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

Written by means of of Gian-Carlo Rota's former scholars, this e-book relies on notes from his classes and on own discussions with him. themes contain units and valuations, in part ordered units, distributive lattices, walls and entropy, matching conception, loose matrices, doubly stochastic matrices, Moebius capabilities, chains and antichains, Sperner idea, commuting equivalence family members and linear lattices, modular and geometric lattices, valuation earrings, producing services, umbral calculus, symmetric services, Baxter algebras, unimodality of sequences, and placement of zeros of polynomials. Many workouts and study difficulties are incorporated, and unexplored parts of attainable examine are mentioned. This booklet can be at the shelf of all scholars and researchers in combinatorics and comparable parts.

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**Example text**

As j is a join-irreducible, j = j ∧ x and j ≤ x, or j = j ∧ y and j ≤ y. We conclude that j ∈ I (x) ∪ I (y). Hence, I (x ∨ y) = I (x) ∪ I (y) and ϕ(x ∨ y) = ϕ(x) ∪ ϕ(y). 3, the set of maximal elements of I (x) equals J (x) and x determines J (x) uniquely. Hence, ϕ is injective. To show that ϕ is surjective, let I be an order ideal in J (L) and x be the join of all the elements in I. Then I ⊆ I (x). Suppose that j ∈ I (x). Then y = j = j ∧x = j ∧ y: y∈I (j ∧ y). y: y∈I Since j is a join-irreducible, j = j ∧ y, and thus, j ≤ y, for some y ∈ I.

Since I is an order ideal, j ∈ I. We conclude that I = I (x) = ϕ(x). 3. Inverting the order, Birkhoff’s theorem says that a finite distributive lattice is isomorphic to the lattice of filters of the partially ordered set of meetirreducibles. 5. Corollary. A finite lattice is distributive if and only if it can be represented as a ring of sets. 16 Birkhoff’s theorem delineates the extent to which a finite lattice may be represented by sets so that meets and joins are intersections and unions. For lattices which are not distributive, one might ask for representations by subsets for which meets are intersections and there is a natural way to construct the joins.

1, n), (2, n), . . , (m, n)}. Then σ and τ are independent. Conversely, given two independent partitions σ and τ, we can display S as a rectangular “chessboard,” with rows the blocks of σ, columns the blocks of τ, and squares the blocks of the meet σ ∧ τ. Put another way, independent partitions arise from the rows and columns of a Cartesian product. 3. The chessboard construction. Let p1 , p2 , . . , pm and q1 , q2 , . . , qn be nonnegative real numbers such that p1 + p2 + · · · + pm = 1 and q1 + q2 + · · · + qn = 1.