Poset Theory
This chapter was written to furnish a starting point for the study of Artinian rings in commutative algebra. We are primarily interested in the Sperner theory of finite posets.
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Poset Theory
This chapter was written to furnish a starting point for the study of Artinian rings in commutative algebra. We are primarily interested in the Sperner theory of finite posets. The basic examples of finite posets to keep in mind are (1) the Boolean lattice 2Œn , (2) the divisor lattice L .n/ or the finite chain product, (3) the order ideal I .s r / of Young diagrams contained in a rectangle .s r /, and (4) the vector space lattice V .n; q/ of finite dimensional vector spaces over a finite field. We are interested in the methods used to prove the Sperner property for these lattices, and how they can be translated into terms of commutative rings, which together with the theory of Artinian rings can be applied to prove that certain classes of such rings have the Sperner property.
1.1 Poset and Dilworth Number We assume that the reader is familiar with the definition of a partially ordered set (poset) and a lattice. In this book it is enough to consider finite posets, but sometimes an infinite poset may appear. The purpose of this section is to introduce to the reader the Dilworth number and the Dilworth theorem of posets. As an application of the Dilworth theorem, we prove Hall’s marriage theorem. For completeness we start with the definition of a poset. (For a definition of a lattice see the first paragraph of Sect. 1.3.) Definition 1.1. A partially ordered set (poset) is a set P together with a binary relation satisfying 1. For all x 2 P , x x. 2. If x y and y x, then x D y. 3. If x y and y z, then x z.
T. Harima et al., The Lefschetz Properties, Lecture Notes in Mathematics 2080, DOI 10.1007/978-3-642-38206-2 1, © Springer-Verlag Berlin Heidelberg 2013
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1 Poset Theory
The binary relation is called a partial order or simply an order. The notation x < y is used in the obvious sense: x y and x ¤ y. Two posets are regarded as isomorphic if there exists an order preserving bijection. We say that two elements x; y 2 P are comparable, if either x y or y x, and otherwise they are incomparable. A subset of a poset is automatically a poset with the induced order. It is called a subposet. Example 1.2. Let Œn D f 1; 2; : : : ; n g, and 2Œn the power set of Œn, i.e., the set of subsets of Œn. Then 2Œn is a poset ordered by inclusion. The elements f 1; 2 g, f 1; 2; 3 g 2 2Œ4 are comparable. The elements f 1; 2 g, f 2; 3 g 2 2Œ4 are incomparable. The empty set ; is a unique minimum element and f 1; 2; : : : ; n g a unique maximum element. A product of two posets is defined as follows: Example 1.3. Let P and Q be posets with orders P and Q , respectively. Then P Q D f .x; y/ j x 2 P; y 2 Q g is ordered by the binary relation : .x; y/ .x 0 ; y 0 / ” x P x 0 and y Q y 0 for .x; y/, .x 0 ; y 0 / 2 P Q. The poset P Q is the product of posets P and Q. Example 1.4. Let P be a poset. For x; y 2 P with x y, we define the interval Œx; y between x and y to be the set f z j x z y g. Each interval is automatically a poset with the induced order. Example 1.5. Let P D 2Œn
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