On the measure of M-rough approximation of L-fuzzy sets

We develop an approach allowing to measure the “quality” of rough approximation of fuzzy sets. It is based on what we call “an approximative quadruple” Q=(L,M,φ,ψ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q=(L,M,\varphi ,\psi )$$\end{document} where L and M are complete lattice commutative monoids and φ:L→M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi : L \rightarrow M$$\end{document}, ψ:M→L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi : M \rightarrow L$$\end{document} are mappings satisfying certain conditions. By realization of this scheme, we get measures of upper and lower rough approximation for L-fuzzy subsets of a set equipped with an M-preoder R:X×X→M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R: X\times X \rightarrow M$$\end{document}. In case R is symmetric, these measures coincide. Basic properties of such measures are studied. Besides, we present an interpretation of measures of rough approximation in terms of LM-fuzzy topologies.