Generalized projections in [formula omitted

Mazurkiewicz trace theory is not powerful enough to describe concurrency paradigms as, for instance, the “Producer / Consumer”. We propose in this paper a generalization of Mazurkiewicz trace monoids which allows to model such problems. We consider quotients of the free monoids by congruences which preserve the commutative images of words. An equivalence class in the quotient monoid consists of all the sequential observations of a distributed computation. In order to characterize congruences which do model concurrency, we study the relationship of this approach and the classical representation of distributed computations with partial orders. We show that the only congruences for which the classes can be represented by partial orders and the concatenation transfers modularly to partial orders are congruences generated by commutations, that is trace congruences. We prove necessary conditions and sufficient conditions on congruences so that their classes can be represented by partial orders. In particular, an important sufficient condition covers both trace congruences and the “Producer / Consumer” congruence.

Three partial orders on the types of points in $\beta N$ are defined and studied in this paper. Their relation to the types of points in $\beta N - N$ is also described.

Continuing the development [4] of an aspect of the approach to the axiomatization of quantum mechanics of G. W. Mackey [3], we consider here the real linear space X of signed measures on the set P of events generated by the states, and the set F0 of linear functionals on X which are induced in a natural way by the bounded observables. A necessary and sufficient condition for two events to be simultaneously measurable is found in terms of the order structure of F0, with the following consequence: if F0 is a lattice, P is deterministic. At the opposite extreme, F0 is said to be an anti-lattice1 if the greatest lower bound exists only for comparable pairs of its elements and we show in this case that the center of P is trivial. Our results extend those of R. V. Kadison [l], in which F0 and P are the self-adjoint operators and projections respectively in a uniformly closed self-adjoint operator algebra. While the framework and plan of the proofs were inspired by Kadison's work, almost none of the apparatus used by him is available here with the result that, in detail, our techniques are quite different from his. Let P be a weakly modular partially ordered set (see [4]). A function x from P to the non-negative real numbers and + oo is said to be a measure if x(0) =0 and x is countably additive in the sense that whenever {a,} is a pairwise orthogonal sequence of elements of P, then x(Ua?) = 2Zxiad- If x is a measure and {bt} EP is an increasing (decreasing) sequence with supremum (infimum) b, then x(&?) —*x(6). A countably additive function x from P to the extended real numbers is a signed measure if x(0) =0 and x takes on at most one of the values + =o and — oo ; x is finite if x(l) is finite. Define the functions 5 and i on the signed measures on P by six) = sup(x(a) : aEP\, fx) =inf {x(a): aEP} and set ||x|| =s(x)—?(x). Clearly ||x|| < oo if and only if x is finite. It iseasy toseethat||x|| = sup {x(a)— xia'):aE P} ■ Lemma 1. Let X be a real linear space of finite signed measures on P. Then the function || || defined above is a norm for X and, under this norm and its natural partial ordering, X is a partially ordered normed linear space. That is,2

Special types of coverings and axiomatization of rough sets based on partial orders

We study the star and minus partial orders on the set B(H,K) of all bounded operators acting from a Hilbert space H to a Hilbert space K. We extend and strengthen some results from matrix case to the case of general operators, which may not possess generalized inverses. By means of some norm inequalities, we give necessary and sufficient condition under which two operators have orthogonal ranges, and thus, we give a characterization of the star partial order. When A = PB for some projection P, we prove the equivalence of A<∗B with f(AA∗)A<∗f(BB∗)B for a wide class of continuous functions f. Also, we prove that A<∗B if and only if A<−B and A2<−B2 when A is a weak EP operator and B is a self-adjoint operator. Finally, we consider the Moore–Penrose invertibility and the ordinary invertibility of a linear combination of operators when they are related with one of these two orders.

On Monotone Strategy Equilibria in Simultaneous Auctions for Complementary Goods

In this paper, we study the concepts of male optimality and uniqueness of stable marriages for partially ordered preferences. We give an algorithm to find a stable marriage that is male optimal, and a sufficient condition on the preferences, which guarantees the uniqueness of stable marriages.

Product partial orders with the sperner property

A matrix inequality and admissibility of linear estimators with respect to the mean square error matrix criterion

Almost all partial order reduction techniques proposed for time Petri nets (TPNs in short) are based on the notion of Partially Ordered Sets. The idea is to explore simultaneously, by relaxing some firing order constraints of persistent transitions (An enabled transition is persistent, if it cannot be disabled until its firing.), several equivalent sequences, while computing the convex hull of the abstract states reached by these equivalent sequences. However, unlike timed automata, in the TPN state space abstractions, the union of the abstract states reached by different interleavings of the same set of non conflicting transitions is not necessarily identical to their convex hull. Moreover, the convex hull over-approximation preserves neither the boundedness nor the reachability properties of the TPN. In this context, the main challenge is to establish sufficient conditions over transitions that ensure, in addition to their persistency, identity between the union and the convex hull of the abstract states reachable by their different interleavings. This paper shows how to weaken the sufficient conditions proposed in the literature, by taking into better account the structure, the marking, the static and the dynamic time parameters of the TPN.

In this paper, we provide a constructive proof for threshold representations of partial orderings in a finite setting and extend the construction to arbitrary sets. This contributes to a long-standing open problem posed in Nakamura (2002) and Aleskerov et al. (2007). We identify and discuss aspects of our proof that do not directly generalize to an infinite setting and, for each, provide intuitive sufficient conditions. • ≻ is a partial order if and only if it is metric representable (finite case). • Constructive proof via the minimum weighted path metric. • The representation is extended to infinite sets under mild conditions. • These resolve an open problem posed in Nakamura (2002) and Aleskerov et al. (2007).

Let X be either the unit interval in ℝ or the unit disk \( \mathbb{D} \) in ℂ. Chalendar, Flattot, and Partington [2] study weighted composition operators Tw,ϒ on L2(X), where w ∈ L∞(X), ϒ : X → X is injective, and Tw,ϒf = wfoϒ for f ∈ L2(X). They introduce a (strict) partial order ≻ on X associated with Tw,ϒ and use it to obtain a sufficient condition for convergence of the sequence (Tn w,ϒyn) where (yn) is a backward minimal-vector sequence for Tw,ϒ. For the L2(\( \mathbb{D} \)) case, they give a detailed analysis of the situation where ϒ is linear-fractional. Through further study of the partial order ≻, we are able to generalize results from [2] that apply when ϒ is linear-fractional, replacing the linear-fractional hypotheses with univalence. In particular, our work yields generalizations of an invariant-subspace theorem in [2].

In this paper we present a theory of relational database systems based on the partition lattice, which represents a new mathematical approach to the structure of relational database systems. A partition lattice can be defined for any given relation. This partition lattice is shown to be a meet-morphic image of the Boolean algebra of subsets of the attribute set. The partial ordering in the lattice is proved to be equivalent to the concept of functional dependency, and thus Armstrong's axioms for functional dependencies are proved. We solve the problem of finding the list of all keys by seeking the prime implicants of the Boolean function associated with the principal ideals generated by the attributes. We demonstrate the properties of the Boyce-Codd Normal Form (BCNF), and give a modified algorithm for synthesizing an information-lossless BCNF based on the principal filter. The necessary and sufficient conditions for multivalued dependency (MVD) are given in terms of a lattice equation, and the inference rules of MVD are proved. The necessary and sufficient conditions for join dependency (JD) are given; consequently, we can prove the known result that acyclic join dependency (AJD) is equivalent to a set of MVDs. The concept of data independence is introduced, and is extended to conditional independence and mutual independence. We established this algebraic theory of relational databases in the same spirit that the theory of probability was constructed. We present a comparison that demonstrates the similarities.

Combinatorics of quasi-hereditary structures

Two partial orders that play an important role in the combinatorics of words, can be defined in a natural way on the free monoid X* generated by the finite alphabet X: the infix and the embedding orders. A set C of nonempty words is called an infix code (hypercode) over X if C is an antichain with respect to the infix (embedding) order. A set of words is said to be e-convex if it is convex with respect to the embedding order. Two characterizations of the e-convex infix codes are given as well as a sufficient condition for such codes to be finite. It is shown that the family EIC(X) of the e-convex infix codes with the empty word forms, under the operation of concatenation, a free submonoid of the free monoid B(X) of the biprefix codes and that the generating alphabet of EIC(X) is a sub-alphabet of the generating alphabet of B(X).