Situations Endowed with Lattice Structure

Order-theoretically connected posets are introduced and applied to create the notion of T-connectivity in ordered topological spaces. As special cases T-connectivity contains classical connectivity, order-connectivity, and link-connectivity.


INTRODUCTION
In the literature on order and topology (ordered topological spaces, topological lat- tices, order-determined topologies, etc.) connectivity has generally been introduced as a topological property.Given some natural compatibility between topology and order, much is known about the impact different connectivity properties have on the order in question.
Classical theory (in case of total order), as well as [2], [4], and [5], have revealed the close relationship between topological connectivity on one hand and the order-theoretical prop- erties "conditionally complete" and "order dense" on the other.The aim of this note is to define a concept of order-theoretical connectivity (using conditional completeness and order density), and then to apply this concept to the theory of ordered topological spaces.First, some relevant remarks on (partial) order and connectivity will be given.
A triple S (S,r,o) consisting of a set, an order o and a topology r on that set is called an ordered topological space.The space S is called order-connected (cf.[1]) if all order-preserving continuous maps from S to the space 2 are constants (2 _--the set {0,1} endowed with discrete topology and natural order); connected (path-connected) if it is connected (path-connected) as a topological space; and link-connected if any two points of S can be joined by a finite set of connected links of S (link between x and y maximal chain of Ix, y]).A class C of subsets of a given set S is called a connectivity system (cf.[2]) if any set A C_ S is a member of C whenever for all x, y E A there are sets C1,C2,...,C, C in A with x Cl,y C,,, and CiNCi+I for 1,2,...,n-1.The classes of order-connected, or connected, or path-connected, or link-connected sets constitute connectivity systems.
An ordered topological space is a Ti-ordered space (an i-space) if every ma:dmal chain inherits a topology which is finer than or equal to (which coincides with) its own interval topology (see [2], [5]).Note that for any lattice all topologies between the interval topology and Birkhoff's order topology are i-topologies (cf.[5])."Conditionally complete" and "order dense" will be abbreviated "cc" and "od", re- spectively.A subchain of an ordered set is called cc (od) if it is cc (od) regarded as an ordered space in its own right.It is important to recall that (A) a chain is connected in its interval topology iff it is od and cc; (B) art ordered set is od (cc) iff all mazimal chains of it are od (cc); (C) art od and cc chain between two points of a given ordered set is always a complete link.
1. ORDER-THEORETICAL CONNECTIVITY Let P be an arbitrary poset.Regard the equivalence relation ,ord defined in P by a ,,ord b === a and b are joined by a finite set of od and cc links in P.
The ,,ord classes are called the order-theoretical components in P. In case of only one equivalence class, P is order-theoretically connected.The order-theoretically connected subsets (defined in the obvious way) of any poset are a connectivity system in the scnse of [2].Morcover, a poset is order-theoretically connected iff it is link-connected in its interval topology (or equivalently, in any of its i-structures).
PROPOSITION 1.For any a E P, the order-theoretical component of a equals the maximal order-theoretically connected subset of P containing a.
LEMMA 2. Irt any product poser P IIPi, all projections of an od and cc link J are od and cc links.
PaOOF.For a topological proof, endow P and the Pi's with Birkhoff's order topology o, note that J is a connected set in o(P) and that all projections are continuous maps o(P) --, o(Pi).Thus pri(J) is a connected chain in o(Pi), and since pr(g) carries a structure which is finer than its own interval topology it is an od and cc chain.
Prtoor.To see the first part for n 2, take a2 q P2, express the product set P x P2 as Ja, epl[(P1 x {a2}) U ({el} x P2)], and use Lemma 2. Then proceed via induction over the natural numbers.The remaining part is straightforward.
EXAMPLE 4 (denumerable products).Consider the Euclidean plane, let l, denote the line segment (y -,0 < x < 1) and m the line segment connecting the right endpoint of l, and the left endpoint of lk+l.Define the lattice L to be the union of all the Ik's and m's (k fi Z+) endowed with the natural order of the plane.Obviously, L is order-theoretically connected.To see that the product lattice P =-L z+ is not, take pairs of points am, b, in L which can be joined by a set of no less than n od and cc links of L and note (a,) (b,) is not satisfied in P (use Lemma 2).EXAMPLE 5 (basic remarks).The od and cc lattices form a class of order-theoretically connected structures which is closed under formation of arbitrary products.A lattice is od and cc iff all maximal chains (or all links) are order-theoretically connected.As was demonstrated in Lemma 2, projections preserve order-theoretical connectivity.However, a lattice with prime ideals (for instance any distributive lattice) allows a lattice-morphisn onto the lattice 2. This shows that order-theoretical connectivity cannot be characterized via morphisms (and, of course, that morphisms do not preserve order-theoretical connec- tivity).Cantor's teepee (cf.[3]) with base interval [0, 1] and top (1,1)is a connected subset of R2; nevertheless, all its order-theoretical components are singletons (which is a quite natural order-theoretical property of this structure).Finally, note that the topo- logical closure of the lattice L of Example 4 (the Euclidean topology) consists of two order-theoretical components, although L is order-theoretically connected.
In ordered topological spaces S, order-theoretical connectedness relates to path- connectedness, as demonstrated below.PROPOSITION 6. Suppose each x E S has an order-theoretically connected nezghborhood.
Then S is order-theoretically connected if S is path-connected.
PROOF" Suppose x, y E S and p" I --.S is a path between x and y.For each z p(I), let Nz be an order-theoretically connected neighborhood of z, and let -lz C_ Nz be an open neighborhood of z.Consider {Ni}'=l where {M'i}'=l is a subcover of p(I) from {Mz z p(I)}.Now (Ui=iNi)k CI (Ll=k+1Ni) # $ for any 1 _< k <_ n, for otherwise p-l(Ui=Mi Up-l(Ui=k+x}Ii) would separate I for some k.Since the order-theoretically connected subsets of S form a connectivity system, it follows that x and y are in the same order-theoretical component and thus S is order-theoretically connected.
An obvious sufficient condition for order-theoretical connectedness to imply path- connectedness is that every cc and od link frown x to y be a path from x to y. Applying this and the proposition we finally get COROLLARY 7.An open subset of R is path-connected iff it is order-theoretically connected, given the usual topology and order on Rn.

T-CONNECTIVITY
Order-theoretical connectedness is now employed to create a natural connectivity con- cept in the category of (partially) ordered topological spaces and order-preserving contin- uous maps (here called morphisms).
Let T denote an ordered topological space.The ordered topological space S is called T-connected if for every morphism S T the image T(S) is an order-theoretically connected set in T. A subset of an arbitrary ordered topological space is a T-connected set if it is a T-connected space (regarded as a subspace).Every ordered topological space can be partitioned into maximal T-connected sets, the T-components of the space.
Obviously, the T-connected sets in any ordered topological space form a connectiv- ity system in the sense of [2].T-connectivity provides the realm for several well-known connectivity concepts.For instance, an ordered topological space is order-connected iff it is 2-connected.As a special case of this, note that a topological space is connected in the classical sense iff it is a 2-connected ordered topological space when equipped with the discrete order.A connected space endowed with any order is always an R-connected ordered topological space.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation