By Joseph P. S. Kung, Gian-Carlo Rota, Catherine H. Yan

Written by means of of Gian-Carlo Rota's former scholars, this ebook is predicated on notes from his classes and on own discussions with him. themes comprise units and valuations, partly ordered units, distributive lattices, walls and entropy, matching concept, unfastened matrices, doubly stochastic matrices, Moebius features, chains and antichains, Sperner conception, commuting equivalence kin and linear lattices, modular and geometric lattices, valuation earrings, producing capabilities, umbral calculus, symmetric capabilities, Baxter algebras, unimodality of sequences, and placement of zeros of polynomials. Many routines and learn difficulties are incorporated, and unexplored parts of attainable learn are mentioned. This publication may be at the shelf of all scholars and researchers in combinatorics and similar components.

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**Example text**

The partial order (B, S) has minimum B and maximum the one-element antichain {S}. (a) Show that the partial order (B, S) is a lattice. An n-partition σ of S is an antichain partitioning the collection of all size-n subsets of S; that is, σ is a collection of subsets of S of size at least n such that every n-subset of S is contained in exactly one subset in σ. Let (n; S) be the lattice of all n-partitions of S. (b) Show that (n; S) is isomorphic to a sublattice of (n + 1; S ∪ {a}), where a is a new element not in S.

1. Lemma. Let L be a lattice, 2L be the Boolean algebra of subsets of all the lattice elements, and I : L → 2L be the function sending an element a to the principal ideal I (a) generated by a. Then I preserves arbitrary meets, x I x: x∈A = I (x), x: x∈A and is “superadditive” on joins, I (x ∨ y) ⊇ I (x) ∪ I (y). A function ϕ : L → M between lattices is a (lattice) homomorphism if ϕ preserves both meets and joins. 1 is not a lattice homomorphism. P1: KAE CUUS456-01 cuus456-Kung 18 978 0 521 88389 4 November 7, 2008 16:33 1 Sets, Functions, and Relations The representation using ideals can be strengthened for distributive lattices.

Xn ) is the table of the values of β at the 2n possible inputs ( 1 , 2 , . . , n ), where i equals 0 or 1. Show that β(x1 , x2 , . . , xn ) = x11 x22 · · · xnn , ( i ): β( i )=1 where the union ranges over all inputs ( 1 , 2 , . . , n ) for which β( 1 , 2 , . . , n ) = 1. In particular, this shows that every Boolean function is expressible as a Boolean polynomial. A Boolean function is monotone if β( 1 , 2 , . . , n ) ≤ β(˜1 , ˜2 , . . , ˜n ) whenever i ≤ ˜i for all i, 1 ≤ i ≤ n. (c) Show that β is monotone if and only if β is a disjunction of monomials not involving complements.