Linear and Affine Sets and Relations

Linear and affine sets and relations have been formerly investigated by a great number of authors. The aim of the present paper is to put the subject into a new perspective by slightly improving and supplementing some of the earlier definitions and results. In the family P ( X ) $$\mathscr {P}(X)$$ of all subsets of a vector space X over a field K, we shall consider the usual extensions of the linear operations to sets. Thus, P ( X ) $$\mathscr {P}(X)$$ , with the ordinary set inclusion, forms a complete, partially ordered generalized vector space with null, zero and infinity elements ∅ $$\emptyset $$ , { 0 } $$\{0\}$$ and X. Now, a subset A of the vector space X may be naturally called (a) affine if λA + ( 1 − λ ) A ⊆ A $$\lambda A +( 1-\lambda ) A\subseteq A$$ for all λ ∈ K $$\lambda \in K$$ ; (b) linear if A + A ⊆ A $$A+A\subseteq A$$ and λA ⊆ A $$\lambda A\subseteq A$$ for all λ ∈ K $$\lambda \in K$$ . Moreover, a relation R on X to another vector space Y over K (i.e., a subset R of the product set X × Y $$X\!\times \!Y$$ ) may be naturally called linear (affine) if it is a linear (affine) subset of the product vector space X × Y $$X\!\times \!Y$$ . These properties can also be expressed in terms of the values R ( x ) $$R\,(x)$$ with x ∈ X $$x\in X$$ . Some of the results obtained will be extended to super and hyper relations on X to Y in a subsequent paper. However, for this, instead of P ( X × Y ) $$\mathscr {P}\,(X\!\times \!Y)$$ the more difficult power sets P ( P ( X ) × Y ) $$\mathscr {P}\,\bigl (\mathscr {P}\,(X)\times Y\,\bigr )$$ and P ( P ( X ) × P ( Y ) ) $$\mathscr {P}\,\bigl (\mathscr {P}\,(X)\times \mathscr {P}\,(Y)\,\bigr )$$ have to be equipped with appropriate inequalities and linear operations.