In this paper, we study the categorical and algebraic properties, such as limits and colimits of the category Pos-S with respect to order DENSE embeddings. Injectivity with respect to this class of monomorphisms has been studied by the author and used to obtain information about injectivity relative to regular monomorphisms. Then, we study three di erent kinds of essentiality, usually used in literature, with respect to the class of all order DENSE embed-dings of S-posets, and investigate their relations to order DENSE injectivity. We will see, among other things, that although all of these essential extensions are not necessarily equivalent, they behave equivalently with respect to order DENSE injectivity. More precisely, it is proved that order DENSE injectivity well behaves regarding these essentialities. Finally, a characterization of these essentialities over pogroups is given.

Let M and P be right R-modules. A submodule K of an R-module M is called P-DENSE if for each m Î M; (K: m) is a P-faithful right ideal of R: PR is nonsingular if and only if, for each R-module M; every essential submodule of M is a P-DENSE submodule. For any R-module M, we obtain P-rational extention of M and equivalent condition in order that M is equal with its P-rational extention is found. An R-module P is called right Kasch if every simple R-module can be embedded in P: Finally, we given some equivalent conditions for an R-module P to be right Kasch.

Introduction: With writing the special code based on Grid SEARCH technique, best method for describing the elements energy spectrum is introduced. This method is suitable for investigation of Atomic, molecular and nuclear Energy spectrum, laser pulses and study of spectrum that obtained by laser interaction with cell and human body.Microraman spectrum of eye lens that fitted by De. Mul and time spectrum of one molecule that excited by 585 nm laser and fitted by Gnu plot software. In a special application, this method is used for 22Na spectrums.Material and Methods: For fitting mathematical function to a set of experimental data, the first step is seeing data plot in every software. Then, an initial mathematical function is guessed and the function parameters change as far as possible for initial fitting. After this step, by using this special code based on Grid SEARCH technique, is the X2 changes between function data and experimental data are compared. If the X2 is high, it will reduce by introducing small change in every parameter of mathematical function. This is done step, by step, so the X2 reaches to the lowest.Results: By this method, we can have the formulation of element energy spectrum. For example, the equation of 22Na energy spectrum’s description with least squares fitted by experiment data is: N=199.0477 +76100EXP (0.00167 (E-508.7539) 2) + 75100EXP (0.00168 (E-509.2428) 2) + 3450.8070 EXP (0.0000831(E-655.356) 2) + 4002.813EXP (0.0001431(E-839.401) 2) + 11000EXP (0.00105 (E-1048.0230) 2) Fitting start with fit parameter X2=1107.6 and finished with X2 =44.16.Conclusion: By using the special code based on Grid SEARCH technique, we try to introduce the best method for describing of the elements energy spectrum.

The submodules with the property of the title (a submodule N of an R -module M is called strongly DENSE in M, denoted by N£sd M, if for any index set I, ΠIN£d ΠIM) are introduced and fully investigated. It is shown that for each submoduleN of M there exists the smallest subsetD′ ÍM such that N+D′ is a strongly DENSE submodule of M and D′ ∩ N=0. We also introduce a class of modules in which the two concepts of strong essentiality and strong density coincide. It is also shown that for any module M, DENSE submodules in M are strongly DENSE if and only if M £sd ~E (M), where ~E (M) is the rational hull of M. It is proved that R has no strongly DENSE left ideal if and only if no nonzeroelement of every cyclic R -module M has a strongly DENSE annihilator in R. Finally, some properties and new concepts related to strong density are studied.

‎In this paper we take A to be the category Pos-S‎ ‎of S-posets‎, ‎for a posemigroup S‎, Mpd to‎ ‎be the class of partially ordered sequantially-DENSE‎ ‎monomorphisms and study the categorical properties‎, ‎such as‎ ‎limits and colimits‎, ‎of this class‎. ‎These properties are usually‎ ‎needed to study the homological notions‎, ‎such as injectivity‎, ‎of‎ S-posets‎. Also we show that it is actually equivalent to Cpd-density resulting from a closure operator.

Let M be a class of (mono) morphisms in a category A. To study mathematical notions, such as injectivity, tensor products and flatness, one needs to have some categorical and algebraic in-formation about the pair (A, M). In this paper we take A to be the category Act-S of S-acts, for a semigroup S, and Msd to be the class of stronglys-DENSE monomorphisms and study the categorical properties, such as limits and colimits, of this class.

In this paper, the notion of injectivity with respect to order DENSE embeddings in the category ofS-posets, posets with a monotone action of a pomonoidS on them, is studied. We give a criterion, like the Baer condition for injectivity of modules, or Skornjakov criterion for injectivity ofS-sets, for the order DENSE injectivity. Also, we consider such injectivity forS itself, and its order DENSE ideals. Further, we define and study some kinds of weak injectivity with respect to order DENSE embeddings, consider their relations with order DENSE injectivity. Also investigate if these kinds of injectivity are preserved or reflected by products, coproducts, and direct sums ofS-posets.