first general theory of posets was developed by him in his 1914 book Grundzüge der mengenlehre. It remained until the 1930's for the subset of lattice theory to blossom as an independent entity with Birkhoff's justly famous text on the subject first published in 1940. It has only been in the last three decades that posets and their relationships to applied areas such as computer science, engineering and the social sciences have been extensively investigated…

The following sections are included:

• Hasse Diagram

• Labeling with a Hasse Diagram

• The Adjacency Matrix of a Hasse Diagram

• Interval Order and Semiorder

• Angle Order and Circle Order

• Other Representations

• Order Geometry (2 dimensional)

As in other structures, algebraic and otherwise, morphisms play a very important role in investigating the structure of posets. In the literature several authors use different terminologies and notations for poset morphisms. Let (X, ≤) and (Y, ≤) be two partially ordered sets. A mapping ƒ from X to Y is called order preserving if for every two elements x and y of X, x ≤ y implies ƒ(x) ≤ ƒ(y)…

The following sections are included:

• Sum and Ordinal Sum

• Product Order and Lexicographic Order

• Exponential Posets

We have already defined the notion of connectedness when we discussed poset morphisms in Chapter 3. Moreover, we also discussed methods of constructing new posets from old ones. Making use of these concepts we will next take a look at connectedness for various classes of posets. We recall the definition of connectedness. Simply, a poset X is called connected if its Hasse diagram is connected as a graph, i.e., if for any two points a and b of X, there is an intermediate set S = {x₁, ⋯ ,x_n} such that a ↔ x₁, x₁ ↔ x₂, ⋯ , x_n−1 ↔ x_n,x_n ↔ b, where u ↔ v means that either u ≤ v or v ≤ u. Hence, as we have already noted above, X is connected if and only if it is not the sum of posets…

The following sections are included:

• Linear Extensions

• Representation Polynomials and Linear Extensions

• Dimension

• Applications of Linear Extensions

The following sections are included:

• Appendix

• REFERENCES

• Notations and Symbols

• Index