Lower and upper semi basis in quasilinear spaces

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Aseev introduced the notion of quasilinear spaces as a generalization of linear spaces, [1]. The fundamental deficiency in the theory of quasilinear spaces is the lack of a satisfactory definition of linear dependenceindepence and basis. Perhaps this is the most important obstacle on the improvement of theory of quasilinear spaces. In this study, we will present the definitions of these important concepts. Also we show that these new definitions are given consistent with counterparts of similar results in linear spaces. Our investigations show that these notions directly depend on the order relation on the quasilinear space and have to split into two ways as lower and upper quasilinear independence. Thus, firstly we introduce lower-upper quasilinear combination and lower-upper quasilinear independence of a finite set {Xk }n k=1 in a quasilinear space X . Finally we give lower and upper span of {Xk }n k=1. These concepts lead us to introduce the notions of lower-upper dimension and lower-upper semi base of a quasilinear space.

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Quasilinear spaces , lower (upper) quasilinear combination , lower (upper) span , lower (upper) quasilinear dependenceindependence , lower (upper) semi basis , lower (upper) dimension

Erciyes University Journal Of The Institute Of Science an‎d Technology

104

Erciyes University Journal Of The Institute Of Science an‎d Technology