We will use Krasner's notion of a graduation introduced 1940’s in his paper:
[1] Krasner, M.: Une généralisation de la notion de corps-corpoide. Un corpoide remarquable de la théorie des corps valués, C.R. Acad. Sci., Paris 219 (1944), 345-347.
but can be found also in his paper:
[2] Krasner, M.; Anneaux gradués généraux, Colloque d'algèbre, Renns (1980), p. 209-308,
and for paragraded structures introduced first by M. Krasner and M. Vuković
[3] Krasner, M.; Vuković, M.: Structures paragraduées (groupes, anneaux, modules), Queen's Papers in Pure and Applied Mathematics, Queen's University Kingston, ONT, CanCanada, (1987), pp. 163.
[4] Vuković, M.: Structures graduées et paragraduées, Prepublication de l’Institut Fourier, Université de Grenoble I, No. 536 (2001), pp. 1-40
From a pattern recognition perspective the answer is: yes. Why? Paramonotone mappings (from convex analysis) are monotone, but the reverse does not hold in general. Hence, paramonotonicity is a stronger property than monotonicity.
Update: I also checked the dictionary - "para" means something like "enlarge". I think that it means "stronger" in terms of properties.
Hi colleague Patriksson,
Thank you for your interest for my question. I know something.concerning my question. There are .two classes of paragraded rings that are also graded.
But I want to know more: Are all paragraded rings at the same time graded? And what about the graded rings? Can e a graded rings at the same time paragraded ?
The main difference between the graded and paragraded rings (groups and modules, as well) is the following: paragraded structures are closed with respect to the direct sum and direct product, which is not the case with graded structures. Searching for structures with this feature, M. Krasner and myself came to the structures that we call paragraded
All best, Mirjana
My question is not in the field of applied mathematics, but in the field of algebra, so pure, theoretical mathematics
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