Unique Minimal Element Research Articles

Abelian ideals of a Borel subalgebra and long positive roots

Let b be a Borel subalgebra of a simple Lie algebra g. Let Ab denote the set of all Abelian ideals of b. It is easily seen that any a ∈ Ab is actually contained in the nilpotent radical of b. Therefore, a is determined by the corresponding set of roots. More precisely, let t be a Cartan subalgebra of g lying in b and let ∆ be the root system of the pair (g, t). Choose ∆, the system of positive roots, so that the roots of b are positive. Then a = ⊕γ∈Igγ, where I is a suitable subset of ∆ and gγ is the root space for γ ∈ ∆. It follows that there are finitely many Abelian ideals and that any question concerning Abelian ideals can be stated in terms of combinatorics of the root system. An amazing result of D. Peterson says that the cardinality of Ab is 2rk . His approach uses a one-to-one correspondence between the Abelian ideals and the so-called minuscule elements of the affine Weyl group Ŵ. An exposition of Peterson’s results is found in [5]. Peterson’s work appeared to be the point of departure for active recent investigations of Abelian ideals, ad-nilpotent ideals, and related problems of representation theory and combinatorics [1, 2, 3, 4, 5, 6, 7, 8]. We consider Ab as poset with respect to inclusion, the zero ideal being the unique minimal element of Ab. Our goal is to study this poset structure. It is easily seen that Ab is a ranked poset; the rank function attaches to an ideal its dimension. It was shown in [8] that there is a one-to-one correspondence between the maximal Abelian ideals and the long simple roots of g. (For each simple Lie algebra, the maximal Abelian ideals were determined in [10].) This correspondence possesses a number of nice properties, but the very existence of it was demonstrated in

On an approximate eigenvector associated with a modulation code

Let S be a constrained system of finite type, described in terms of a labeled graph M of finite type. Furthermore, let C be an irreducible constrained system of finite type, consisting of the collection of possible code sequences of some finite-state-encodable, sliding-block-decodable modulation code for S. It is known that this code could then be obtained by state splitting, using a suitable approximate eigenvector. In this correspondence, we show that the collection of all approximate eigenvectors that could be used in such a construction of C contains a unique minimal element. Moreover, we show how to construct its linear span from knowledge of M and C only, thus providing a lower bound on the components of such vectors. For illustration we discuss an example showing that sometimes arbitrary large approximate eigenvectors are required to obtain the best code (in terms of decoding-window size) although a small vector is also available.

Minimal Affinizations of Representations of Quantum Groups: the Rank 2 Case

If U_q (\mathfrak g) is a finite-dimensional complex simple Lie algebra, an affinization of a finite-dimensional irreducible representation V of U_q (\mathfrak g) is a finite-dimensional irreducible representation \hat V of U_q (\mathfrak g) which contains \hat V with multiplicity one, and is such that all other U_q (\mathfrak g) -types in \hat V have highest weights strictly smaller than that of V . We define a natural partial ordering ≼ on the set of affinizations of V . If \mathfrak g is of rank 2 , we show that there is a unique minimal element with respect to this order and give its U_q (\mathfrak g) -module structure when \mathfrak g is of type A_2 or C_2 .

The Complexity of Default Reasoning Under the Stationary Fixed Point Semantics

Stationany default extensions have recently been introduced by Przymusinska and Przymusinsky as an interesting alternative to classical extensions in Reiter′s default logic. An important property of this new approach is that the set of all stationary extensions has a unique minimal element. In this paper we investigate the computational complexity of the main reasoning tasks in propositional stationary default logic, namely, cautious and brave reasoning. We show that cautious reasoning is complete for Δ P 2 while brave reasoning is Σ P 2 -complete. We also show that for normal default theories, the least fixed point iteration used to compute the unique minimal stationary extension is noticeably simplified. Based on this observation we show that cautious reasoning with normal defaults is complete for P NP[log n] . This is the class of decision problems solvable in polynomial time with a logarithmic number of queries to an oracle in NP.

Towards the reconstruction of posets

The reconstruction conjecture for posets is the following : every finite poset P of more than three elements is uniquely determined - up to isomorphism - by its collection of (unlabelled) one-element-deleted subposets [ P - {x} : x V (P) ]. This conjecture belongs to the list of open problems in Order. We show that disconnected posets, posets with unique minimal (respectively, maximal) element and interval orders are reconstructible and that N-free orders are recognizable. We show that the following parameters are reconstructible : the number of minimal (respectively, maximal) elements, the level structure, the ideal-size sequence of the maximal elements, the ideal size (respectively, filter-size) sequence of any fixed level of the HASSE-diagram and the number of edges of the HASSE-diagram. This is considered to be a first step towards a proof of the reconstruction conjecture for posets.

A free resolution of the module of derivations for generic arrangements

Let Y be an I-dimensional vector space over an arbitrary field F and d an arrangement of n hyperplanes H,, . . . . i7,, in V. We always assume that n > I > 1. For each i, 1~ id ~1, we fix once and forever aj E V* such that ker(cl,) = Hi. The arrangement d is generic if its every I elements intersect only at 0 or equivalently every I functionals ai are linearly independent in V*. Let S be the symmetric algebra of V* and Der(S) = Der,(S, S) the S-module of derivations of S. The following graded S-module was introduced by H. Terao [4] and studied in many papers on arrangements: D = D(d) = (0 E Der(S) ( f3(ai) E Sa,, i= 1, . . . . n}. At the NSF-CBMS conference on “Arrangements of Hyperplanes” in 1988 the author conjectured that for a generic arrangement with II > 1 we have hd, D = 1 2. Soon after that L. Rose and H. Terao [2] proved this conjecture. Moreover they constructed minimal free resolutions of all modules of logarithmic forms with poles along d and used the fact that D is isomorphic to one of them. The aim of the present paper is to give another somewhat more straightforward construction of a minimal free resolution of D. More precisely we put Do= D,(d)= (BEDJO =O) and note that D= Does MO, where t$, is the Euler derivation defined by &,(a) = CI for every CI E V*. Then we construct a minimal free resolution of Do which of course has the same length as a minimal free resolution of D. The notation introduced above is used throughout the paper. Besides we denote by L the intersection lattice of & (including V as the unique minimal element of L), put Li=(X~L/dimX=i}, and for every XEL put cQzx= (Xn HiI X$ Hi}, viewing it as an arrangement in X Also for every XEL we put L~‘={YEL~(YcX), L,=(YEL(XC Y>, and a(X)=

Axioms for consensus functions based on lower bounds in posets

Let (X, ≤) be a finite partially ordered set with a unique minimal element, and consider the problem of associating with each subset of X a corresponding set of consensus objects. When based on lower bound operators, such consensus functions are interesting because their consensus objects can be interpreted as conservative representations of structure inherent in the objects being studied. We extend early results of Pfaltz to obtain an axiomatic characterization of a consensus function based on an iterated maximal lower bound operator, and we relate this function to a recent consensus proposal of Bonacich.

Partitions into Even and Odd Block Size and Some Unusual Characters of the Symmetric Groups

For each n and k, let ∏ ¯ ( i , k ) denote the poset of all partitions of n having every block size congruent to i mod k. Attach to ∏ ¯ n ( i , k ) a unique maximal or minimal element if it does not already have one, and denote the resulting poset ∏ n ( i , k ) Results of Björner, Sagan, and Wachs show that ∏ n ( 0 , k ) and ∏ n ( 1 , k ) are lexicographically shellable, and hence Cohen–Macaulay. Let β n ( 0 , k ) and β n ( 1 , k ) denote the characters of Sn acting on the unique non-vanishing reduced homology groups of ∏ n ( 0 , k ) and ∏ n ( 1 , k ) . This paper is divided into three parts. In the first part, we use combinatorial methods to derive defining equations for the generating functions of the character values of the β n ( i , k ) The most elegant of these states that the generating function for the characters β n 1 + 1 ( 1 , k ) (t = 0, 1,…) is the inverse in the composition ring (or plethysm ring) to the generating function for the corresponding trivial characters ɛnt$1. In the second part, we use these cycle index sum equations to examine the values of the characters β n ( 1 , 2 ) and β n ( 0 , 2 ) . We show that the values of β n ( 0 , 2 ) are simple multiples of the tangent numbers and that the restrictions of the β n ( 0 , 2 ) to Sn−1 are the skew characters examined by Foulkes (whose values are always plus or minus a tangent number). In the case β n ( 1 , 2 ) a number of remarkable results arise. First it is shown that a series of polynomials {Pα(λ): σ € Sn} which are connected with our cycle index sum equations satisfy β n ( 1 , 2 ) (σ) = pσ(0) or pσ(1) depending on whether n is odd or even. Next it is shown that the pσ(λ) have integer roots which obey a simple recursion. Lastly it is shown that the pσ(λ) have a combinatorial interpretation. If the rank function of ∏ n ( 1 , 2 ) is naturally modified to depend on σ then the polynomials pσ(λ) are the Birkhoff polynomials of the fixed point posets ( ∏ n ( 1 , 2 ) ) σ . In the last part we prove a conjecture of R. P. Stanley which indentifies the restriction of β n ( 0 , 2 ) to Sn−1 as a skew character. A consequence of this result is a simple combinatorial method for decomposing β n ( 0 , k ) into irreducibles.

The prime spectrum of a two-dimensional affine domain

From a distance, the two partially ordered sets spec Z[x] and spec Q[x, y] appear rather similar: They each have a unique minimal element and denumerable sets of elements of heights one and two. Moreover, there are several restrictions placed on the orderings by the fact that both rings are Noetherian. We will show that the partial ordering of spec7/[x] is much simpler than that of specQ[x,y]. In fact, U= spec Z[x] is characterized among countable partially ordered sets by the follow

Homomorphisms of 3-chromatic graphs

This paper examines the effect of a graph homomorphism upon the chromatic difference sequence of a graph. Our principal result (Theorem 2) provides necessary conditions for the existence of a homomorphism onto a prescribed target. As a consequence we note that iterated cartesian products of the Petersen graph form an infinite family of vertex transitive graphs no one of which is the homomorphic image of any other. We also prove that there is a unique minimal element in the homomorphism order of 3-chromatic graphs with non-monotonic chromatic difference sequences (Theorem 1). We include a brief guide to some recent papers on graph homomorphisms.

Maximum antichains in the partition lattice

Furthermore, this bound on t can only be achieved by taking the Sk to be all the [~] -element subsets of S, or, n is odd, by taking all the ([~] +l)-element when subsets. Sperner's theorem, like Schur's work [15] on the solutions of x m + ym = z m van der Waerden's theorem [17] on arithmetic progressions, Ramsey's fundamental result [12] on partitions of the subsets o f a set, and Polya's approach [11 ] to the theory o f enumeration, has been the seed f r o m wh ich a major branch of combinatorial theory has grown during the past 50 years. This branch, often called extremal set theory, has been especially active during the past 10 years. In particular, one of the outstanding open problems, which was first raised by G. -C. Rota nearly 15 years ago and which was responsible for much of this activity, has just been settled within the past year by E. Rodney Canfield of the University o f Georgia. What is even more intriguing is that Canfield showed that the answer that everyone had expected (and was trying to prove) was wrong*. In this note I would like to give a brief sketch of the background of Rota's problem and its resolution. By a chain C in a (finite) partially ordered set P we mean a totally ordered subset of P; the length of C is just the number o f elements in it. We say that P is graded if P has a unique minimal element 0 and for every p e P, all maximal chains from 0 to p have the same length, called the rank of p. We denote the elements of P having rank k by ek, also called the k th level. By an anfichain in P we mean a subset of mutually incomparable elements of P. For example, for any k the set e k forms an antichain.

Weak cutpoint ordering on hereditarily unicoherent continua

1. Introduction. We present here two criteria for determining when an hereditarily unicoherent continuum admits the structure ot a partially ordered space which is closed, and has a unique minimal element (definitions below). Our interest in such partial orderings stems from the fact (Hunter [1]) that a one-dimensional compact connected semigroup with zero and unit admits such a partial ordering. Ward [5] has studied a class of partially ordered spaces, called generalized trees, which he characterized as hereditarily unicoherent continua which admit a closed, order-dense partial with unique minimal element, We improve his characterization by replacing order dense by the weaker monotone, and we give two other characterizations, one of which is completely topological. 2. Definitions. A partial < on a space X is a relation on X (a subset of X XX) which is reflexive, transitive, and antisymmetric. We assume throughout that X is a Hausdorff space, but not necessarily metrizable. The term arc is used to denote a continuum irreducibly connected between two points. For xCX we set L(x) = {yX:y<x}, and for ACX, L(A)=U{L(x):xCA}. We say that < is monotone if L(x) is connected for each xEX or continuous