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Difference between revisions 1430 and 19719 on enwiki<b>FORREST LEE HORN</b> ---- [email protected] Address and telephone number available upon request. <b>OBJECTIVES</b>: Temporary or part time [[Life coaching]] contracts; contract, temporary or part time [[dispute resolution]] assignments as mediator and/or arbitrator. <b>SUMMARY OF QUALIFICATIONS</b> :<b>[[Human Resources]]</b> ::<b>*</b> Excellent human resources skills. Possess over 14 years experience in all aspects of human resources management and development. ::<b>*</b> Experienced in wide variety of organizational cultures ranging from Fortune 500 corporations to small shops and covering diverse product and service environments. ::<b>*</b> Conceived, developed and implemented training and team building programs for both military and civilian organizations. ::<b>*</b> Experienced in labor contract [[Negotiation]]. ::<b>*</b> Conceived and developed highly successful employee participation programs that provided both manufacturing and administrative staff with the opportunity to effect organizational change. Programs resulted in lower operational costs, decrease in quality defects, and increases in productivity, profitability, and personnel retention. ::<b>*</b> Assisted in the development of a supervisor assessment center. ::<b>*</b> Conceived, implemented and managed an organizational communication program which greatly facilitated communication between management, administration, engineering and manufacturing employees at all levels. ::<b>*</b> Redesigned compensation structure for an organization of over 3,000 people. :<b>Conflict Management</b> ::<b>*</b> Researched, designed and managed my own organization providing conflict management, including [[mediation]], [[conciliation]], and [[arbitration]] for both organizations and individuals. ::<b>*</b> Designed and implemented conflict resolution programs which included procedural guidelines, internal complaint procedures and independent arbitration options. ::<b>*</b> Managed and participated as mediator/arbitrator in over 200 conflict resolution cases. Several of these cases involved dollar amounts of over $100,000. :<b>Additional Experience and Skills</b> ::<b>*</b> Highly creative and experienced writer. ::<b>*</b> Directly managed organizations of up to 250 people. ::<b>*</b> Excellent presentation skills, including small group facilitation and public speaking. ::<b>*</b> Highly organized, creative, results-oriented. ::<b>*</b> Computer and Internet proficient; have taken the <b>Microsoft Certified Systems Engineer</b> course. Professional history includes [[United States Army]], [[General Electric]] Company and [[Exxon]] Corporation. <B>EDUCATION</b> :<b>* Masters Degree</b> in [[Industrial Relations]], secondary concentration in [[Organizational Development]], University of Cincinnati, Ohio :<b>* Bachelor of Science</b> in Business Administration, secondary concentration in Social Science, minors in Public Speaking and [[Economics]], [[Geneva College]], Beaver Falls, [[Pennsylvania]] :<b>* Employee Relations Management Training Program</b> graduate, General Electric Company :<b>* Continuing Seminars and Workshops</b>: interpersonal relations, training and development, equal employment practices, organizational policy development, organizational productivity planning, facilitator training, mediation and arbitration training, automated human resources systems, and others. References available upon request :''See also :'' [[F. Lee HornIn [[mathematics]], a '''filter''' is a special subset of a [[partially ordered set]]. A frequently used special case is the situation that the ordered set under consideration is just the [[power set]] of some set, ordered by set inclusion. Filters appear in [[order theory|order]] and [[lattice theory]], but can also be found in [[topology]]. The [[duality (order theory)|dual]] notion of a filter is an [[ideal (order theory)|ideal]]. Filters were introduced by [[Henri Cartan]] in [[1937]] and subsequently used by [[Bourbaki]] in their book ''[[Topologie Gènèrale]]''. An equivalent notion called [[net (topology)|net]] was developed in [[1922]] by [[Eliakim Hastings Moore|E. H. Moore]] and [[H. L. Smith]]. == General definition == A [[non-empty]] subset ''F'' of a partially ordered set (''P'',≤) is a '''filter''', if the following conditions hold: # For every ''x'', ''y'' in ''F'', there is some element ''z'' in ''F'', such that ''z'' ≤ ''x'' and ''z'' ≤ ''y''. (''F'' is a '''filter base''') # For every ''x'' in ''F'' and ''y'' in ''P'', ''x'' ≤ ''y'' implies that ''y'' is in ''F''. (''F is an upper set'') A filter is '''proper''' if it is not equal to the whole set ''P''. While the above definition is the most general way to define a filter for arbitrary posets, it was originally defined for [[lattice (order)|lattice]]s only. In this case, the above definition can be characterized by the following equivalent statement: A non-empty subset ''F'' of a lattice (''P'',≤) is a filter, [[iff]] it is an upper set that is closed under finite meets ([[infimum|infima]]), i.e., for all ''x'', ''y'' in ''F'', we find that ''x'' ^ ''y'' is also in ''F''. The smallest filter that contains a given element ''p'' is a '''principal filter''' and ''p'' is a '''principal element''' in this situation. The principal filter for ''p'' is just given by the set {''x'' in ''P'' | ''p'' ≤ ''x''} and is denoted by prefixing ''p'' with an upward arrow. The dual notion of a filter, i.e. the concept obtained by reversing all ≤ and exchanging ^ with v, is '''ideal'''. Because of this duality, the discussion of filters usually boils down to the discussion of ideals. Hence, most additional information on this topic (including the definition of '''maximal filters''' and '''prime filters''') is to be found in the article on [[ideal (order theory)|ideals]]. There is a separate article on [[ultrafilter]]s. == Filters of sets == An important special case of order filters are filters of sets, which are obtained by taking the powerset of a set ''S'' as a partial order, ordered by subset inclusion. Thus, a filter ''F'' on a [[set]] ''S'' is a set of [[Subset|subsets]] of ''S'' with the following properties: # ''S'' is in ''F''. (''F is non-empty'') # The empty set is not in ''F''. (''F is proper'') # If ''A'' and ''B'' are in ''F'', then so is their intersection. (''F is closed under finite joins'') # If ''A'' is in ''F'' and ''A'' is a subset of ''B'', then ''B'' is in ''F'', for all subsets ''B'' of ''S''. (''F is an upper set'') Note that this definition is in absolute correspondence with the general notion introduced above, since the powerset clearly forms a lattice. An important notion for filters of sets is that of a filter base. Given any subset ''T'' of ''P(S)'' such that the intersection of any finite subset of ''T'' is non-empty, there is a unique smallest filter ''F'' containing ''T'', called the filter generated by ''T''. If ''T'' is closed under finite intersections then ''F'' takes the simple form <math>F = \{ x : \exists y \in T \ y \subseteq x \}</math> and ''T'' is called a '''filter base''' for ''F'' A simple example of a filter is the set of all subsets of ''S'' that include a particular subset ''C'' of ''S''. Such a filter is called the "principal filter" generated by ''C''. The [[Frechet filter|Fréchet filter]] on an infinite set ''S'' is the set of all subsets of ''S'' that have finite complement. For any filter ''F'' on a set ''S'', the set function defined by :<math> m(A)=\left\{ \begin{matrix} \,1 & \mbox{if }A\in F \\ \,0 & \mbox{if }S\setminus A\in F \\ \,\mbox{undefined} & \mbox{otherwise} \end{matrix} \right. </math> is finitely additive -- a "[[measure (mathematics)|measure]]" if that term is construed rather loosely. Therefore the statement :<math>\left\{\,x\in S: \varphi(x)\,\right\}\in F</math> can be considered somewhat analogous to the statement that φ holds "almost everywhere". That interpretation of membership in a filter is used (for motivation, although it is not needed for actual ''proofs'') in the theory of [[ultraproduct]]s in [[model theory]], a branch of [[mathematical logic]]. ==Filters in topology== Filters are used in [[topology]] and analysis. They are a good way of talking about convergence, in a manner similar to the role of [[sequence|sequences]] in a [[metric space]]. Given a point ''x'' the set of all neighbourhoods of ''x'' is a filter, <math>N_x</math>. A (proper) filter which is a superset of <math>N_x</math> is said to converge to ''x'', written <math>F \to x</math>. Note that if <math>F \to x</math> and <math>F \subseteq G</math> then <math>G \to x</math>. Given a filter ''F'' on a set ''X'' and a function <math>f : X \to Y</math>, the set <math> \{ f(A) : A \in F \} </math> forms a filter base for a filter which, in a slight abuse of notation, we denote by <math>f(F)</math>. The following useful results hold: # ''X'' is [[Hausdorff]] iff every filter on ''X'' has at most one limit. # ''f'' is continuous at ''x'' iff <math> F \to x </math> implies <math>f(F) \to f(x)</math> # ''X'' is compact iff every filter on ''X'' is a subset of a convergent filter. # ''X'' is compact iff every ultrafilter on ''X'' converges. ==Filters in uniform spaces== Given a [[uniform space]] X, a filter ''F'' on ''X'' is said to be Cauchy if for every ''U'' in the entourage, there is an <math>A \in F</math> with for every <math>x, y \in A \ \ (x, y) \in U</math>. In a metric space this takes the form F is Cauchy if for every <math> \epsilon > 0 \ \ \exists A \in F \ \ \mathrm{diam}(A) < \epsilon </math>. ''X'' is said to be complete if every Cauchy filter converges. Let <math>F \subseteq G , \ \ G \to x, \ F</math> Cauchy. Then <math>F \to x</math>. Thus every compact uniformity is complete. Further, a uniformity is compact iff it is complete and [[totally bounded]]. == See also == * [[Filtration (abstract algebra)]] == References == *Cartan, H. (1937) "Thèorie des filtres". ''CR Acad. Paris'', '''205''', 595–598. *Cartan, H. (1937) "Filtres et ultrafiltres" ''CR Acad. Paris'', '''205''', 777–779 ==External links== [http://www.srcf.ucam.org/~drm39/filters.pdf An introductory account of the theory of filters in metric and topological spaces] [[Category:Order theory]] [[fr:Filtre (mathématiques)]] [[de:Filter (Mathematik)]] All content in the above text box is licensed under the Creative Commons Attribution-ShareAlike license Version 4 and was originally sourced from https://en.wikipedia.org/w/index.php?diff=prev&oldid=19719.
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