Linear operators and orthomorphisms on the order dual and second order dual of f-algebras and f-modules

In the first chapter of this thesis, which consists of six chapters, we establish an introduction. In the second chapter, the definitions and notations, which are going to be used throughout the thesis, are given and Riesz spaces, f-algebras, orthomorphisms and Banach lattices are mentioned, shortly. In the third chapter, Aren's products are given, which are defined on the order bidual of an Archimedean f-algebra. By means of these products, a mapping is defined and by the way of showing this mapping is a lattice homomorphism, some results are obtained. Also, the relationship between f-orthomorphisms and the orthomorphisms on the order dual of f-algebras, is revealed. In the fourth chapter, some results are given, which are about order dual of an fmodule over an f-algebra. In the fifth chapter, it is shown that, order bidual of an f-module over an f-algebra has similar properties with the order dual of it. In the last chapter, if X is a vector lattice and order dual of X is X', it is investigated when Orth(X') is in the ideal center of X'. xii Keywords: f-algebra, f-module, orthomorphism, linear operator, order dual, Riesz space, vector lattice, ideal center