^{}Department of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan, Iran

Abstract

lthough fuzzy set theory and sheaf theory have been developed and studied independently, Ulrich Hohle shows that a large part of fuzzy set theory is in fact a subfield of sheaf theory. Many authors have studied mathematical structures, in particular, algebraic structures, in both categories of these generalized (multi)sets. Using Hohle's idea, we show that for a (universal) algebra $A$, the set of fuzzy algebras over $A$ and the set of subalgebras of the constant sheaf of algebras over $A$ are order isomorphic. Then, among other things, we study the category of fuzzy acts over a fuzzy semigroup, so to say, with its universal algebraic as well as classic algebraic definitions.

bibitem{Adamek} J. Ad'amek, H. Herrlich and G. E. Strecker, {it Abstract and concrete categories}, John Wiley and Sons, 1990.

bibitem{3} B. Banaschewski, {it Equational Compactness of $G$-sets}, Canadian Mathematical Bulletin, {bf 17}textbf{(1)} (1974), 11-18.

bibitem{Bel1} R. Belohlavek, {it Birkhoff variety theorem and fuzzy logic}, Archive for Mathematical Logic, {bf 42} (2003), 781-790.

bibitem{Bel2} R. Belohlavek and V. Vychodil, {it Algebras with fuzzy equalities}, Fuzzy Sets and Systems, {bf 157} (2006), 161-201.

bibitem{Bos} I. Bosnjak, R.Madarasz and G.Vojvodic, {it Algebra of fuzzy sets}, Fuzzy Sets and Systems, {bf 160} (2009), 2979-2988.

bibitem{univalg} S. Burris and H. P. Sankapanavar, {it A course in universal algebra}, Springer-Verlag, 1981.

bibitem{7} S. Burris and M. Valeriote, {it Expanding varieties by monoids of endomorphisms}, Algebra Universalis, {bf 17}textbf{(2)} (1983), 150-169.

bibitem{Me1} M. M. Ebrahimi, {it Algebra in a Grothendieck topos: injectivity in quasi-equational classes}, Pure and Applied Algebra, {bf 26}textbf{(3)} (1982), 269-280.

bibitem{Me2} M. M. Ebrahimi, {it Equational compactness of sheaves of algebras on a notherian local}, Algebra Universalis, {bf 16} (1983), 318-330.

bibitem{Me-MH2} M. M. Ebrahimi and M. Haddadi, {it Essential pure monomorphisms of sheaves of group actions}, Semigroup Forum, {bf 80} (2010), 440-452.

bibitem{Me-MH1} M. M. Ebrahimi, M. Haddadi and M. Mahmoudi, {it Equational compactness of $G$-sheaves}, Communucations in Algebra, {bf 40} (2012), 666-680.

bibitem{Mset} M. M. Ebrahimi and M. Mahmoudi, {it The category of $M$-sets}, Italian Journal of Pure and Applied Mathematics, {bf 9} (2001), 123-132.

bibitem{G-pure} M. M. Ebrahimi and M. Mahmoudi, {it Purity of $G$-sheaves}, Submitted, 2009.

bibitem{process} H. Ehrig, F. Parisi-Presicce, P. Boehm, C. Rieckhoff, C. Dimitrovici and M. Grosse-Rhode, {it Algebraic data type and process specifications based on projection Spaces}, Lecture Notes in Computer Science, {bf 332} (1988), 23-43.

bibitem{combine} H. Ehrig, F. Parisi-Presicce, P. Boehm, C. Rieckhoff, C. Dimitrovici and M. Grosse-Rhode, {it Combining data type and recursive process specifications using projection algebras}, Theoretical Computer Science, {bf 71} (1990), 347-380.

bibitem{pro} H. Herrlich and H. Ehrig, {it The construct PRO of projection spaces: its internal structure}, Lecture Notes in Computer Science, {bf 393} (1988), 286-293.

bibitem{hohle1} U. Hohle, {it Fuzzy sets and sheaves. Part text{I} Basic concepts}, Fuzzy Sets and Systems, {bf 158} (2007), 1143-1174.

bibitem{hohle2} U. Hohle, {it Fuzzy sets and sheaves. Part text{II}: Sheaf-theoretic foundations of fuzzy set theory with applications to algebra and topology}, Fuzzy Sets and Systems, {bf 158} (2007), 1175-1212.

bibitem{KKM} M. Kilp and U. Knauer and A. Mikhalev, {it Monoids, Acts and Categories}, New York, 2000.

bibitem{Kura} T. Kuraoka, {it Formulas on the lattice of fuzzy subalgebras in universal algebra}, Fuzzy Sets and Systems, {bf 158} (2007), 1767-1781.

bibitem{12} S. Maclane, {it Categories for the working Mathematicians}, Springer Verlag, 1971.

bibitem{11} S. Maclane and I. Moerdijk, {it Sheaves in Geometry and Logic}, Springer Verlag, 1992.

bibitem{tennison} R. Tennison, {it Sheaf Theory}, Cambridge University Press, 1975.