Actions, Norms, Subactions and Kernels of (Fuzzy) Norms

Document Type : Research Paper


1 Department of Applied Mathematics, Hanyang University , Ahnsan, 426-791, Korea

2 Department of Mathematics, Hanyang University , Seoul, 133-791, Korea

3 Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, U. S. A


In this paper, we introduce the notion of an action $Y_X$
as a generalization of the notion of a module,
and the notion of a norm $vt: Y_Xto F$, where $F$ is a field and $vartriangle(xy)vartriangle(y') =$ $ vartriangle(y)vartriangle(xy')$ as well as the notion of fuzzy norm, where $vt: Y_Xto [0, 1]subseteq {bf R}$, with $bf R$  the set of all real numbers. A great many standard mappings on algebraic systems can be modeled on norms as shown in the examples and it is seen that $mathrm{Ker}vt ={y|vt(y)=0}$ has many useful properties. Some are explored, especially in the discussion of fuzzy norms as they relate to the complements of subactions $N_X$ of $Y_X$.


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