In this paper we consider the enumeration of ordered set partitions avoiding a permutation pattern of length 2 or 3. We provide an exact enumeration for certain special cases, and a recursive technique to exactly enumerate the appropriate set partitions in general. We also give some asymptotic results for the growth rates of the number of ordered set partitions avoiding a single pattern; includ...

In this paper, we investigate the notion of partition of a finite partially ordered set (poset, for short). We will define three different notions of partition of a poset, namely, monotone, regular, and open partition. For each of these notions we will find three equivalent definitions, that will be shown to be equivalent. We start by defining partitions of a poset in terms of fibres of some su...

We study the ordered set of rough sets determined by relations which are not necessarily reflexive, symmetric, or transitive. We show that for tolerances and transitive binary relations the set of rough sets is not necessarily even a semilattice. We also prove that the set of rough sets determined by a symmetric and transitive binary relation forms a complete Stone lattice. Furthermore, for the...

1 Department of Computer Science. University of Seville. Spain. {aguilar, riquelme, mtoro}@lsi.us.es Abstract. In this paper, an editing algorithm based on the projection of the examples in each dimension is presented. The algorithm, that we have called EOP (Editing by Ordered Projection) has some interesting characteristics: important reduction of the number of examples from the database; lowe...

If a decision maker prefers x to y to z, would he choose orderd set [x, z] or [y, x]? This article studies extension of preferences over individual alternatives to an ordered set which is prevalent in closed ballot elections with proportional representation and other real life problems where the decision maker is to choose from groups with an associated hierarchy inside. I introduce five ordina...

Let $S= \{e_1,\,e_2‎, ‎\ldots,\,e_m\}$ be an ordered subset of edges of a connected graph $G$‎. ‎The edge $S$-representation of an edge set $M\subseteq E(G)$ with respect to $S$ is the‎ ‎vector $r_e(M|S) = (d_1,\,d_2,\ldots,\,d_m)$‎, ‎where $d_i=1$ if $e_i\in M$ and $d_i=0$‎ ‎otherwise‎, ‎for each $i\in\{1,\ldots‎ , ‎k\}$‎. ‎We say $S$ is a global forcing set for maximal matchings of $G$‎ ‎if $...

We consider quasi-martingales indexed by a linearly order set. We show that such processes are isomorphic to a given class of (finitely additive) measures. From this result we easily derive the classical theorem of Stricker as well as the decompositions of Riesz, Rao and the supermartingale decomposition of Doob and Meyer.

The paper is devoted to knowledge discovery from data, taking into account prior knowledge about preference semantics in patterns to be discovered. The data concern a set of situations (objects, states, examples) described by a set of attributes (properties, features, characteristics). The attributes are, in general, divided into condition and decision attributes, corresponding to input and out...

Let P = {pi|i = 1, 2, . . . , n} be a set of n elements with partial order defined as follows: pi ≤ pj if and only if i divides j. Assume real numbers are assigned to all members of P in an order preserving manner, this results in a set A of n real numbers. Given an unknown x ∈ R, we want to determine whether x is belong to A or not, by comparing x with as few elements of A as possible. We pres...