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

Document Type: Research Paper

Authors

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

Abstract

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 $Y_X$.

Keywords


bibitem{BS}
    T. Bag and S. K. Samata, textit{A comparative study of fuzzy norms on a linear space}, Fuzzy Sets and Systems
textbf{159} (2008), 670-684.

bibitem{K}
A. K. Katsaras, textit{Fuzzy topological vector space} II, Fuzzy Sets and Systems textbf{12} (1984), 143-154.

bibitem{O}
O. T. O'meara, textrm{Introduction to quadratic forms}, Springer-Verlag, Berlin, 1963.

bibitem{RS}
J. R. Raftery and T. Sturm, textit{On completions of pseudo-normed $BCK$-algebras and pseudo-metric universal algebras}, Math. Japonica textbf{33} (1988), 919-929.

bibitem{ZS}
O. Zariski and P. Samuel, textrm{Commutative algebra}, D. Van Nostrand, Toronto, textbf{I, II} (1958).