Computing on-line the lattice of maximal antichains of posets

We consider the on-line computation of the lattice of maximal antichains of a finite poset . This on-line computation satisfies what we call the linear extension hypothesis: the new incoming vertex is always maximal in the current subposet of . In addition to its theoretical interest, this abstraction of the lattice of antichains of a poset has structural properties which give it interesting practical behavior. In particular, the lattice of maximal antichains may be useful for testing distributed computations, for which purpose the lattice of antichains is already widely used. Our on-line algorithm has a run time complexity of , where |P| is the number of elements of the poset, , |MA(P)| is the number of maximal antichains of and (P) is the width of . This is more efficient than the best off-line algorithms known so far.     

Homotopy type of smooth weighted complete intersections

If the total degree d has no prime divisors less than(n+3)/2,then we prove that the homotopy type of complex odd dimensional smooth weighted complete intersection Xn(d;w) is determined by the dimension n,the total degree d,the Euler characteristic and the Kervaire invariant,provided that the weights w =(ω0,...,ωn+r) is pairwise relatively prime.     

N-free posets as generalizations of series-parallel posets

N-Free posets have recently taken some importance and motivated many studies. This class of posets introduced by Grillet [8] and Heuchenne [11] are very related to another important class of posets, namely the series-parallel posets, introduced by Lawler [12] and studied by Valdes et al. [21]. This paper shows how N-free posets can be considered as generalizations of series-parallel posets, by giving a recursive construction of N-free posets. Furthermore we propose a linear time algorithm to recognize and decompose any N-free poset. This yields some very naturel problems, namely: which are the properties(such as linear time algorithm for some invariant) of series-parallel posets that are kept for N-free posets?     

Incidence posets of trees in posets of large dimension

In this paper, we investigate substructures of partially ordered sets which must be present whenever the dimension is large. We show that for eachn1, ifT is a tree onn vertices and ifP is any poset having dimension at least 4n ⁶, then eitherP or its dual contains the incidence poset ofT as a suborder.     

Homotopy type of the boolean complex of a Coxeter system

In any Coxeter group, the set of elements whose principal order ideals are boolean forms a simplicial poset under the Bruhat order. This simplicial poset defines a cell complex, called the boolean complex. In this paper it is shown that, for any Coxeter system of rank n, the boolean complex is homotopy equivalent to a wedge of (n−1)-dimensional spheres. The number of such spheres can be computed recursively from the unlabeled Coxeter graph, and defines a new graph invariant called the boolean number. Specific calculations of the boolean number are given for all finite and affine irreducible Coxeter systems, as well as for systems with graphs that are disconnected, complete, or stars. One implication of these results is that the boolean complex is contractible if and only if a generator of the Coxeter system is in the center of the group.     

Modular posets and semigroups

Posets and poset homomorphisms (preserving both order and parallelism) have been shown to form a category which is equivalent to the category of pogroupoids and their homomorphisms. Among the posets those posets whose associated pogroupoids are semigroups are identified as being precisely those posets which are (C ₂+1)-free. In the case of lattices this condition means that the lattice is alsoN ₅-free and hence modular. Using the standard connection: semigroup to poset to pogroupoid, it is observed that in many cases the image pogroupoid obtained is a semigroup even if quite different from the original one. The nature of this mapping appears intriguing in the poset setting and may well be so seen from the semigroup theory viewpoint.     

Homotopy type of gauge groups of quaternionic line bundles over spheres

Let P be a principal S³-bundle over a sphere Sn, with n?4. Let GP be the gauge group of P. The homotopy type of GP when n=4 was studied by A. Kono in [A. Kono, A note on the homotopy type of certain gauge groups, Proc. Roy. Soc. Edinburgh Sect. A 117 (1991) 295-297]. In this paper we extend his result and we study the homotopy type of the gauge group of these bundles for all n?25.     

Canonical extensions of posets

We study a construction that produces a variety of completions of a given poset. The denseness and compactness properties of the completions obtained in this way are investigated. Next we focus our attention on three specific completions of a given poset that can be obtained through this construction—two of which have been called ‘the canonical extension’ of the poset in the literature. We investigate extensions of maps to these three completions. Although the extensions of unary operators need not be operators on the completions, we show that the extensions of unary residuated maps are residuated. We also investigate extensions of n-ary maps. In particular, we have a closer look at order-preserving n-ary maps and binary residuated maps.     

Parity representations of posets

Parity representations, introduced in this paper, comprise a new method of representation of posets that yields insight into the combinatorics of the poset of all intervals of a poset. Results here generalize some results previously obtained for the face lattices of binary partition polytopes.     

Homotopy Groups of Spheres and Lipschitz Homotopy Groups of Heisenberg Groups

We provide a sufficient condition for the nontriviality of the Lipschitz homotopy group of the Heisenberg group, ${\pi_m^{\rm Lip}(\mathbb{H}_n)}$ , in terms of properties of the classical homotopy group of the sphere, ${\pi_m(\mathbb{S}^n)}$ . As an application we provide a new simplified proof of the fact that ${\pi_n^{\rm Lip}(\mathbb{H}_n)\neq \{0\}, n=1,2,\ldots}$ , and we prove a new result that ${\pi_{4n-1}^{\rm Lip}(\mathbb{H}_{2n})\neq \{0\}}$ for n = 1,2,… The last result is based on a new generalization of the Hopf invariant. We also prove that Lipschitz mappings are not dense in the Sobolev space ${W^{1,p}(\mathcal{M},\mathbb{H}_{2n})}$ when ${\dim \mathcal{M} \geq 4n}$ and 4n?1 ≤  p < 4n.     

Counting finite posets and topologies

A refinement of an algorithm developed by Culberson and Rawlins yields the numbers of all partially ordered sets (posets) with n points and k antichains for n11 and all relevant integers k. Using these numbers in connection with certain formulae derived earlier by the first author, one can now compute the numbers of all quasiordered sets, posets, connected posets etc. with n points for n14. Using the well-known one-to-one correspondence between finite quasiordered sets and finite topological spaces, one obtains the numbers of finite topological spaces with n points and k open sets for n11 and all k, and then the numbers of all topologies on n14 points satisfying various degrees of separation and connectedness properties, respectively. The number of (connected) topologies on 14 points exceeds 10²³.