Applied Mathematical Sciences, Vol. 7, 2013, no. 115, 5711 - 5739 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2013.37419 Order Relations of Sets and its Application in Socio-Economics Norman Neukel Department of Mathematics, Applied Mathematics 2 University of Erlangen-Nuremberg, 91058 Erlangen, Germany [email protected] Copyright © 2013 Norman Neukel. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The order relations of comparative sets and an application to emission influences on sets of urban building conflict situations are investigated with non-ordered as-built properties and their effects on the property market. Sets of points, independent of convexity properties and varying according to the consideration of socio-economic properties, are comparatively examined, concerning minima and maxima. Keywords: set optimization, order relations, socio-economics 1 Introduction The technical records of engineering specialists justify the practice of public administrations working according to the optimality concept of the minimizer and the deviations from this for urban building issues in Germany many times over, similar to scientific publications in the mathematical field of set optimization. Here, the definition of a set-valued mapping FS: ⊂ X Ywith real linear spaces X and Y and a nonempty subset S considers only a minimal element y in the set Fx( ) and then defines the set Fx( ) with y∈ Fx( ) for arbitrary xS∈ xS∈ as the "best". However, in general a minimal element y does not mean 5712 Norman Neukel that the entire set Fx( ) is minimal for all sets Fx( ) with xS∈ is (e.g. the most favourable value in a rent index says nothing about whether the entire range in which this value occurs is also the least expensive). This concept, or this understanding of the minimizer, is found again in socio-economics. The Federal Association of German Realtors (IVD) [9] analyzes the real estate market over a defined period. Individual data from the residential building structure, purchasing power and real estate price index, without set-valued structure, are given separately and the "best" result always indicated. Hagedorn [6] investigates the extent to which the building of another runway at the Munich Airport (Germany) has influenced real estate prices in the surrounding area. Here again, the predicted set inventory is individually viewed as "minimized" and positive and negative quality values accordingly assigned. The Institute for Organization Communication (IFOK) [7] investigates the relationship between noise disturbance and quality of life in the region surrounding the Frankfurt Airport (Germany). With these data, the present paper studies new optimal sets with order relationships for sets with a view to an individually chosen thematic focus. The as-built properties of this urban building conflict situation are explored on the basis of set-valued considerations. This paper is structured as follows. Chapter 2 studies specific order relationships of Jahn and Duc-Ha [11] on the basis of a "setless" order relationship and investigates the strength of the different order relationships according to implications. Their intersection and as-built properties find application in a new practical, socio-economic example, the building conflict situation in the surroundings of the Frankfurt Airport. The order relationships allow a new, scientifically-founded, optimal, comparative set consideration. 2 Order Relations in Set Optimization Order relations of sets and their use in the power set (Y) ={ A ⊂ YA is non-empty} with Y as an arbitrary real vector space and in a certain subset relating to one of the minima and maxima will now be introduced. Definition 2.1 Let AB, ∈ ( Y) be arbitrarily chosen sets. (a) The setless or Kuroiwa-Nishnianidze-Young (KNY) – order relation s is defined as As B :⇔∀∈∃∈ ( aAbBab : ≤) and ( ∀∈∃∈ bBaAab : ≤) . Order relations of sets and its application in socio-economics 5713 (b) The l-typeless order relation l is defined as Al B :⇔ ( ∀∈ b Ba ∃∈ Aa: ≤ b). (c) The u-typeless order relation u is defined as Au B :⇔ ( ∀∈ a Ab ∃∈ Ba: ≤ b). The order relation s was introduced independently by Young [20] in algebra and Nishnianidze [14] by using this in fixed point theory. Chiriaev and Walster [3] use this order relation in the interval arithmetic and SUN Microsystems [15] implements this concept in the f95 FORTRAN compiler. Kuroiwa [12] describes this order relation as a natural criterion. In general s is not anti-symmetric. Example 2.2 The sets {1,2,4} and {1, 3, 4} are given as subsets of . If ≤ denotes the natural order, then {1,2,4}s { 1,3,4} and {1,3,4}s { 1,2,4}, and the two sets are not identical. The disadvantage of an arbitrary spread of the setless order relation becomes evident in 2 . We shall examine the "peripheral regions" which cannot be further considered with a chosen optimization problem and accordingly can vary arbitrarily. Fig. 1: "Variable peripheral regions" in s The definition of the setless order relation considers only one minimal point and one maximal point and then characterizes the two sets. This is a disadvantage. The other value regions of these sets can be arbitrarily distributed, leading to an infinite number of arrangement models. The question arises, how and whether these outlier values must be considered in a dominant comparison. A stronger order relation than that last mentioned is referred to as a certainly less order relation. 5714 Norman Neukel Definition 2.3 Let AB, ∈ ( Y) be arbitrarily chosen sets. The certainly less order relation c is defined as Ac B :⇔ ( A = B) or ( A ≠ B, ∀∈ a A ∀∈ b Ba: ≤ b) . The presentation of a weaker concept follows next. Definition 2.4 Let AB, ∈ ( Y) be arbitrarily chosen sets. The possibly less order relation p is defined as Ap B :⇔∃∈ a Ab ∃∈ Ba: ≤ b. Possibly less thus means that one set is "possibly" smaller than the other. The following order relations of Jahn and Duc-Ha [11] utilize the mininimal and maximal elements of a set. It is known that in a real linear topological space Y minimal and maximal elements exist for every compact set in (Y ) . This leads us to the definition of the minmax less order relation. We shall consider sets A of , where :={AY ∈ ( ) min( A) and max( A) are non-empty} is assumed to be nonempty. There min ( A) and max ( A) mean the sets of minimal and maximal elements of A. The following definitions refer specifically to this. Definition 2.5 Let AB, ∈ be arbitrarily chosen sets. The minmax less order relation m is defined as ABms :⇔ min( A) min( B) and max( A) s max( B) . The lower index m stands for the term minmax. Definition 2.6 Let AB, ∈ be arbitrarily chosen sets. The minmax certainly less order relation mc is defined as ABmc :⇔ ( AB=) or ( AB ≠ , min( A) cc min( B) and max( A) max( B)) . The lower index mc stands for the term minmax certainly. In practice the minmax certainly less order relation is extremely strong: min ( A) must in any case be less than min (B) ; the same procedure applies for the maximum of both sets. Order relations of sets and its application in socio-economics 5715 Proposition 2.7 Let AB, ∈ ( Y) with AB≠ be arbitrarily given. Then: ABABABsl ⇒⇒ p and ABABABsu ⇒⇒ p. Proof (by [11]): Let C be a convex cone. The implications ABsl ⇒ AB and ABsu ⇒ AB follow directly from Definition 2.1. The implications ABlp ⇒ A Band ABup ⇒ AB result from the fact that A B ⇔ B ⊂+ AC ⇒∩ B( AC +) ≠∅⇔∩ A( BC −) ≠∅⇔ A B lp and ABABCABCup ⇔ ⊂ − ⇒ ∩( −) ≠∅ ⇔ BAC ∩( +) ≠∅ ⇔ AB are true. Definition 2.8 Let C be a convex cone in the real vector space Y. A set A∈ is called quasi-dominated iff min ( ACAC) +=+ and max ( ACAC) −=−. Proposition 2.9 Let AB, ∈ with AB≠ be arbitrarily given, let C be a convex cone in the real vector space Y, A⊂+min ( AC) and A⊂−max( AC) . Assuming that A and B have the quasi-dominated property, then ABABABABcm ⇒ cms ⇒⇒ . Proof (by [11]): "ABc ⇒ Amc B": By definition, ABc with AB≠ means BAC−⊂. As min( AA) , max ( ) are subsets of A and min(BB) , max ( ) subsets of B. min(B) −⊂ min ( AC) and max(B) −⊂ max ( AC) are equivalent to ABmc . "Amc B ⇒ ABm ": ABmc with AB≠ implies that min(B) −⊂ min ( AC) and max(B) −⊂ max( AC) . Thus, min(B) ⊂+ min( AC) , min( A) ⊂− min ( BC) and max(B) ⊂+ max( AC) , max( A) ⊂− max ( BC) are equivalent to ABm . "Ams B ⇒ AB": For arbitrary sets AB, ∈ the order relation ABm means with the proberty of the set less order relation which follows from its definition min( AB) s min( ) ⇔ min( A) ⊂−∧⊂+=+ min( B) C min( B) min ( A) C AC and 5716 Norman Neukel max( AB) s max( ) ⇔ max( A) ⊂−=−∧⊂+ max ( B) C BC max( B) max( A) C . This implies with Definition 2.9 that min (B) ⊂+ AC and max ( A) ⊂− BC, so that BC+=min( B) + C and AC −=max ( A) −⊂− C BC. We therefore have B⊂+ AC and A ⊂−⇔ BC As B. The above implications of the order relations can be applied as to a socio-economic problem given in the following section. 3 Socio-economic Application of Order Relations: the conflict situation For historical reasons the term conflict situation has a number of meanings. In the following the meaning of "conflict situation" in relation to the present model will be explained and accordingly applied. A conflict situation in the meaning of the present paper refers to a region in which different, opposing, occasionally conflicting uses come together.