Tableau method for residuated logic

Residuated logic is a generalization of intuitionistic logic, which does not assume the idempotence of the conjunction operator. Such generalized conjunction operators have proved important in expert systems (in the area of Approximate Reasoning) and in some areas of Theoretical Computer Science. Here we generalize the intuitionistic tableau procedure and prove that this generalized tableau method is sound for the semantics (the class of residuated algebras) of residuated propositional calculus (RPC). Since the axioms of RPC are complete for the semantics we may conclude that whenever a formula 0 is tableau provable, it is deducible in RPC. We present two different approaches for constructing residuated algebras which give us countermodels for some formulas φ which are not tableau provable. The first uses the fact that the theory of residuated algebras is equational, to construct quotients of free algebras. The second uses finite algebras. We end by discussing a number of open questions.     

如何利用单纯形表上的信息

本文分析了求解线性规划的基本方法－－单纯形法所使用的单纯形表，将表中所提供的信息分为直接信息和间接信息两类，论述了如何充分利用这些信息的方法。例如如何由最终表求原问题、如何利用表中的数据互相推演和校正等。这是一篇教学经验的总结，对初学者可能有一定的帮助。     

Tableau stabilization and rectangular tableaux fixed by promotion powers

Journal of Algebraic Combinatorics - We introduce tableau stabilization, a new phenomenon and statistic on Young tableaux based on jeu de taquin. We investigate bounds for tableau stabilization,...     

A Tableau Approach to the Representation Theory of 0-Hecke Algebras

A 0-Hecke algebra is a deformation of the group algebra of a Coxeter group. Based on work of Norton and Krob-Thibon, we introduce a tableau approach to the representation theory of 0-Hecke algebras of type A, which resembles the classic approach to the representation theory of symmetric groups by Young tableaux and tabloids. We extend this approach to types B and D, and obtain a correspondence between the representation theory of 0-Hecke algebras of types B and D and quasisymmetric functions and noncommutative symmetric functions of types B and D. Other applications are also provided.     

A Tableau Formula of Double Grothendieck Polynomials for 321-Avoiding Permutations

Annals of Combinatorics - In this article, we prove a tableau formula for the double Grothendieck polynomials associated to 321-avoiding permutations. The proof is based on the compatibility of the...     

Second structure relation for q-semiclassical polynomials of the Hahn Tableau

The q-classical orthogonal polynomials of the q-Hahn Tableau are characterized from their orthogonality condition and by a first and a second structure relation. Unfortunately, for the q-semiclassical orthogonal polynomials (a generalization of the classical ones) we find only in the literature the first structure relation. In this paper, a second structure relation is deduced. In particular, by means of a general finite-type relation between a q-semiclassical polynomial sequence and the sequence of its q-differences such a structure relation is obtained.     

Blockbusting: Or How to Handle Complex Constraints in the Transportation Tableau

This paper considers how far it is possible to manipulate the layout of a Transportation (LP) matrix to handle the following characteristics:

1. i)
   Any combination of =, ?, ?, and (? and ? combined) constraints, henceforward referred to collectively as "mixed constraints".
    
2. ii)
   The application of mixed constraints to (a) single cells, (b) whole rows or columns of cells, (c) groups of cells within a row or column, (d) groups of several rows or groups of several columns.
    

It is shown how most models which possess some or all of these characteristics can be represented in a compact tableau and a solution found by applying a well-known algorithm. The advantages of manipulating the layout in this way are then reviewed.     

A Direct Way to Find the Right Key of a Semistandard Young Tableau

The right key of a semistandard Young tableau is a tool used to find Demazure characters for ${sl_n(\mathbb{C})}$ . This paper gives methods to obtain the right and left keys by inspection of the semistandard Young tableau.     

A Proof of the Murnaghan–Nakayama Rule Using Specht Modules and Tableau Combinatorics

Annals of Combinatorics - The Murnaghan–Nakayama rule is a combinatorial rule for the character values of symmetric groups. We give a new combinatorial proof by explicitly finding the trace...     

Wakamatsu-silting complexes

We introduce Wakamatsu-silting complexes (resp., Wakamatsu-tilting complexes) as a common generalization of both silting complexes (resp., tilting complexes) and Wakamatsu-tilting modules. Characterizations of Wakamatsu-silting complexes are given. In particular, we show that a complex T is Wakamatsu-silting if and only if its dual DT is Wakamatsu-silting. It is conjectured that compact Wakamatsu-silting complexes are just silting complexes. We prove that the conjecture lies under the finitistic dimension conjecture.     

Semi-tilting complexes

We introduce the notion of semi-tilting complexes, which is a small generalization of tilting complexes. Interesting examples include APR-semitilting complexes, etc. Note that non-trivial semi-tilting complexes exist for any non-semisimple non-local artin algebras, while tilting complexes may not. We extend interesting results in the tilting theory to semi-tilting complexes. As corollaries, we also obtain some new characterizations of tilting complexes.     

Balanced complexes and complexes without large missing faces

The face numbers of simplicial complexes without missing faces of dimension larger than i are studied. It is shown that among all such (d−1)-dimensional complexes with non-vanishing top homology, a certain polytopal sphere has the componentwise minimal f-vector; and moreover, among all such 2-Cohen–Macaulay (2-CM) complexes, the same sphere has the componentwise minimal h-vector. It is also verified that the l-skeleton of a flag (d−1)-dimensional 2-CM complex is 2(d−l)-CM, while the l-skeleton of a flag piecewise linear (d−1)-sphere is 2(d−l)-homotopy CM. In addition, tight lower bounds on the face numbers of 2-CM balanced complexes in terms of their dimension and the number of vertices are established.     

Threshold graphs, shifted complexes, and graphical complexes

We consider a variety of connections between threshold graphs, shifted complexes, and simplicial complexes naturally formed from a graph. These graphical complexes include the independent set, neighborhood, and dominance complexes. We present a number of structural results and relations among them including new characterizations of the class of threshold graphs.     

Subword complexes, cluster complexes, and generalized multi-associahedra

In this paper, we use subword complexes to provide a uniform approach to finite-type cluster complexes and multi-associahedra. We introduce, for any finite Coxeter group and any nonnegative integer k, a spherical subword complex called multi-cluster complex. For k=1, we show that this subword complex is isomorphic to the cluster complex of the given type. We show that multi-cluster complexes of types A and B coincide with known simplicial complexes, namely with the simplicial complexes of multi-triangulations and centrally symmetric multi-triangulations, respectively. Furthermore, we show that the multi-cluster complex is universal in the sense that every spherical subword complex can be realized as a link of a face of the multi-cluster complex.     

Pre-balanced dualizing complexes

We study pre-balanced dualizing complexes over noncommutative complete semilocal algebras and prove an analogue of Van den Bergh’s theorem [VdB, 6.3]. The relationship between pre-balanced dualizing complexes and Morita dualities is studied. Some immediate applications to classical ring theory are also given.     

Coxeter cochain complexes

We define the Coxeter cochain complex of a Coxeter group (G, S) with coefficients in a ?[G]-module A. This is closely related to the complex of simplicial cochains on the abstract simplicial complex I(S) of the commuting subsets of S. We give some representative computations of Coxeter cohomology and explain the connection between the Coxeter cohomology for groups of type A, the (singular) homology of certain configuration spaces, and the (Tor) homology of certain local Artin rings.