actions, norms, subactions and kernels of (fuzzy) norms

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 $y_x$.