Preface.- Introduction.- Glossary of Notation.- I First Concepts.- 1 Two Definitions of Lattices.- 2 How to Describe Lattices.- 3 Some Basic Concepts.- 4 Terms, Identities, and Inequalities.- 5 Free Lattices.- 6 Special Elements.- II Distributive Lattices.- 1 Characterization and Representation Theorems.- 2 Terms and Freeness.- 3 Congruence Relations.- 4 Boolean Algebras.- 5 Topological Representation.- 6 Pseudocomplementation.- III Congruences.- 1 Congruence Spreading.- 2 Distributive, Standard, and Neutral Elements.- 3 Distributive, Standard, and Neutral Ideals.- 4 Structure Theorems.- IV Lattice Constructions.- 1 Adding an Element.- 2 Gluing.- 3 Chopped Lattices.- 4 Constructing Lattices with Given Congruence Lattices.- 5 Boolean Triples.- V Modular and Semimodular Lattices.- 1 Modular Lattices.- 2 Semimodular Lattices.- 3 Geometric Lattices.- 4 Partition Lattices.- 5 Complemented Modular Lattices.- VI Varieties of Lattices.- 1 Characterizations of Varieties 397.- 2 The Lattice of Varieties of Lattices.- 3 Finding Equational Bases.- 4 The Amalgamation Property.- VII Free Products.- 1 Free Products of Lattices.- 2 The Structure of Free Lattices.- 3 Reduced Free Products.- 4 Hopfian Lattices.- Afterword.- Bibliography.

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Lattice Theory: Foundation

This is a direct continuation of Shellable Nonpure Complexes and Posets. I, which appeared in Transactions of the American Mathematical Society 348 (1996), 1299-1327. 8. Interval-generated lattices and dominance order In this section and the following one we will continue exemplifying the applicability of lexicographic shellability to nonpure posets. Let F = {I1, I2, . . . , In} be a family of intervals of integers, by which is meant sets of the form [a, b] = {a, a + 1, . . . , b}, a ≤ b. We assume that there are no containments among these intervals, and that they are ordered so that their left and right endpoints are increasing. Let L(F) be the lattice of all sets that are unions of subfamilies of F , ordered by inclusion. Such interval-generated lattices L(F) were introduced and studied by Greene [G]. Define an edge-labeling λ of L(F) as follows. If A → B is a covering and a = max(B \A), then λ(A→ B) = { −a, if (a+ 1) ∈ A and a is the left endpoint of some I ∈ F ,

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Shellable nonpure complexes and posets. II

Counting rooted maps by genus II

The search for a theory of the S-Matrix over the past five decades has revealed surprising geometric structures underlying scattering amplitudes ranging from the string worldsheet to the amplituhedron, but these are all geometries in auxiliary spaces as opposed to the kinematical space where amplitudes actually live. Motivated by recent advances providing a reformulation of the amplituhedron and planar $$ \mathcal{N} $$
 = 4 SYM amplitudes directly in kinematic space, we propose a novel geometric understanding of amplitudes in more general theories. The key idea is to think of amplitudes not as functions, but rather as differential forms on kinematic space. We explore the resulting picture for a wide range of massless theories in general spacetime dimensions. For the bi-adjoint ϕ3 scalar theory, we establish a direct connection between its “scattering form” and a classic polytope — the associahedron — known to mathematicians since the 1960’s. We find an associahedron living naturally in kinematic space, and the tree level amplitude is simply the “canonical form” associated with this “positive geometry”. Fundamental physical properties such as locality and unitarity, as well as novel “soft” limits, are fully determined by the combinatorial geometry of this polytope. Furthermore, the moduli space for the open string worldsheet has also long been recognized as an associahedron. We show that the scattering equations act as a diffeomorphism between the interior of this old “worldsheet associahedron” and the new “kinematic associahedron”, providing a geometric interpretation and simple conceptual derivation of the bi-adjoint CHY formula. We also find “scattering forms” on kinematic space for Yang-Mills theory and the Non-linear Sigma Model, which are dual to the fully color-dressed amplitudes despite having no explicit color factors. This is possible due to a remarkable fact—“Color is Kinematics”— whereby kinematic wedge products in the scattering forms satisfy the same Jacobi relations as color factors. Finally, all our scattering forms are well-defined on the projectivized kinematic space, a property which can be seen to provide a geometric origin for color-kinematics duality.

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Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet

A topological representation theory for lattices

Problems of associativity: A simple proof for the lattice property of systems ordered by a semi-associative law

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Monoïdes préordonnés et chaînes de Malcev

Problèmes d'associativité: Une structure de treillis finis induite par une loi demi-associative

Publisher Summary This chapter discusses the associativity problem for monoids and the word problem for semigroups and groups. The chapter also discusses the nonvalidity of the one-mountain theorem for general chains, associativity chains, homomorphic images, reduction of the word problem of semigroups to the associativity problem for monoids, the converse reduction, unsolvability degrees, and open questions. The chapter also discusses symmetrical monoids and groups.

Dov Tamari

Papers

Problems of associativity: A simple proof for the lattice property of systems ordered by a semi-associative law

Monoïdes préordonnés et chaînes de Malcev

Problèmes d'associativité: Une structure de treillis finis induite par une loi demi-associative

The Associativity Problem for Monoids and the Word Problem for Semigroups and Groups