Fuzzy Acts over Fuzzy Semigroups and Sheaves

Document Type: Research Paper

Author

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.

Keywords


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.