scholarly journals Some lattices of closure systems on a finite set

2004 ◽  
Vol Vol. 6 no. 2 ◽  
Author(s):  
Nathalie Caspard ◽  
Bernard Monjardet
Keyword(s):  
Systematic Study ◽  
Closed Set ◽  
Covering Relation ◽  
Closure Systems ◽  
Irreducible Elements ◽  
Finite Set ◽  
International Audience ◽  
Convex Geometries

International audience 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 \prec 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 \prec C'. We also do a systematic study of these lattices of closure systems, characterizing for instance their join-irreducible and their meet-irreducible elements.

scholarly journals The Prime Stems of Rooted Circuits of Closure Spaces and Minimum Implicational Bases

10.37236/3068 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Masataka Nakamura ◽  
Kenji Kashiwabara
Keyword(s):  
Convex Geometry ◽  
Closure Operators ◽  
Optimal Basis ◽  
Closed Set ◽  
Closed Sets ◽  
Closure Systems ◽  
Closure Spaces ◽  
Convex Geometries ◽  
Necessary And Sufficient ◽  
Special Case

A rooted circuit is firstly introduced for convex geometries (antimatroids). We generalize it for closure systems or equivalently for closure operators. A rooted circuit is a specific type of a pair $(X,e)$ of a subset $X$, called a stem, and an element $e\not\in X$, called a root. We introduce a notion called a 'prime stem', which plays the key role in this article. Every prime stem is shown to be a pseudo-closed set of an implicational system. If the sizes of stems are all the same, the stems are all pseudo-closed sets, and they give rise to a canonical minimum implicational basis. For an affine convex geometry, the prime stems determine a canonical minimum basis, and furthermore  gives rise to an optimal basis. A 'critical rooted circuit' is a special case of a rooted circuit defined for an antimatroid. As a precedence structure, 'critical rooted circuits' are necessary and sufficient to fix an antimatroid whereas critical rooted circuits are not necessarily sufficient to restore the original antimatroid as an implicational system. It is shown through an example.

scholarly journals Counting occurrences for a finite set of words: an inclusion-exclusion approach

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Frédérique Bassino ◽  
Julien Clément ◽  
J. Fayolle ◽  
P. Nicodème
Keyword(s):  
Generating Function ◽  
Exclusion Principle ◽  
Complete Proof ◽  
Detailed Proof ◽  
Finite Set ◽  
International Audience ◽  
The One ◽  
Maple Package

International audience In this paper, we give the multivariate generating function counting texts according to their length and to the number of occurrences of words from a finite set. The application of the inclusion-exclusion principle to word counting due to Goulden and Jackson (1979, 1983) is used to derive the result. Unlike some other techniques which suppose that the set of words is reduced (<i>i..e.</i>, where no two words are factor of one another), the finite set can be chosen arbitrarily. Noonan and Zeilberger (1999) already provided a MAPLE package treating the non-reduced case, without giving an expression of the generating function or a detailed proof. We give a complete proof validating the use of the inclusion-exclusion principle and compare the complexity of the method proposed here with the one using automata for solving the problem.

scholarly journals Graphs with many vertex-disjoint cycles

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Dieter Rautenbach ◽  
Friedrich Regen
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Linear Time ◽  
Simple Extension ◽  
Disjoint Cycles ◽  
Cyclomatic Number ◽  
Finite Set ◽  
Extension Rule ◽  
International Audience ◽  
Vertex Disjoint

Graph Theory International audience We study graphs G in which the maximum number of vertex-disjoint cycles nu(G) is close to the cyclomatic number mu(G), which is a natural upper bound for nu(G). Our main result is the existence of a finite set P(k) of graphs for all k is an element of N-0 such that every 2-connected graph G with mu(G)-nu(G) = k arises by applying a simple extension rule to a graph in P(k). As an algorithmic consequence we describe algorithms calculating minmu(G)-nu(G), k + 1 in linear time for fixed k.

scholarly journals The lattices of closure systems, closure operators, and implicational systems on a finite set: a survey

2003 ◽  
Vol 127 (2) ◽  
pp. 241-269 ◽  
Author(s):  
Nathalie Caspard ◽  
Bernard Monjardet
Keyword(s):  
Closure Operators ◽  
Closure Systems ◽  
Finite Set

scholarly journals Alignments, crossings, cycles, inversions, and weak Bruhat order in permutation tableaux of type $B$

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Soojin Cho ◽  
Kyoungsuk Park
Keyword(s):  
Bruhat Order ◽  
Type B ◽  
Coxeter System ◽  
Covering Relation ◽  
Signed Permutations ◽  
Weak Bruhat Order ◽  
International Audience

International audience Alignments, crossings and inversions of signed permutations are realized in the corresponding permutation tableaux of type $B$, and the cycles of signed permutations are understood in the corresponding bare tableaux of type $B$. We find the relation between the number of alignments, crossings and other statistics of signed permutations, and also characterize the covering relation in weak Bruhat order on Coxeter system of type $B$ in terms of permutation tableaux of type $B$. De nombreuses statistiques importantes des permutations signées sont réalisées dans les tableaux de permutations ou ”bare” tableaux de type $B$ correspondants : les alignements, croisements et inversions des permutations signées sont réalisés dans les tableaux de permutations de type $B$ correspondants, et les cycles des permutations signées sont comprises dans les ”bare” tableaux de type $B$ correspondants. Cela nous mène à relier le nombre d’alignements et de croisements avec d’autres statistiques des permutations signées, et aussi de caractériser la relation de couverture dans l’ordre de Bruhat faible sur des systèmes de Coxeter de type $B$ en termes de tableaux de permutations de type $B$.

scholarly journals Are even maps on surfaces likely to be bipartite?

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Guillaume Chapuy
Keyword(s):  
Asymptotic Behaviour ◽  
Planar Map ◽  
Generating Series ◽  
Finite Set ◽  
Positive Genus ◽  
International Audience ◽  
Special Case

International audience It is well known that a planar map is bipartite if and only if all its faces have even degree (what we call an even map). In this paper, we show that rooted even maps of positive genus $g$ chosen uniformly at random are bipartite with probability tending to $4^{−g}$ when their size goes to infinity. Loosely speaking, we show that each of the $2g$ fundamental cycles of the surface of genus $g$ contributes a factor $\frac{1}{2}$ to this probability.We actually do more than that: we obtain the explicit asymptotic behaviour of the number of even maps and bipartite maps of given genus with any finite set of allowed face degrees. This uses a generalisation of the Bouttier-Di Francesco-Guitter bijection to the case of positive genus, a decomposition inspired by previous works of Marcus, Schaeffer and the author, and some involved manipulations of generating series counting paths. A special case of our results implies former conjectures of Gao.

scholarly journals NBC Complexes of Convex Geometries

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Kenji Kashiwabara ◽  
Masataka Nakamura
Keyword(s):  
Convex Geometry ◽  
Decomposition Theorem ◽  
International Audience ◽  
Generating Theorem ◽  
Convex Geometries ◽  
Broken Circuit

International audience We introduce a notion of a $\textit{broken circuit}$ and an $\textit{NBC complex}$ for an (abstract) convex geometry. Based on these definitions, we shall show the analogues of the Whitney-Rota's formula and Brylawski's decomposition theorem for broken circuit complexes on matroids for convex geometries. We also present an Orlik-Solomon type algebra on a convex geometry, and show the NBC generating theorem.

scholarly journals Finitely Presented Heyting Algebras

1998 ◽  
Vol 5 (30) ◽  
Author(s):  
Carsten Butz
Keyword(s):  
Simple Proof ◽  
Heyting Algebra ◽  
Distributive Lattices ◽  
Heyting Algebras ◽  
Model Property ◽  
Disjunction Property ◽  
Irreducible Elements ◽  
Finite Set ◽  
Finite Distributive Lattices ◽  
Finitely Presented

In this paper we study the structure of finitely presented Heyting<br />algebras. Using algebraic techniques (as opposed to techniques from proof-theory) we show that every such Heyting algebra is in fact co- Heyting, improving on a result of Ghilardi who showed that Heyting algebras free on a finite set of generators are co-Heyting. Along the way we give a new and simple proof of the finite model property. Our main technical tool is a representation of finitely presented Heyting algebras in terms of a colimit of finite distributive lattices. As applications we construct explicitly the minimal join-irreducible elements (the atoms) and the maximal join-irreducible elements of a finitely presented Heyting algebra in terms of a given presentation. This gives as well a new proof of the disjunction property for intuitionistic propositional logic.<br />Unfortunately not very much is known about the structure of Heyting algebras, although it is understood that implication causes the complex structure of Heyting algebras. Just to name an example, the free Boolean algebra on one generator has four elements, the free Heyting algebra on one generator is infinite.<br />Our research was motivated a simple application of Pitts' uniform interpolation theorem [11]. Combining it with the old analysis of Heyting algebras free on a finite set of generators by Urquhart [13] we get a faithful functor J : HAop<br />f:p: ! PoSet; sending a finitely presented Heyting algebra to the partially ordered set of its join-irreducible elements, and a map between Heyting algebras to its leftadjoint<br />restricted to join-irreducible elements. We will explore on the induced duality more detailed in [5]. Let us briefly browse through the contents of this paper: The first section<br />recapitulates the basic notions, mainly that of the implicational degree of an element in a Heyting algebra. This is a notion relative to a given set of generators. In the next section we study nite Heyting algebras. Our contribution is a simple proof of the nite model property which names in particular a canonical family of nite Heyting algebras into which we can<br />embed a given finitely presented one.<br />In Section 3 we recapitulate the standard duality between nite distributive lattices and nite posets. The `new' feature here is a strict categorical<br />formulation which helps simplifying some proofs and avoiding calculations. In the following section we recapitulate the description given by Ghilardi [8]<br />on how to adjoin implications to a nite distributive lattice, thereby not destroying a given set of implications. This construction will be our major technical ingredient in Section 5 where we show that every nitely presented<br />Heyting algebra is co-Heyting, i.e., that the operation (−) n (−) dual to implication is dened. This result improves on Ghilardi's [8] that this is true<br />for Heyting algebras free on a finite set of generators. Then we go on analysing the structure of finitely presented Heyting algebras<br />in Section 6. We show that every element can be expressed as a finite join of join-irreducibles, and calculate explicitly the maximal join-irreducible elements in such a Heyting algebra (in terms of a given presentation). As a consequence we give a new proof of the disjunction property for propositional intuitionistic logic. As well, we calculate the minimal join-irreducible elements, which are nothing but the atoms of the Heyting algebra. Finally, we show how all this material can be used to express the category of finitely presented Heyting algebras as a category of fractions of a certain category with objects morphism between finite distributive lattices.

Finite Clones Containing All Permutations

1994 ◽  
Vol 46 (5) ◽  
pp. 951-970 ◽  
Author(s):  
L. Haddad ◽  
I. G. Rosenberg
Keyword(s):  
Closed Set ◽  
Finite Set

AbstractLet A be a finite set with |A| > 2. We describe all clones on A containing the set SA of all permutations of A among its unary operations. (A clone on A is a composition closed set of finitary operations on A containing all projections). With a few exceptions such a clone C is either essentially unary or cellular i.e. there exists a monoid M of self-maps of A containing SA such that either C = (= all essentially unary operations agreeing with some f ∊ M) or C = ∪ Гh where 1 < h ≤ |A| and Гh consists of all finitary operations on A taking at most h values. The exceptions are subclones of Burle's clone or of its variant (provided |A| is even).

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