رده بندی (شبه) ابرحلقه های کراسنر تا مرتبۀ 4

در حد یکریختی به تعداد 10 ابرگروه کانونی و 12 نیم گروه با عضو صفر از مرتبۀ 3 وجود دارند. در این مقاله، به بررسی توزیع پذیری عمل نیم گروه ها نسبت به ابرعمل ابرگروه های کانونی تا مرتبۀ 4 به کمک محاسبات کامپیوتری پرداخته میشود. سپس ابرحلقه ها و ابرمیدان های کراسنر و کلاسهای جدیدی از ابرحلقه ها (ابرحلقه کراسنر باتوزیع پذیری ضعیف، ابرحلقه کراسنر شمولی چپ، ابرحلقه کراسنر شمولی راست، شبه ابرحلقه کراسنر چپ و شبه ابرحلقه کراسنر راست) با مرتبه های کمتر از 4 شمارش و در حد یکریختی رده بندی می شوند. در پایان، گروه های خودریختی ابرحلقه های کراسنر به دست آمده را می یابیم.

on computation of some types krasner hyperrings of order less than 4 and their automorphisms

hyperstructure theory was founded in 1934 at the 8th congress of scandinavian mathematicians. marty introduced the hypergroups as ageneralization of groups in order to study problems in non-commutativealgebra, such as cosets determined by non-invariant subgroups and then heproved its utility in solving some problems of groups, algebraic functions andrational fractions. now this field of modern algebra is widely studied from thetheoretical and applied viewpoints because of their applications can be usedin the following areas: optimization theory, theory of discrete event dynamicalsystems, formal language theory, geometry, graphs, fuzzy sets, cryptography,automata, lattices binary relations, analysis of computer programs, codes andartificial intelligence have been extensively studied in the literature.m. krasner was the first to expand hypercompositional structures via thecreation of structures containing composition and hypercompositions. thus in1956, he replaced the additive group of a field with a special hypergroup,thereby introducing the hyperfield. he then used the hyperfield as the properalgebraic tool, in order to define a certain approximation of complete valuedfields by sequences of such fields. later, he introduced a more generalstructure, which relates to hyperfields in the same way rings relate to fields.he called this structure hyperring. additional hypercompositional structures,similar to the above, introduced by various researchers, soon followed.examples of those are the superring and the superfield, in which both theaddition and the multiplication are hypercompositions. additionally, thestudy of formal languages introduced structures in which thehypercompositional component is a join hypergroup. hyperstructures aremuch more flexible and varied than classical algebraic structures. forexamples if 𝐻 is of prime order 𝑝 , there are a large number of nonisomorphichypergroups on 𝐻 while up to isomorphism, there is only onegroup ℤ𝑝. this becomes clear in this paper, which enumerates the krasnerand new classes of hyperrings of order less than 4 and their automorphismgroups. an important case in algebraic structures is characterize of them. ouraim in this paper is characterizing of krasner hyperrings and new classes of hyperrings of order less than 4 and then determine their automorphismsgroups that is useful for researchers, ph.d. students and et al.