DOCSLIB.ORG

Lecture Notes on Algebraic Combinatorics Jeremy L. Martin

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture Notes on Algebraic Combinatorics Jeremy L. Martin Lecture Notes on Algebraic Combinatorics Jeremy L. Martin [email protected] December 3, 2012 Copyright c 2012 by Jeremy L. Martin. These notes are licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 2 Foreword The starting point for these lecture notes was my notes from Vic Reiner's Algebraic Combinatorics course at the University of Minnesota in Fall 2003. I currently use them for graduate courses at the University of Kansas. They will always be a work in progress. Please use them and share them freely for any research purpose. I have added and subtracted some material from Vic's course to suit my tastes, but any mistakes are my own; if you find one, please contact me at [email protected] so I can fix it. Thanks to those who have suggested additions and pointed out errors, including but not limited to: Logan Godkin, Alex Lazar, Nick Packauskas, Billy Sanders, Tony Se. 1. Posets and Lattices 1.1. Posets. Definition 1.1. A partially ordered set or poset is a set P equipped with a relation ≤ that is reflexive, antisymmetric, and transitive. That is, for all x; y; z 2 P : (1) x ≤ x (reflexivity). (2) If x ≤ y and y ≤ x, then x = y (antisymmetry). (3) If x ≤ y and y ≤ z, then x ≤ z (transitivity). We'll usually assume that P is finite. Example 1.2 (Boolean algebras). Let [n] = f1; 2; : : : ; ng (a standard piece of notation in combinatorics) and let Bn be the power set of [n]. We can partially order Bn by writing S ≤ T if S ⊆ T . 123 123 12 12 13 23 12 13 23 1 2 1 2 3 1 2 3 The first two pictures are Hasse diagrams. They don't include all relations, just the covering relations, which are enough to generate all the relations in the poset. (As you can see on the right, including all the relations would make the diagram unnecessarily complicated.) Definition 1.3. Let P be a poset and x; y 2 P . • x is covered by y, written x l y, if x < y and there exists no z such that x < z < y. • The interval from x to y is [x; y] := fz 2 P j x ≤ z ≤ yg: (This is nonempty if and only if x ≤ y, and it is a singleton set if and only if x = y.) 3 The Boolean algebra Bn has a unique minimum element (namely ;) and a unique maximum element (namely [n]). Not every poset has to have such elements, but if a poset does, we'll call them 0^ and 1^ respectively (or if necessary 0^P and 1^P ). Definition 1.4. A poset that has both a 0^ and a 1^ is called bounded.1 An element that covers 0^ is called an atom, and an element that is covered by 1^ is called a coatom. (For example, the atoms in Bn are the singleton subsets of [n].) We can make a poset P bounded: define a new poset P^ by adjoining new elements 0^; 1^ such that 0^ < x < 1^ for every x 2 P . Meanwhile, sometimes we have a bounded poset and want to delete the bottom and top elements. Definition 1.5. A subset C ⊂ P is called a chain if its elements are pairwise comparable. Thus every chain is of the form C = fx0; : : : ; xng, where x0 < ··· < xn. An antichain is a subset of P in which no two of its elements are comparable.2 123 123 123 123 12 13 23 12 13 23 12 13 23 12 13 23 1 2 3 1 2 3 1 2 3 1 2 3 φ φ φ φ chain antichain antichain neither 1.2. Ranked Posets. One of the many nice properties of Bn is that its elements fall nicely into horizontal slices (sorted by their cardinalities). Whenever S l T , it is the case that jT j = jSj + 1. A poset for which we can do this is called a ranked poset. However, it would be tautological to define a ranked poset to be a poset in which we can rank the elements! The actual definition of rankedness is a little more subtle, but makes perfect sense after a little thought. 3 Definition 1.6. A chain x0 < ··· < xn is saturated if it is not properly contained in any other chain from x0 to xn; equivalently, if xi−1 l xi for every i 2 [n]. In this case, the number n is the length of the chain A poset P is ranked if for every x 2 P , all saturated chains with top element x have the same length; this number is called the rank of x and denoted r(x). It follows that (1.1) x l y =) r(y) = r(x) + 1: A poset is graded if it is ranked and bounded. Note: (1) \Length" means the number of steps, not the number of elements | i.e., edges rather than vertices in the Hasse diagram. 1This has nothing to do with the more typical metric-space definition of \bounded". 2To set theorists, \antichain" means something stronger: a set of elements such that no two have a common lower bound. This concept does not typically arise in combinatorics, where many posets are bounded. 3Sometimes called \maximal", but that word can easily be misinterpreted to mean \of maximum size". 4 (2) The literature is not consistent on the usage of the term \ranked". Sometimes \ranked" is used for the weaker condition that for every pair x; y 2 P , every chain from x to y has the same length. Under this definition, the implication (1.1) fails (proof left to the reader). (3) For any finite poset P (and some infinite ones) one can define r(x) to be the supremum of the lengths of all chains with top element x | but if P is not a ranked poset, then there will be some pair a; b such that b m a but r(y) > r(x) + 1. For instance, in the bounded poset shown below (known as N5), we have 1^ m y, but r(1)^ = 3 and r(y) = 1. z y x Definition 1.7. Let P be a ranked poset with rank function r. The rank-generating function of P is the formal power series X r(x) FP (q) = q : x2P Thus, for each k, the coefficient of qk is the number of elements at rank k. We can now say that the Boolean algebra is ranked by cardinality. In particular, X jSj n FBn (q) = q = (1 + q) : S⊂[n] The expansion of this polynomial is palindromic, because the coefficients are a row of Pascal's Triangle. That ∗ is, Bn is rank-symmetric. In fact, much more is true. For any poset P , we can define the dual poset P by reversing all the order relations, or equivalently turning the Hasse diagram upside down. It's not hard ∼ ∗ to prove that the Boolean algebra is self-dual, i.e., Bn = Bn, from which it immediately follows that it is rank-symmetric. Example 1.8 (The partition lattice). Let Πn be the poset of all set partitions of [n]. E.g., two elements of Π5 are S = f1; 3; 4g; f2; 5g (abbr.: S = 134j25) T = f1; 3g; f4g; f2; 5g (abbr.: T = 13j4j25) The sets f1; 3; 4g and f2; 5g are called the blocks of S. We can impose a partial order on Πn by putting T ≤ S if every block of T is contained in a block of S; for short, T refines S. 1234 123 123|4 124|3 134|2 234|1 12|3413|24 14|23 12|3 1|23 13|2 12|3|4 13|2|4 23|1|4 14|2|3 24|1|3 34|1|2 1|2|3 1|2|3|4 5 • The covering relations are of the form \merge two blocks into one". • Πn is graded, with 0^ = 1j2j · · · jn and 1^ = 12 ··· n. The rank function is r(S) = n − jSj. • The coefficients of the rank-generating function of Πn are the Stirling numbers of the second kind: S(n; k) = number of partitions of [n] into k blocks. That is, n X n−k Fn(q) = FΠn (q) = S(n; k)q : k=1 2 2 3 For example, F3(q) = 1 + 3q + q and F4(q) = 1 + 6q + 7q + q . Example 1.9 (Young's lattice.). A partition is a sequence λ = (λ1; : : : ; λ`) of weakly decreasing positive integers: i.e., λ1 ≥ · · · ≥ λ` > 0. If n = λ1 + ··· + λ`, we write λ ` n and/or n = jλj. For convenience, set λi = 0 for all i > `. Let Y be the set of all partitions, partially ordered by λ ≥ µ if λi ≥ µi for all i = 1; 2;::: . This is an infinite poset, but it is locally finite, i.e., every interval is finite. Its rank-generating function X X X qjλj = qn λ n≥0 λ`n is given by the justly celebrated formula 1 Y 1 : 1 − qk k=1 There's a nice pictorial way to look at Young's lattice. Instead of thinking about partitions as sequence of numbers, view them as their corresponding Ferrers diagrams: northwest-justified piles of boxes whose ith row contains λi boxes. For example, 5542 is represented by the following Ferrers diagram: Then λ ≥ µ if and only the Ferrers diagram of λ contains that of µ. The bottom part of the Hasse diagram of Y looks like this: 6 Definition 1.10.
Load more

Recommended publications
  • Contractions of Polygons in Abstract Polytopes
    Contractions of Polygons in Abstract Polytopes by Ilya Scheidwasser B.S. in Computer Science, University of Massachusetts Amherst M.S. in Mathematics, Northeastern University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy March 31, 2015 Dissertation directed by Egon Schulte Professor of Mathematics Acknowledgements First, I would like to thank my advisor, Professor Egon Schulte. From the first class I took with him in the first semester of my Masters program, Professor Schulte was an engaging, clear, and kind lecturer, deepening my appreciation for mathematics and always more than happy to provide feedback and answer extra questions about the material. Every class with him was a sincere pleasure, and his classes helped lead me to the study of abstract polytopes. As my advisor, Professor Schulte provided me with invaluable assistance in the creation of this thesis, as well as career advice. For all the time and effort Professor Schulte has put in to my endeavors, I am greatly appreciative. I would also like to thank my dissertation committee for taking time out of their sched- ules to provide me with feedback on this thesis. In addition, I would like to thank the various instructors I've had at Northeastern over the years for nurturing my knowledge of and interest in mathematics. I would like to thank my peers and classmates at Northeastern for their company and their assistance with my studies, and the math department's Teaching Committee for the privilege of lecturing classes these past several years.
    [Show full text]
  • On Finding Two Posets That Cover Given Linear Orders
    algorithms Article On Finding Two Posets that Cover Given Linear Orders Ivy Ordanel 1,*, Proceso Fernandez, Jr. 2 and Henry Adorna 1 1 Department of Computer Science, University of the Philippines Diliman, Quezon City 1101, Philippines; [email protected] 2 Department of Information Sytems and Computer Science, Ateneo De Manila University, Quezon City 1108, Philippines; [email protected] * Correspondence: [email protected] Received: 13 August 2019; Accepted: 17 October 2019; Published: 19 October 2019 Abstract: The Poset Cover Problem is an optimization problem where the goal is to determine a minimum set of posets that covers a given set of linear orders. This problem is relevant in the field of data mining, specifically in determining directed networks or models that explain the ordering of objects in a large sequential dataset. It is already known that the decision version of the problem is NP-Hard while its variation where the goal is to determine only a single poset that covers the input is in P. In this study, we investigate the variation, which we call the 2-Poset Cover Problem, where the goal is to determine two posets, if they exist, that cover the given linear orders. We derive properties on posets, which leads to an exact solution for the 2-Poset Cover Problem. Although the algorithm runs in exponential-time, it is still significantly faster than a brute-force solution. Moreover, we show that when the posets being considered are tree-posets, the running-time of the algorithm becomes polynomial, which proves that the more restricted variation, which we called the 2-Tree-Poset Cover Problem, is also in P.
    [Show full text]
  • A Combinatorial Analysis of Finite Boolean Algebras
    A Combinatorial Analysis of Finite Boolean Algebras Kevin Halasz [email protected] May 1, 2013 Copyright c Kevin Halasz. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license can be found at http://www.gnu.org/copyleft/fdl.html. 1 Contents 1 Introduction 3 2 Basic Concepts 3 2.1 Chains . .3 2.2 Antichains . .6 3 Dilworth's Chain Decomposition Theorem 6 4 Boolean Algebras 8 5 Sperner's Theorem 9 5.1 The Sperner Property . .9 5.2 Sperner's Theorem . 10 6 Extensions 12 6.1 Maximally Sized Antichains . 12 6.2 The Erdos-Ko-Rado Theorem . 13 7 Conclusion 14 2 1 Introduction Boolean algebras serve an important purpose in the study of algebraic systems, providing algebraic structure to the notions of order, inequality, and inclusion. The algebraist is always trying to understand some structured set using symbol manipulation. Boolean algebras are then used to study the relationships that hold between such algebraic structures while still using basic techniques of symbol manipulation. In this paper we will take a step back from the standard algebraic practices, and analyze these fascinating algebraic structures from a different point of view. Using combinatorial tools, we will provide an in-depth analysis of the structure of finite Boolean algebras. We will start by introducing several ways of analyzing poset substructure from a com- binatorial point of view.
    [Show full text]
  • The Number of Graded Partially Ordered Sets
    JOURNAL OF COMBINATORIAL THEORY 6, 12-19 (1969) The Number of Graded Partially Ordered Sets DAVID A. KLARNER* McMaster University, Hamilton, Ontario, Canada Communicated by N. G. de Bruijn ABSTRACT We find an explicit formula for the number of graded partially ordered sets of rank h that can be defined on a set containing n elements. Also, we find the number of graded partially ordered sets of length h, and having a greatest and least element that can be defined on a set containing n elements. The first result provides a lower bound for G*(n), the number of posets that can be defined on an n-set; the second result provides an upper bound for the number of lattices satisfying the Jordan-Dedekind chain condition that can be defined on an n-set. INTRODUCTION The terminology we will use in connection with partially ordered sets is defined in detail in the first few pages of Birkhoff [1]; however, we need a few concepts not defined there. A rank function g maps the elements of a poset P into the chain of integers such that (i) x > y implies g(x) > g(y), and (ii) g(x) = g(y) + 1 if x covers y. A poset P is graded if at least one rank function can be defined on it; Birkhoff calls the pair (P, g) a graded poset, where g is a particular rank function defined on a poset P, so our definition differs from his. A path in a poset P is a sequence (xl ,..., x~) such that x~ covers or is covered by xi+l, for i = 1 ...
    [Show full text]
  • Modular and Semimodular Lattices Ph.D. Dissertation
    Modular and semimodular lattices Ph.D. Dissertation Benedek Skublics Supervisor: Prof. G´abor Cz´edli Doctoral School in Mathematics and Computer Science University of Szeged, Bolyai Institute 2013 Szeged Contents Introduction 1 1 Von Neumann frames 5 1.1 Basic definitions and notions . 7 1.2 The product frame . 12 1.3 The ring of an outer von Neumann frame . 17 1.3.1 A pair of reciprocal mappings . 19 1.3.2 Addition and further lemmas . 22 1.3.3 Multiplication . 23 2 Isometrical embeddings 28 2.1 Motivation: the finite case . 30 2.1.1 Matroids . 32 2.1.2 Embeddings with matroids . 35 2.2 The general case . 38 2.2.1 Basic concepts and lemmas . 39 2.2.2 The main proofs . 45 2.2.3 Examples . 48 3 Mal'cev conditions 50 3.1 Definition of a Mal'cev condition . 52 3.2 Congruences of algebras with constants . 53 Summary 60 Osszefoglal´o¨ 65 Bibliography 70 i Acknowledgment First of all, I would like to express my sincere gratitude to my supervisor, G´abor Cz´edli,who gave me my first research problem and also taught me how to write mathematics. I am thankful for his constant support and patience. He never got tired of correcting the returning mathematical and grammatical mistakes in my manuscripts. I hope I do not bring discredit upon him with this dissertation. I am also grateful to P´eterP´alP´alfy, my former supervisor at ELTE, who sug- gested coming to the Bolyai Insitute. Both his mathematical thinking and his per- sonality meant a lot to me.
    [Show full text]
  • Confluent Hasse Diagrams
    Confluent Hasse diagrams David Eppstein Joseph A. Simons Department of Computer Science, University of California, Irvine, USA. October 29, 2018 Abstract We show that a transitively reduced digraph has a confluent upward drawing if and only if its reachability relation has order dimension at most two. In this case, we construct a confluent upward drawing with O(n2) features, in an O(n) × O(n) grid in O(n2) time. For the digraphs representing series-parallel partial orders we show how to construct a drawing with O(n) fea- tures in an O(n) × O(n) grid in O(n) time from a series-parallel decomposition of the partial order. Our drawings are optimal in the number of confluent junctions they use. 1 Introduction One of the most important aspects of a graph drawing is that it should be readable: it should convey the structure of the graph in a clear and concise way. Ease of understanding is difficult to quantify, so various proxies for readability have been proposed; one of the most prominent is the number of edge crossings. That is, we should minimize the number of edge crossings in our drawing (a planar drawing, if possible, is ideal), since crossings make drawings harder to read. Another measure of readability is the total amount of ink required by the drawing [1]. This measure can be formulated in terms of Tufte’s “data-ink ratio” [22,35], according to which a large proportion of the ink on any infographic should be devoted to information. Thus given two different ways to present information, we should choose the more succinct and crossing-free presentation.
    [Show full text]
  • Arxiv:1508.05446V2 [Math.CO] 27 Sep 2018 02,5B5 16E10
    CELL COMPLEXES, POSET TOPOLOGY AND THE REPRESENTATION THEORY OF ALGEBRAS ARISING IN ALGEBRAIC COMBINATORICS AND DISCRETE GEOMETRY STUART MARGOLIS, FRANCO SALIOLA, AND BENJAMIN STEINBERG Abstract. In recent years it has been noted that a number of combi- natorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such structures. In a recent pa- per, the authors established a close connection between algebraic and combinatorial invariants of a left regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band. The purpose of the present monograph is to further develop and deepen the connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all simple mod- ules of unital left regular band algebras over fields and much more. In the process, we are led to define the class of CW left regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the examples that have arisen in the literature belong to this class. A new and important class of ex- amples is a left regular band structure on the face poset of a CAT(0) cube complex.
    [Show full text]
  • Operations on Partially Ordered Sets and Rational Identities of Type a Adrien Boussicault
    Operations on partially ordered sets and rational identities of type A Adrien Boussicault To cite this version: Adrien Boussicault. Operations on partially ordered sets and rational identities of type A. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2013, Vol. 15 no. 2 (2), pp.13–32. hal- 00980747 HAL Id: hal-00980747 https://hal.inria.fr/hal-00980747 Submitted on 18 Apr 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Discrete Mathematics and Theoretical Computer Science DMTCS vol. 15:2, 2013, 13–32 Operations on partially ordered sets and rational identities of type A Adrien Boussicault Institut Gaspard Monge, Universite´ Paris-Est, Marne-la-Valle,´ France received 13th February 2009, revised 1st April 2013, accepted 2nd April 2013. − −1 We consider the family of rational functions ψw = Q(xwi xwi+1 ) indexed by words with no repetition. We study the combinatorics of the sums ΨP of the functions ψw when w describes the linear extensions of a given poset P . In particular, we point out the connexions between some transformations on posets and elementary operations on the fraction ΨP . We prove that the denominator of ΨP has a closed expression in terms of the Hasse diagram of P , and we compute its numerator in some special cases.
    [Show full text]
  • LNCS 7034, Pp
    Confluent Hasse Diagrams DavidEppsteinandJosephA.Simons Department of Computer Science, University of California, Irvine, USA Abstract. We show that a transitively reduced digraph has a confluent upward drawing if and only if its reachability relation has order dimen- sion at most two. In this case, we construct a confluent upward drawing with O(n2)features,inanO(n) × O(n)gridinO(n2)time.Forthe digraphs representing series-parallel partial orders we show how to con- struct a drawing with O(n)featuresinanO(n)×O(n)gridinO(n)time from a series-parallel decomposition of the partial order. Our drawings are optimal in the number of confluent junctions they use. 1 Introduction One of the most important aspects of a graph drawing is that it should be readable: it should convey the structure of the graph in a clear and concise way. Ease of understanding is difficult to quantify, so various proxies for it have been proposed, including the number of crossings and the total amount of ink required by the drawing [1,18]. Thus given two different ways to present information, we should choose the more succinct and crossing-free presentation. Confluent drawing [7,8,9,15,16] is a style of graph drawing in which multiple edges are combined into shared tracks, and two vertices are considered to be adjacent if a smooth path connects them in these tracks (Figure 1). This style was introduced to re- duce crossings, and in many cases it will also Fig. 1. Conventional and confluent improve the ink requirement by represent- drawings of K5,5 ing dense subgraphs concisely.
    [Show full text]
  • Some Lattices of Closure Systems on a Finite Set Nathalie Caspard, Bernard Monjardet
    Some lattices of closure systems on a finite set Nathalie Caspard, Bernard Monjardet To cite this version: Nathalie Caspard, Bernard Monjardet. Some lattices of closure systems on a finite set. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2004, 6 (2), pp.163-190. hal-00959003 HAL Id: hal-00959003 https://hal.inria.fr/hal-00959003 Submitted on 13 Mar 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Discrete Mathematics and Theoretical Computer Science 6, 2004, 163–190 Some lattices of closure systems on a finite set Nathalie Caspard1 and Bernard Monjardet2 1 LACL, Universite´ Paris 12 Val-de-Marne, 61 avenue du Gen´ eral´ de Gaulle, 94010 Creteil´ cedex, France. E-mail: [email protected] 2 CERMSEM, Maison des Sciences Economiques,´ Universite´ Paris 1, 106-112, bd de l’Hopital,ˆ 75647 Paris Cedex 13, France. E-mail: [email protected] received May 2003, accepted Feb 2004. In this paper we study two lattices of significant particular closure systems on a finite set, namely the union stable closure systems and the convex geometries. Using the notion of (admissible) quasi-closed set and of (deletable) closed set we determine the covering relation ≺ of these lattices and the changes induced, for instance, on the irreducible elements when one goes from C to C ′ where C and C ′ are two such closure systems satisfying C ≺ C ′.
    [Show full text]
  • Single-Element Extensions of Matroids
    JOURNAL OF RESEARCH of the National Bureau of Standards-B. Mathematics and Mathematical Physics Vol. 69B, Nos. 1 and 2, January-June 1965 Single-Element Extensions of Matroids Henry H. Crapo Assistant Professor of Mathematics, Northeastern University, Boston, Mass. (No ve mbe r 16, 1964) Extensions of matroids to sets containing one additional ele ment are c ha rac te ri zed in te rms or modular cuts of the lattice or closed s ubsets. An equivalent characterizati o n is given in terms or linear subclasse s or the set or circuits or bonds or the matroid. A scheme for the construc ti o n or finit e geo· me tric lattices is deri ved a nd th e existe nce of at least 2" no ni somorphic matroids on an II- ele ment set is established. Contents Page 1. introducti on .. .. 55 2. Differentials .. 55 3. Unit in crease runcti ons ... .... .. .... ........................ 57 4. Exact diffe re ntia ls .... ... .. .. .. ........ " . 58 5. Sin gle·ele me nt exte ns ions ... ... 59 6. Linear subclasses.. .. .. ... .. 60 7. Geo metric la ttices ................... 62 8. Numeri ca l c lassification of matroid s .... 62 9. Bibl iography.. .. ............... 64 1. Introduction I of ele ments Pi of a lattice L, in whic h Pi covers Pi - t for i = 1, . .. , n, is a path (of length n) from x to y. In order to facilitate induc tive proofs of matroid 2 theore ms, we shall set forth a characte rization of single­ A step is a path of length J. element extensions of matroid structures.
    [Show full text]
  • Friday 1/18/08
    Friday 1/18/08 Posets Definition: A partially ordered set or poset is a set P equipped with a relation ≤ that is reflexive, antisymmetric, and transitive. That is, for all x; y; z 2 P : (1) x ≤ x (reflexivity). (2) If x ≤ y amd y ≤ x, then x = y (antisymmetry). (3) If x ≤ y and y ≤ z, then x ≤ z (transitivity). We'll usually assume that P is finite. Example 1 (Boolean algebras). Let [n] = f1; 2; : : : ; ng (a standard piece of notation in combinatorics) and let Bn be the power set of [n]. We can partially order Bn by writing S ≤ T if S ⊆ T . 123 123 12 12 13 23 12 13 23 1 2 1 2 3 1 2 3 The first two pictures are Hasse diagrams. They don't include all relations, just the covering relations, which are enough to generate all the relations in the poset. (As you can see on the right, including all the relations would make the diagram unnecessarily complicated.) Definitions: Let P be a poset and x; y 2 P . • x is covered by y, written x l y, if x < y and there exists no z such that x < z < y. • The interval from x to y is [x; y] := fz 2 P j x ≤ z ≤ yg: (This is nonempty if and only if x ≤ y, and it is a singleton set if and only if x = y.) The Boolean algebra Bn has a unique minimum element (namely ;) and a unique maximum element (namely [n]). Not every poset has to have such elements, but if a poset does, we'll call them 0^ and 1^ respectively.
    [Show full text]