In Memoriam Mefodie Teodor Rațiu
Mitrofan Ciobanu, Ilie Burdujan, Petru Soltan, Vladimir Izbaş, Olga Izbaș, Anca Veronica Ion
i-xx
Introduction to Neutrosophic Hypergroups A.A.A. Agboola, B. Davvaz
1-10
Stiffness of the linear diffusion and wave-type Partial Differential Equations Elisabete Alberdi Celaya, Juan José Anza Aguirrezabala
11-25
The cocartesian product and pairs of conjugate subcategories Botnaru Dumitru, Baes Elena
27-33
On thermal stresses in anisotropic inhomogeneous microstretch cylinders Emilian Bulgariu
35-50
On a higher order fractional differential inclusion with multi-strip fractional integral boundary conditions
Aurelian Cernea
51-60
On dense subspaces of the spaces of continuous pseudometrics Mitrofan M. Choban, Dorin I. Pavel
61-74
On the impact of explicit or semi-implicit integration methods over the stability of real-time numerical simulations
Teodor Cioacă, Horea Cărămizaru
75-88
The implicit reducibility in the chain super-intutionistic logics Ion Cucu, Mefodie Rațiu
89-94
Some properties of finite order solutions of a class of linear differential equations with entire coefficients
Saada Hamouda, Benharrat Belaïdi
95-106
Some sandwich results associated with a generalized linear operator S. Sivaprasad Kumar, Virendra Kumar
107-118
On the lifetime as the maximum or minimum of the sample with power series distributed size
Alexei Leahu, Bogdan Gheorghe Munteanu, Sergiu Cataranciuc
119-128
On spectral problem for Laplace operator in domain with perturbed boundaries Boris V. Loginov, Davran G. Rakhimov
129-141
Algerian wind data
Zine Labidine Mahri, Said Zid, Rouabah Mohamed Salah
Parameters estimation for the bivariate Sarmanov distribution with normal-type marginals Elena Pelican, Raluca Vernic
155-165
Some remarks on eigenvalue problems for the Laplace operator Valentina Proytcheva, Snezhana Hristova
167-172
The Interval Lattice Boltzmann Method for transient heat transfer in a silicon thin film Alicja Piasecka-Belkhayat, Anna Korczak
173-179
About the maximal number of algebraically independent Lyapunov quantities for the differential system s(1,4)
Victor Pricop
181-195
Common fixed points for Banach-Caristi contractive pairs Mihai Turinici
197-203
i
RAŢIU (RAŢĂ) MEFODIE TEODOR
Corresponding Member of the Academy of Sciences of Moldova, Corresponding Member of the
American Romanian Academy of Arts and Sciences,
Doctor Habilitatus on Physical and Mathematical Sciences, Professor 01 octombrie 1935 - 12 august 2013
ii
„L'histoire géologique nous montre que la vie n'est qu'un court épisode entre deux etérnités de mort, et que, dans cet épisode même, la pensée consciente n'a duré et ne duréra qu'un moment. La pensée n'est qu'un éclair au milieu d'une longue nuit, mais c'est cet eclair qui est tout.”
Henri Poincaré „C'est avec la logique que nous prouvons et avec l'intuition que nous trouvons.”
Henri Poincaré
The distinguished mathematician Mefodie Raţiu would have been be 78 years old at 01 October 2013. He leaved for posterity a durable scientific creation and the image of a remarkable scientist. The ideas and methods created in his scientific works represent an important moment in the development and confirmation of the Moldavian mathematics and science in the scientific world.
Professor Mefodie Raţiu was an irrefutable leader of the Moldavian school of mathematical logic, which have had an important contribution to the mathematical logic, to the organization of mathematical research in the Republic of Moldova and to the education of new generations of highly-qualified specialists.
A short life chronology of Professor Mefodie Raţiu:
• October 01, 1935 – born in village Şipoteni, Romania (now Călăraşi district, Republic of Moldova);
• 1943 - 1954, pupil of secondary school, Republic of Moldova;
• 1954-1955, school-teacher in village Volcineţ, Călăraşi district, Republic of Moldova;
• 1955 - 1959, student at “I.Creangă” State Pedagogical University, Chişinău;
• 1959 - 1962, school-teacher in village Bravicea, Călăraşi district;
• 1960 - graduated at “I.Creangă” University, Chişinău;
• 1962 - August 12, 2013, research worker at the Academy of Sciences of Republic of Moldova, holding several positions;
• 1964 – 1967 – PhD student of the Institute of Mathematics of the Academy of Sciences of R. Moldova;
• 1968 – gained the PhD Degree, at Institute of Mathematics of the Siberian Branch of the Academy of Sciences of USSR, Novosibirsk, Russian Federation;
• 1984 - 2013, leader of the Scientific Seminary on Mathematical logic and Algorithmic theory, member of the Scientific Council of the Institute of Mathematics and Computer Science;
• 1985 - 2013, scientific coordinator for doctoral theses;
• 1986 – habilitatus doctor in sciences, Moscow M.V. Lomonosov State University, Russian Federation;
• 1991 - academic rank of professor, Superior Certifying Commission of the USSR, Moscow, USSR;
• 1994 – laureat of the State Prise of R.Moldova in the fields of science, technique and production;
• 1999 - decorated with State order “Gloria Muncii” of Republic of Moldova;
iii
of Republic of Moldova;
• 2001 - laureat of prize „Academician Constantin Sibirschi”;
• 2010 - elected Corresponding Member of the American Romanian Academy of Arts and Science;
• August 12, 2013 – died, after painful non-long illness.
Mefodie Raţiu (Raţă, Ratsa) was born at October 01, 1935 in village Şipoteni, Romania (now Călăraşi district, Republic of Moldova). Şipoteni is one of the old Romanian localities and is composed of two villages: Podul Lung and Şipoteni. It is now a commune in Călăraşi district, Republic of Moldova.
In 1954 he successfully finished the local secondary (ten-year) school. Mefodie's teachers at the school were so impressed with his abilities that persuaded him to attend to a university. The mathematical formulas and logical deduction charmed him, and without any doubt he decided to continue the mathematical studies. But his financial status (that of a Moldavian peasant) did not permit him to continue the studies. One year he worked as a teacher in the school of village Volcineţ, Călăraşi district, Republic of Moldova. That was not a pleasant year. He felt the absence of mathematical and special psycho-pedagogical knowledge. In 1955 he started his universitary education at the Faculty of Physics and Mathematics of "Ion Creangă" Pedagogical State Institute Chişinău (now Chişinău "Ion Creangă" Pedagogical State University). He was fascinated by the abstract algebraic construction. But the program of the algebra in the pedagogical institutions was scarce.
Mefodie Raţiu began independently to explore additional facts about rings, algebraic operations, algebraic equations, algebraic identities. In 1959 he was appointed as a teacher in the school of village Bravicea, Călăraşi district, Republic of Moldova. After graduating the university in 1960, he continued exploring of additional algebraic facts.
In 1962 he heard that the Academy of Science of Moldova was founded in Chişinău.
In that year he was appointed as a scientific worker of the Institute of Mathematics and Computer Sciences (named at that time Institute of Mathematics and Physics) of the Academy of Science of Moldova. Thus in 1962 Professor M.Raţiu, steady and full of energy, began the scholar activity and didactic carrier in university. His whole future life was associated with the Academy of Science of Moldova, occupying successively the positions of lower researcher, researcher, senior researcher, coordinator scientific researcher, scientific secretary of the Institute of Mathematics and Computer Sciences, founder and head of the Section of mathematical logic, principal scientific researcher, scientific consultant.
In the mathematical logic and the theoretical informatics it is well known the problem of the determination of the functions (operations) of a given logic, which can be, in a sense or other, expressible through other beforehand given functions. The expressibility relation of the functions (functional expressibility) is interpreted as the possibility to obtain some functions from certain initial functions, through the method of superposition. In the semantic construction of the classical propositional logic, the fundamental objects of the analysis are the Boolean functions, understood as functions of the Boolean bivalent algebra (the propositional algebra). The study of expressibility in the classical logic was initiated in the papers of the American mathematician E. Post, which refers to the description of all closed (relative to expressibility) classes of Boolean functions. These classes, named subsequently Post classes, constitute, relative to the inclusion, a countable lattice, illustrated by a not very complicated diagram. Any Post class can be considered as a finite-generated algebra by
iv
difficult to build a recognition algorithm of the expressibility in the classical logic, which, for any Boolean functions {f, f1, f2, ... ,fm} given by formulas or tables, allows to recognize if the function f can be expressed in terms of the system {f1, f2, ... ,fm}. It can also be obtained an algorithm for recognition the completeness in this logic, allowing to recognize, for any finite Boolean functions system, if the system is included in a finite number of Post classes.
A new interest, linked to the problem of the expressibility of the functions, appeared in the 50es of the past century, in connection with the applications of the mathematical logic in Informatics (Computer Sciences). The investigation of the expressibility, both in the classical logic and in its multi-valued generalizations, i.e. the general k–valued (k = 2, 3,…) logics was stimulated by prof. P.S. Novikov’s lectures at the Moscow "M.V. Lomonosov" University.
The general k-valued logic is considered mostly from a functional point of view, being interpreted as the class of all operations on the set of truth values of cardinal k. In more detail it was expressed the problem of completeness (relative to expressibility), which requires to be clarified the necessary and sufficient conditions (preferably uncomplicated and algorithmically) to express all the operations of the researched logic. The existence of the completeness criterion of the functions systems in the general k-valued logic was obtained by A.V. Kuznetzov. S.V. Jablonskii has established the completeness criteria in the 2-and 3- valued logics in terms of pre-complete classes and described some similar families in the k- valued logic, for k> 3. I. Rosenberg has established the completeness criterion in the k-valued logic, for any k > 3, based on six families of pre-complete classes. The connection of these problems with the algebra and their algebraic treatment were investigated by A.V. Kuznetzov and especially by A.I.Maltzev, which proposed in 1966 to research the iterative Post algebras.
Such an algebra has as a support a closed class of k-valued logic functions, and the superposition in its various variations played the role of signature. In this way, we can ascertain that the iterative algebras possess a finite signature. Consequently, we will see that, from a functional point of view, the research of the iterative algebras, in most cases, is equivalent to the investigation of their supports, that are the closed classes of functions of the k-valued logic. The general problem of description of closed classes in the k-valued logic for k> 3, which are exactly the iterative algebras supports, become more complicated in essence through the presence in this logic of closed classes with infinite countable basis, as well as of closed classes, which generally do not have a basis. Consequently, the set of these classes forms, relatively to inclusion, a lattice of continuum cardinality. For the case of the general k- valued logic (k> 3) these properties have been rendered by I.I. Ivanov and A.A. Mucinik. One should remark as well the efforts of many authors for the descriptions of some separate classes of functions from the k-valued logic, or of the families of such classes. But the problem of the complete description of these closed classes, even the problem of recognition of the completeness in the k-valued logic for k> 3 remains open.
Beginning with the papers of L. Brouwer, V.I. Glivenko, A. Heyting, A.N.
Kolmogorov, C. Lewis, Gr. Moisil and others, the investigation of logical calculus and of non-classical logics defined through them, were initiated. These investigations appear in the context of the mathematical study of constructive notions, in the formalization of the natural language, in the demonstration theory, in the programming theory, and so on. In 1930 the intuitionistic logic was formalized and recognized as being the most important among the propositional logics used in the foundations of mathematics. We should remember that this logic, as it was demonstrated in 1932 by K. Gödel, cannot be adequately defined through a finite truth matrix, but as S. Jaskowski has shown in 1936, it can be approximated through an infinite series of such matrices.
v
By suitable specifications, A.V. Kuznetzov has generalized the notion of expressibility for the formulas of the so-called super-intuitionistic logics and he has initiated the research of these logics from expressibility point of view. Professor Mefodie Rațiu succeeded to solve the completeness problem relative to the expressibility in the intuitionistic logic and, in general, in any super-intuitionistic logic. In this sense, the approach consisted in the research of similar problems for certain intermediary logics between the intuitionistic and classical logics. The simplest among these intermediary logics is Jaskowski’s First Matrix Logic. It coincides also with the logic of 3-valued pseudo-Boolean algebra, denoted by Mefodie Raţiu with the symbol LT. The LT logic functions are interpreted as functions of the mentioned pseudo- Boolean algebra, termed as pseudo-Boolean functions. Professor Mefodie Raţiu has proved that the class LT of all 3-valued pseudo-Boolean functions includes families of closed sub- classes of continuous cardinality, with bases functionally independent of countable cardinality, as well as families of continuum cardinality of closed sub-classes, which generally do not accept independent bases. He initiated and developed the theory of chain iterative classes of pseudo-Boolean functions. Professor Mefodie Raţiu built and described all those chains in T closed classes of pseudo-Boolean functions, each of which comprises the constant function 0. In the last 60-70 years an increasing interest was paid for modal logics, that characterizes not only logic operations, but also modalities, as would be "necessity",
"possibility", "distinguishes", etc. C. Lewis and K. Gödel formulated the modal calculus, which successfully formalizes these logics. Moreover, K. Gödel, A. Tarski, P.S. Novikov, S Kripke developed the interpretation of modal logics, showing their connections with algebra, topology and foundations of mathematics. The logic S5 and the logic S4, which is closely related with intuitionistic logic and its interpretations, are the most known among the various modal logics. Logics S4 and S5 are defined by known logical calculus and are non tabular.
In 1981 Professor Mefodie Raţiu establishied algorithmical undecidability of the expressibility problem in modal logic S4 and in any its non-locally tabular extension, that is succeeded to demonstrate that neither for the logic S4 nor for its extensions of the mentioned type expressibility recognition algorithms exist. A few years later he demonstrated the algorithmical undecidability of the expressibility problem in the Gödel-Lob provability logic and in some of its extensions. For this end Professor Mefodie Raţiu created the modeling method of relationship derivation of words in Post’s productions systems by the syntactical expressibility relationship in modal logics and provability logics. He, for the first time, launched the concept of non-existence of recognition algorithms of syntactical expressibility in propositional logic calculus and demonstrated the non-existence of such algorithms.
Hence, the new mathematical theories proposed by Professor Mefodie Raţiu from the logical and philosophical points of view are based on fundamental concepts of mathematical logic, theory of algorithms, discrete mathematics, modern algebra, applied mathematics.
Therefore, the mathematical theories of Professor Mefodie Raţiu correspond to the modern spirit of Mathematics and Computer Science, embrace both classical branches of mathematical logic and new arose conceptions and problems, as well as offer the richest arsenal of well-elaborated methods of investigations.
Professor Mefodie Raţiu published more than 140 research papers and 4 monographs.
Having a good prestige in the world of mathematics, Professor Mefodie Raţiu has been invited at more them 40 prestigious international conferences in Algebra and Mathematical Logic (Russia, Belarus, Ukraine, Poland, Romania, etc). For example, he actively participated in the International Congress of Mathematicians held in Warsaw (1983), at the Eighth International Congress of Logic, Methodology and Philosophy of Science in Moscow (1988), the IV Congress with the same name held in Bucharest (1971), at the 5th Congress of
vi
name in Bucharest (2007), International Congress of ARA in Chişinau (1993), Braşov (2007), Sibiu (2009), Bari (2012) and Chişinău (2013). He passionately and skillfully organized in collaboration with his colleagues several national and international conferences on Algebra and Mathematical Logic.
The contribution of Professor Mefodie Raţiu to the education of new generations of highly-qualified mathematicians is enormous. He has trained 8 doctors of sciences and Ph.D's.
He had a good influence on his colleagues and former students not only as a mathematician, but as a human being. He was simultaneous a professor of the State University of Moldova.
Professor Mefodie Raţiu was an active member of many state communities and commissions. He was a member of the Chairman Commission of Experts and of the Scientific Council of the Higher Certifying Committee of USSR for the academic degree and rank of the Republic of Moldova (1995-2006), vice-President of ROMAI (1992-2005), Head of the Seminar on Algebra, Mathematical Logic and Theory of Numbers of the Republic of Moldova (2012-2013), President of the Trade Union of AŞM (1999-2005), member of the Trade Union of Education and Science of Moldova and member of the Confederation of Trade Unions of Moldova (2000-2005) etc.
Professor Mefodie Raţiu was awarded the State Prize of the Moldova, the prize
"Academician Constantin Sibirschi" (2001), the "Honour Diploma of the AŞM, medal
"Distinction in Labour", medal "Dimitrie Cantemir" and order "Glory of Labour".
Professor Mefodie Raţiu was a member of the Moldavian Mathematics Society, American Mathematical Society, of the Editorial Board of the Bulletin of the Academy of Science of Moldova - Mathematics, and of ROMAI Journal.
In 2000 Professor Mefodie Raţiu was elected Corresponding Member of the Academy of Sciences of Republic of Moldova, the highest scientific forum of the Republic of Moldova and the highest recognition which a scholar may receive in the native country. In 2010 Professor Mefodie Raţiu was elected Corresponding Member of the American Romanian Academy of Arts and Sciences.
As a message to the succeeding generations we mention following words of the Academician Octav Mayer "It is possible to bring for the great disappeared scientists varied homages. Sometimes, the strong wind of progress erases the trace of their steps. To not forget them means to continue their works, connecting them to the living present", translated in English by Academician Radu Miron in his book "The Geometry of Myller Configurations.
Applications to Theory of Surfaces and Nonholonomic Manifoldfs", Ed. Academiei Române, Bucureşti, 2010.
Mitrofan Ciobanu, Ilie Burdujan, Petru Soltan, Vladimir Izbaş, Olga Izbaș, Anca Veronica Ion.
vii
viii
ix
1. M.T. Ratsa, About the class of functions of the logic, corresponding to Jaskowski’s First matrix, In: Issled. po Ob. Algebre, Kishinev, 1965, p. 99-110 (in Russian).
2. M.T. Ratsa, Equivalence criterion of universal algebra to three-element pseudo- Boolean algebra, VII Vses. Kolloq. po Ob. Algebre, Abs., Kishinev, 1965, p.89-90 (in Russian).
3. M.T. Ratsa, A criterion for functional completeness in the logic, corresponding to Jaskowski’s First matrix, DAN SSSR 168, nr.3, 1966, p.524-527 (in Russian).
4. M.F. Ratsa, A test for functional completeness in the logic corresponding to First Jaskowskis matrix, Soviet Math.Dokl. 7 (1966), p.683-687.
5. M.T. Ratsa, Questions of functional completeness in the logic, corresponding to Jaskowski’s First matrix, (PhD thesis). Kishinev: AS MSSR, IM, 1967, 120 p. (in Russian).
6. M.T. Ratsa, Questions of functional completeness in the logic, corresponding to Jaskowski’s First matrix, (PhD thesis’s synopsis). RIO AS MSSR, Kishinev, 1967, 11 p. (in Russian).
7. M.T. Ratsa, About a class of functions of three-valued logic, corresponding to Jaskowski’s First matrix, Problemy Kibernetiki 21, Moscow: Nauka, 1969, p.185-214 (in Russian).
8. M.T. Ratsa, On functional completeness in some logics, intermediate between classical and intuitionistic, Mat. Issledovania, v. 5, is. 4, Kishinev, 1970, p.171-176 (in Russian).
9. M.T. Ratsa, A criterion for functional completeness in a free pseudo-Boolean algebra, XI Vses, Alg. Koll., Abstracts, Kishinev, 1971, p.267-268 (in Russian).
10. M.T. Ratsa, A criterion for functional completeness in the intuitionistic propositional logic, DAN SSSR 201, nr. 4, 1971, p.794-797 (in Russian).
11. M.F. Ratsa, A criterion for functional completeness in the intuitionistic propositional logic, Soviet Math. Dokl. 12 (1971), Nr.6, p.1732-1737.
12. A.V. Kuznetsov, M.T. Ratsa, V.I. Gherciu, An investigation of superintuitionistic logics (functional completeness, finitary approximability and hiperindependence), IV-th Intern. Congress for Logic, Methodology and Philosophy of Science, Abs., Bucharest, 1971, p. 32.
13. M.T. Ratsa, On functional completeness in a family of universal algebras, XII Vses.
Alg. Koll., Abstracts of reports, Notebook , Sverdlovsk, 1973, p. 302 (in Russian).
14. M.T. Ratsa, On the criterion for functional completeness in the intuitionistic logic, Tret. Vses. Conf. po Mat. Logike, Abstracts, 2, Novosibirsk, 1974, p.183-185 (in Russian).
x
15. M.T. Ratsa, The criterion for functional completeness in S5, IV Vses. Conf. po Mat.
Logike, Abstracts, Kishinev, 1976, p.124 (in Russian).
16. M.T. Ratsa, About the operations, expressed through pseudo-boolean terms, XIV Vses. Alg. Conf., Abstracts, 2, Novosibirsk, 1977, p.117-118 (in Russian).
17. M.T. Ratsa, About some subalgebras of iterative Post algebra of rank 3, XV Vses.
Alg. Conf., Abstracts, 2, Krasnoiarsk, 1979, p. 125 (in Russian).
18. A.V. Kuznetsov, M.T. Ratsa, The criterion for functional completeness in classical first-order predicate logic, DAN SSSR 249, nr. 3, 1979, p.540-544 (in Russian).
19. A.V. Kuznetsov, M.T. Ratsa, The algorithm for recognition of the functional completeness in classical predicate logic, 5-ia Vses. Conf. po Mat. Logike, Abstracts, Novosibirsk, 1979, p.78 (in Russian).
20. M.T. Ratsa, The lack of free topo-Boolean algebra the finitary approximability on the functional completeness, XVI Vses. Alg. Conf., Abstracts, 2, Leningrad, 1981, p.110-111 (in Russian).
21. M.T. Ratsa, About functional completeness in the intuitionistic propositional logic, Problemy Kibernetiki, 39, 1982, Moscow: Nauka, p.107-150 (in Russian).
22. M.T. Ratsa, Non-table of the logic S4 on the functional completeness}, Algebra i logika 21, nr. 3, 1982, p.283-320 (in Russian).
23. M.T. Ratsa, The functional completeness in modal logics, GDR, Rostock. Math.
Kolloq., 19, 1982, p .19-28.
24. M.T. Ratsa, Algorithmic questions about functional constructions in modal logics, Int.
Congress of Mathematician, Abs., v.1, Warsaw, 1982, p.19.
25. M.T. Ratsa, Unsolvability of the logic S4 on Expressibility, 6-ia Vses. Conf. po Mat.
Logike, Abstracts, Tbilisi, 1982, p.153 (in Russian).
26. M.T. Ratsa, On functional completeness in modal logics, Bull. AS MSSR, Ser. Phys.
Tecn. and Math. Sci., 2, Kishinev: Ştiinţa, 1983, p.37-39 (in Russian).
27. M.T. Ratsa, On functional completeness in modal logic S5, Issled. Po Neclas.
Logikam I Formalinym systemam, Moscow: Nauka, 1983, p.222-280 (in Russian).
28. M.T. Ratsa, Undecidability of the problem of functional expressibility in the modal logic S4, DAN SSSR 268, nr. 4, 1983, p.814-817 (in Russian).
29. M.F. Ratsa, Undecidability of the problem of functional expressibility in the modal logic S4, Soviet Math. Dokl. 27, 1983, nr. 1, p. 182-186.
30. M.T. Ratsa, The main problems of expressibility of formulas of propositional logics, (doctoral dissertation), Kishinev: AS MSSR, Inst. of Math., 1983, 273 p. (in Russian).
xi
31. M.T. Ratsa, The main problems of expressibility of formulas of propositional logics, (doctoral dissertation’s synopsis), Moscow: Moscow State University, Fac. Mech.-Math., 1985, 28 p. (in Russian).
32. V.P. Malai, M.T. Ratsa, On chain pre-completive sets of pseudo-Boolean algebra operations, XVIII Vses. Alg. Conf., Abstracts, 2, Kishinev, 1985, p.8 (in Russian).
33. I.V. Cucu, M.T. Ratsa, The criterion for completeness to implicit reducibility in the Jaskowski’s First matrix logic, 8-ia Vses. Conf. po Mat. Logike, Abstracts, Moscow, 1986, p.96 (in Russian).
34. M.T. Ratsa, On the completeness of systems of formulas in dual chain logics, Mat.
Issled. 98, Kishinev: Ştiinţa, 1987, p.71-93 (in Russian).
35. M.T. Ratsa, The main problems of expressibility of formulas in non-classical logics, Mat. Issled. 98, Kishinev: Ştiinţa, 1987, p.94-120 (in Russian).
36. M.T. Ratsa, On chain weak completivity in 3-element pseudo-Boolean algebra, XIX Vses. Alg. Conf., Abstracts, 1, Lvov, 1987, p.235 (in Russian).
37. V.P. Malai, M.T. Ratsa, About two pre-completive classes of functions of Jaskowski’s First matrix logic, V.I.N.I.T.I., Dep. manuscript nr. 6052-Â87, Ì., 1987, 34 p. (in Russian).
38. M.F. Ratsa, The problems of expressibility of modal and predicate formulae, 8 Intern. Congress of Logic, Methodology and Phil. of Sci., Abs., Vol. 1, Moscow, 1987, p.
317-320.
39. I.V. Cucu, M.T. Ratsa, On the completeness to implicit reducibility in Jaskowski’s First matrix logic, Bull. AS MSSR, Ser. Phys. Tecn. and Math. Sci. 1, Kishinev Ştiinţa, 1988, p.23-28 (in Russian).
40. M.T. Ratsa, Iterative chain classes of pseudo-boolean functions, I, V.I.N.I.T.I., Dep.
manuscript nr. 4323-88, Ì., 1988, 277 p. (in Russian).
41. A.V. Kuznetsov, M.T. Ratsa, On sheffer criterion in the intuitionistic logic, Bull. AS MSSR, Ser. Phys. Tecn. and Math. Sci., nr. 3, Kishinev: Ştiinţa, 1988, p.3-9 (in Russian).
42. M.T. Ratsa, The criterion of completivity of chain systems pseudo-Boolean functions, containing the constant 0, 2-ia Vses. Conf. po Prikl. Logike, Abstracts, Novosibirsk, 1988, p.198-199 (in Russian).
43. M.A. Coban, M.T. Ratsa, The criterion for formulaic completeness in a extension of the logic S4, 9-ia Vses. Conf. po Mat. Logike, Abstracts, Leningrad, 1988, p.76 (in Russian).
44. M.T. Ratsa, Iterative chain classes of pseudo-Boolean functions, containing the constant 0, 9-ia Vses. Conf. po Mat. Logike, Abstracts, Leningrad, 1988, p.138 (in Russian).
45. M.F. Ratsa, Concerning description of iterative systems of pseudo-Boolean functions, Intern. Conf. in Computer logic. Papers, Part 1, Tallin, 1988, p. 85-87.
xii
46. M.T. Ratsa, Questions of the expressibility formulas in logical calculi, Proceedings of the Workshop on Discrete Math. and Its Applications, Moscow: Publ. Moscow State
University, 1989, p.38-44 (in Russian).
47. M.T. Ratsa, The algorithmic undecidability of the problem of expressibility in modal logics, Matem. Voprosy kibernetiki 2, Moscow: Nauka, 1989, p. 71-99 (in Russian).
48. M.A. Coban, M.T. Ratsa, The conditions of completeness of formulas in the logic of 4- element topological Boolean algebra with three open elements, V.I.N.I.T.I., Dep. manuscript nr. 4299-89, Moscow, 1989, p.1-35 (in Russian).
49. V.P. Malai, M.T. Ratsa, About one chain class of sectorial functions of Jaskowski’s First matrix logic, Bull. AS MSSR, Ser. Phys. Tech. and Math. Sci. 2, Kishinev, Ştiinţa, 1989, p.9-14 (in Russian).
50. M.A. Coban, M.T. Ratsa, About formulaic completeness in 4-valued extension of pre- tabular modal logic EM4, Bull. AS MSSR, Ser. Phys. Tecn. and Math. Sci. 3, Kishinev, Ştiinţa, 1989, p.13-16 (in Russian).
51. M.T. Ratsa, About chain sub-algebras of Post iterative algebra of pseudo-Boolean functions, Intern. Conf. on Algebra , Abstracts of talks on the model theory and algebraic systems, Novosibirsk, 1989, p.112 (in Russian).
52. M.T. Ratsa, Iterative chain classes of pseudo-Boolean functions, Chişinău, Ştiinţa, 1990, 238 p.
53. M.T. Ratsa, On the impossibility of an algorithm for the recognition of the
expressibility of modal formulae, Bull. AS MSSR. Mathematics, 1, Chişinău, Ştiinţa, 1990, p. 69-71 (in Russian).
54. V.P. Malai, M.T. Ratsa, On the bases of the chain classes of pseudo-Boolean
functions, 10-ia Vses. Conf. po Mat. Logike, Abstracts, Alma-Ata, 1990, p.106 (in Russian).
55. M.A. Coban, M.T. Ratsa, About weak completeness in the logic of 8-element topological Boolean algebra with one open atom, 10-ia Vses. Conf. po Mat. Logike, Abstracts, Alma-Ata, 1990, p.83 (in Russian).
56. M.T. Ratsa, Expressibility in the propositional calculi, Chişinău, Ştiinţa, 1991, 204 p.
(in Russian).
57. I.V. Cucu, M.T. Ratsa, Conditions of parametrical completeness in the logic of 5- element pseudo-Boolean algebra with two incomparable elements, V.I.N.I.T.I., Dep.
manuscript nr. 1887-A91, 1991, 41 p. (in Russian).
58. M.A. Coban, M.T. Ratsa, About expressibility terms in Post iterative algebras of topological Boolean functions, Intern. conf. on algebra , Abstracts of talks on the logic and univ. algebra, Novosibirsk, 1991, p.65 (in Russian).
xiii
VI-th Tiraspol Symposium on general topology and its applications, Chişinău, 1991, p. 61- 62 (in Romanian).
60. V.P. Malai, M.T. Ratsa, Two descriptions of classical complete chain systems of pseudo-Boolean functions, VI-th Tiraspol Symposium on General Topology and its Applications, Chişinău, 1991, p. 132-133 (in Romanian).
61. I.V. Cucu, M.T. Ratsa, Parametrical completeness in the logic of the simplest pseudo- Boolean algebra with two incomparable elements, Bull. A.S. R.M. Mathematics, 1 (7), Chişinău, Ştiinţa, 1992, p. 46-51 (in Russian).
62. M.T. Ratsa, The criterion for completeness with respect to expressibility in the three- valued extension of provability-intuitionistic logic, V.I.N.I.T.I., Dep. manuscript nr. 1782- A92, Moscow, 1992, 22 p. (in Russian).
63. V.P. Malai, M.T. Ratsa, Description of chain classically complete systems of pseudo- Boolean functions, Bull. A.S. R.M. Mathematics, 2 (8), Chişinău, Ştiinţa, 1992, p.15-21 (in Russian).
64. Ya.N. Gkhaliekh, M.T. Ratsa, The conditions for completeness with respect to expressibility in the simplest non-chain extension of dual-intuitionistic logic, XI Inter. Conf.
on Math. Logic, Abstracts, Kazan, 1992, p. 45 (in Russian).
65. M.A. Coban, M.T. Ratsa, The criterion for completeness with respect to expressibility in the pretabular modal logic EM4, XI Inter. Conf. on Math. Logic, Abstracts, Kazan, 1992, p. 76 (in Russian).
66. I.V. Cucu, M.T. Ratsa, The criterion for parametrical completeness in the logics of pseudo-Boolean chain algebras with a lower thickening, XI Inter. Conf. on Math. Logic, Abstracts, Kazan, 1992, p. 85 (in Russian).
67. M.T. Ratsa, A.G. Russu, An example of a countable family of pre-complete classes of formulas in the provability logic, XI Inter. conf. on math. logic, Abstracts, Kazan, 1992, p.
120 (in Russian).
68. I.V. Cucu, M.T. Ratsa, The parametrical completeness in 5-valued pseudo-Boolean algebra with two incomparable elements, Proc. of the Conf. on Algebra, Preprint 1, Babeş- Bolyai Univ., Cluj-Napoca, Romania, 1992, p.23-24.
69. M.T. Ratsa, Chain systems of pseudo-boolean functions with autodual or linear boolean images, Proc. of the Conf. on Algebra, Preprint 1, Babeş- Bolyai Univ., Cluj- Napoca, Romania, 1992, p. 107-108.
70. M.A. Coban, M.T. Ratsa, On functional completeness in 8-element topological Boolean algebra with one open atom, Proc. of the Conf. on Algebra, Preprint nr. 1, Babeş- Bolyai Univ., Cluj-Napoca, Romania, 1992, p. 51-54.
xiv
non-chain extension of dual-intuitionistic logic, V.I.N.I.T.I., Dep. manuscript nr. 2951-A92, Moscow, 1992, 72 p. (in Russian).
72. Ya.N. Gkhaliekh, M.T. Ratsa, Solution of the problem of completeness with respect to expressibility in a 5-valued non-chain extension of duaity intuitionistic logic, Bull. A.S. R.M.
Mathematics, 4, Conf. on Math. Logic, 1992, p. 43-51 (in Russian).
73. M.A. Coban, M.T. Ratsa, On weak completeness in Post iterative algebras of topological Boolean functions, 3-th Intern. Conf. on algebra memory M.I. Kargapolov, Abstracts, Krasnoyarsk, 1993, p. 154-155 (in Russian).
74. M.T. Ratsa, The analogue of Slupecki’s theorem for a family of lattices with two incomparable elements, 3-th Intern. Conf. on algebra memory M.I. Kargapolov, Abstracts, Krasnoyarsk, 1993, p. 280-281, (in Russian).
75. M.T. Ratsa, A.G. Russu. The construction of an infinite family of maximal
subalgebras of the Post's iterative algebra of diagonalizable algebra functions, 3-th Intern.
Conf. on Algebra, Memory M.I. Kargapolov, Abstracts, Krasnoyarsk, 1993, p. 282-283 (in Russian).
76. O. Covalgiu, M.T. Ratsa, Modeling of self-dual and commuting with 1 Boolean functions in a 3-valued extension of provability-intuitionistic logic, Bul. A.S. R.M.
Mathematics, Chişinău: Ştiinţa, 2, Conf. on Math. Logic, 1993, p. 72-78, (in Russian).
77. M.A. Coban, M.T. Ratsa, Criterion of weak completeness in a 8-valued extension of pretabular modal logic EM4, V.I.N.I.T.I., Dep. manuscript nr. 276-93, Moscow, 1993, 39 p.
(in Russian).
78. O. Covalgiu, M.T. Ratsa, Conditions of modeling of self-dual and commuting with 1 Boolean functions in a 3-valued extension of provability-intuitionistic logic, V.I.N.I.T.I., Dep.
manuscript nr. 2417-93,
Moscow, 1993, 45 p. (in Russian).
79. V.P. Malai, M.T.Ratsa, The bases of the pre-completive chain-systems of pseudo- Boolean functions, The 18-th Congr. of the American-Romanian Academy of Arts and Sciences, Abs.,vol.2, Chişinău, 1993, p.27.
80. M.A. Coban, M.T. Ratsa, On completeness concerning expressibility in the logics of topological Boolean algebras with one open element, The 18-th Congr. of the American- Romanian Academy of Arts and Sciences, Abs., vol. 2, Chişinău,, 1993, p. 12.
81. M.T. Ratsa, On expressibility in the logical calculus of Hilbert's type, The 18th Congr. of the American-Romanian Academy of Arts and Sciences, Abs., vol. 2, Chişinău, 1993, p. 34.
82. O. Covalgiu, M.T. Ratsa, The Modeling of boolean functions in the 3-valued extension of provability-intuitionistic logic, The 18-th Congr. of the American-Romanian Academy of Arts and Sciences, Abs.,vol. 2, Chișinău, 1993, p. 15.
xv
83. M.A. Coban, M.T. Ratsa, On the problem of expressibility in propositional modal logics, Prepar. Conf. for the Congr. of Romanian Mathematicians Everywhere, vol. 1, Bucharest, 1993, p. 63-65 (in Romanian).
84. M.T. Ratsa, A.G. Russu, Construction of an example of the infinite set of classes of formulas, pre-complete in provability logic, Prepar. Conf. for the Congr. of Romanian Mathematicians everywhere, vol 2, Bucharest, 1993, p. 56-57 (in Romanian).
85. M.A. Coban, M.T. Rata, The analogue of Slupecki’s theorem in the logic of 8-valued topological Boolean algebras with one open element, Prepar. Conf. for the Congr. of
Romanian Mathematicians everywhere, vol 2, Bucharest, 1993, p. 47-49 (in Romanian).
86. I.V. Cucu, M.T. Ratsa, Parametrical completeness for the logics of linear ordered algebras with a lower thickening, Prepar. Conf. for the Congr. of Romanian Mathematicians everywhere, vol 2, Bucharest, 1993, p. 53-54 (in Romanian).
87. M.T. Ratsa, A.G. Russu, The constructions of a numerable collections of pre-complete classes of formulae in the provability logic, Bull. A.S. a R.M. Mathematics, 1 (14), Chişinău Ştiinţa, 1994, p. 66-74.
88. M.A. Coban, M.T. Ratsa, On functional expressibility in topological Boolean algebras with one open element, Proc. of X-th Nation.Conf. on algebra, Univ. Timişoara, Romania, 1994, p. 67-68.
89. M.T. Ratsa, A.G. Russu, One infinite consequence of maximal subalgebras of free diagonalizable algebra, Proc. of the X-th Nation. Conf. on algebra, Univ. Timişoara, Romania, 1994, p. 93-94.
90. I.V. Cucu, M.T. Ratsa, On completeness relative to implicit reducibility in pseudo- boolean linear ordered algebras, Proc. of the X-th Nation. Conf. on algebra, Univ. Timişoara, Romania, 1994, p. 45-46.
91. O. Covalgiu, M.T. Ratsa, On model completeness for the classes of the functions of 3- valued σ-pseudo-Boolean algebra, Anal. St. ale Univ. "Ovidius", Seria Mat., vol. 2,
Constanţa, Romania, 1994, p. 76-79.
92. M.A. Coban, M.T. Ratsa, About maximal subalgebras of Post’s iterative algebras of topological Boolean functions, National Conf. on Algebra, Abs., Constanţa, Romania, 1994, p. 18 (in Romanian).
93. O. Covalgiu, M.T. Ratsa, About model completeness in classes of functions of 3- valued σ-pseudo-Boolean algebra, National Conf. on Algebra, Abs., Constanţa, Romania, 1994, p.20 (in Romanian).
94. M.A. Coban, M.T. Ratsa, Weak completeness in logics of 4-valued and 8-valued Boolean algebras with one open atom, Jub. Scien.-Techn. Conf., Tehn. Univ. of Moldova, Chişinău, 1994, p. 151-152. (in Romanian).
xvi
Boolean functions, Jub. scien.-techn. conf., Tehn. Univ. of Moldova, Chişinău, 1994, p. 148- 149 (in Romanian).
96. M.A. Coban, M.T. Ratsa, The propositional logic EM4 and its 32-valued extension, Scien. Simp.: Forming efficient economy by market forces, Publ. of Academy of Economic Studies of Moldova, Chişinău, 1995, p. 153-156 (in Romanian).
97. O. Covalgiu, M.T. Ratsa, The criterion of Model completeness in the closed classes of 3-valued functions, Applied and Industrial Math., Abs., Oradea- Chişinău, 1995, p. 13.
98. M.T.Ratsa, Results of functional completeness in logical calculi, Bull. A.S. R.M., Mathematics, Chişinău: Ştiinţa, 1996, p. 73-104 (in Romanian).
99. M.A. Coban, M.T. Ratsa, About derivative operations of 8-element topological Boolean algebra with one open atom, Symposium Septimum Tiraspolense generalis
topologiae et suae applicationum, Publ. „Tehnica”, Tehn. Univ. of Moldova, Chişinău, 1996, p. 73 (in Romanian).
100. O. Covalgiu, M.T. Ratsa, The conditions of model completeness in closed classes of many-valued logic, Intern. Conf. on Math. And Informatics, Abs., Chişinău, 1996, p. 18 (in Romanian).
101. O. Covalgiu, M.T. Ratsa, On the existence of a criterion of Model completeness for the closed classes of 3-valued functions, Bull. A.S. R.M. Mathematics, 3, Chişinău: Ştiinţa, 1997, p. 101-104.
102. I.V. Cucu, M.T. Ratsa, A criterion for parametrical completeness in logic of the simplest pseudo-Boolean algebra with incomparable elements, Proceedings of the Workshop on Discrete Math. And Its Applications, Moscow: Publ. House of the Faculty of Mechanics and Mathematics, Moscow State University, 1997, p. 52-53 (in Russian).
103. V.P. Malai, M.T.Ratsa, On the bases of chain classical complete classes of pseudo- Boolean functions, Proceedings of the Workshop on Discrete Math. and Its Applications, Moscow: Publ. House of the Faculty of Mechanics and Mathematics, Moscow State University, 1997, p. 55-56 (in Russian).
104. O. Covalgiu, M.T. Ratsa, A criterion of model completeness for the closed classes of 3-valued logic, Bul. Sci. of Univ. "Politehnica", Timişoara, Romania, t. 43, nr. 1, 1998, p. 31- 34.
105. M.T. Ratsa, Undecidability of the problem of functional expressibility in the provability logic, Proceedings of the Intern. Conf. on Math. Logic, dedicated to the 90-th anniversary of A.I. Maltsev, Novosibirsk, 1999, pgs.107-108.
106. M.T. Ratsa, Undecidable of the expressibility problem in a free diagonalizable
algebra, The second Intern. Conf. on Algebra Ukraine, dedicated to the memory of prof. L.A.
Kaluzhnin (1914-1990), Kiev-Vinitza, 1999, p.40-41.
xvii
Gödel-Lob, Bull. Sci., Ser. Math. and Computer Science 3, Univ. of Piteşti, Romania, 1999, p. 399.
108. M.T. Ratsa, Algoritmical undecidability of the expressibility problem in free diagonalizable algebra, Bull. of the Polytechnic Inst. of Iaşi, t. XLVI (L), fasc. 1 – 2, Mathematics, Theoretical mechanics, Physics, Romania, 2000, p. 18-22.
109. M.T. Ratsa, A.G. Russu, Some properties of complete with respect to expressibility systems of formulas in Gödel -Lob provability logic, Discretnaya Matematica 12, is.4, Moscow, 2000, p.63-82 (in Russian).
110. I.V. Cucu, M.T. Ratsa, The reduction of parametrical completeness in pseudo-boolean algebras defined on the linear ordered sets completed with 2 atoms, First Conf. of the Math.
Society of the R. Moldova, Abs., Chişinău, 2001, p. 42-43.
111. M.T. Ratsa, Formal reduction of the general problem of expressibility of formulas in Godel-Lob provability logic, Discretnaya matematica, v.14, is.2, Moscow: Nauka, 2002, p.95- 106 (in Russian).
112. M.T. Ratsa, Conditions for functional completeness in the simplest extensions of Dual Intuitionistic Logic, 5th Congress of Romanian Mathematicians, Abs., Piteşti, Romania, 2003, p. 120-121.
113. M.T. Ratsa, Functional completeness in the simplest extensions of Dual Intuitionistic Logic, 11th Conf. of Applied and Industrial math., Proc., Oradea, Romania, 2003, p. 190.
114. M. Ciobanu, Anca-Veronica Ion, M. Ratsa, Professor Adelina Georgescu – profile at the age of 60, Bull. Sci., Ser. Math. and Computer Science, nr. 9, Univ. of Piteşti, Romania, 2003, p. V – VIII.
115. V. Cebotari, M. Ratsa, A criterion for implicit reductibility of all the formulas in the 4-valued extension of modal logic S5 , Bull. Sci., Ser. Math. and Computer Science, nr. 9, Univ. of Piteşti, Romania, 2003, p. 73-89 (in Romanian).
116. M.T. Ratsa, Inexistence of the algorithms for the recognition of syntactical expressibility in the logical calculi, Publ. The Flower Power, Piteşti, Romania, 2004, 106 p.
(in Romanian).
117. V. Cebotari, M. Ratsa, A completeness to implicit reductibility in the 4-valued extension of modal logic S5, Second Conf. of Math. Soc. of R.Moldova, Comm., Chişinău, 2004, p. 92-95.
118. M.A. Coban, M.T. Ratsa, A criterion for functional completeness in some free
topological Boolean algebra, Second Conf. of Math. Soc. of R. Moldova, Comm., Chişinău, 2004, p. 114-115.
119. I. Cucu, M. Ratsa, On cardinality of the implicit bases in chain super-intuitionistic logics, Second Conf. of Math. Soc. of R.Moldova, Comm., Chişinău, 2004, p. 121-123.
xviii
of 4-elemente boolean topological algebra with two open elements, 5th Intern. Algebraic Conference in Ukraine, Abs., Odessa, 2005, p. 44.
121. I. Cucu, M. Ratsa, The lengths of implicit bases on chain pseudo-boolean algebras, 5th Intern. Algebraic Conference in Ukraine, Abs., Odessa, 2005, p. 54.
122. M.A. Coban, M.T. Ratsa, Conditions for functional completeness in one pre-tabular extension of Lewis logic S4, 5th Intern. Algebraic Conference in Ukraine, Abs., Odessa, 2005, p. 53.
123. M.A. Coban, M.T. Ratsa, Existence of an algorithm for recognition of the
completeness with respect to expressibility in modal pre-tabular logic EM4, Proc. of the 4th Annual Symposium on Mathematics Applied in Biology & Biophysics, Sci. Annals of USAMU, t. XLVIII, vol. 2, Iasi, 2005, p. 389-402.
124. M.T. Ratsa, Loss of the property to be maximal formula centralizer attached to transition from the logic of first Jaskowsk’s matrix to the Intuitionistic logic, Proc. of the 5th Annual Symposium on Mathematics Applied in Biology & Biophysics, Scientific papers of Univ. of Agricultural Sciences and Veterinary Medicine, t. XLIX, vol. 2, Iaşi, 2006, p. 269- 273.
125. M.T. Ratsa, Disappearence of some parametrical pre-complete systems attached to transition from First Jaskowsk’s matrix Logic to the Intuitionistic Logic, The XIVth Conf. on Applied and Industrial Math., Comm., CAIM 2006, Chişinău, 2006, p. 300-301.
126. M.T. Ratsa, Research on mathematical logic. Elements of the history of mathematics and mathematics in Moldova, Tiraspol State Univ., State Univ. of Moldova, Chişinau, 2006, p. 252 – 291 (in Romanian).
127. M.T. Ratsa, Recursive unsolvability of a problem of expressibility in the Logic of Provability, ROMAI Journal, vol. 3, Romanian Soc. of Applied and Industrial Math., nr. 1, 2007, p. 185-188.
128. M.T. Ratsa, Unsolvability of a problem of expressibility in the logic of provability, The 31st Annual Congress of the American Romanian Academy of Arts and Sciences, Presses Internationals Polytechnique, Transilvania University of Braşov, 2007, p. 487-488.
129. M.T. Ratsa, Algorithmic Unsovability of a problem of expressibility in the Godel-Lob Logic of Provability, 6th Congress of Romanian Matematicians, Abs., Bucharest, Romania, 2007, p. 133-134.
130. M.T. Ratsa, Iterative chain algebras of 3-valued pseudo-Boolean functions, The 33 rd Annual Congress of the American Romanian Academy of Arts and Sciences (ARA), Alma Mater University of Sibiu, Romania, Proceedings, vol. 2, Polytechnique Intern. Press, Montreal, Quebec, 2009, p. 321-323 (in Romanian).
131. M. T. Raţiu, Iterative chain algebras of trivalent pseudo-Boolean functions, Iaşi : Editura Alexandru Myller, 2010. 276 p. ISBN 978-973-88565-3-0. (in Romanian).
xix
order 16 with an open atom, In: The 34th Annual Congress, American Ramanian Academy of Arts and Sciences (ARA), May 18th-23rd, Bucharest, Romania, 2010, Proceedings. Sci. Ed.:
FRUNZETI T.; HANGANU M. –Presses
Internationales Polytechnique, Montreal, Quebec, 2010, p. 579-581 (in Romanian).
133. M. Raţiu, I. Cucu, A criterion for parametrical completeness in the 5-valued non- linear algebraic model of intuitionistic logic, ROMAI Journal 6, nr. 1, Bucharest, 2010. p.
153-154.
134. M. Raţiu, An example of maximal iterative algebra of order 16 with 3 open elements, Abstracts of the 18th Conference on Applied and Industrial mathematics, CAIM, 2010, Iaşi, Romania, 2010, p.77.
135. M. Raţiu, On functional completeness in the Gr. Moisil dual intuitionistic logic, In:
The 35th Annual Congress, American Romanian Academy of Arts and Sciences (ARA), July 6th-11rd, Timişoara, Romania, 2010, Proceedings. Sci. Ed.: IONEL, I.; FLESER, T.;
VETRES, I. Presses Internationales Polytechnique, Montreal, Quebec, 2011, p. 301-303.
136. M. Raţiu, Criterion for functional completeness in the 16 elements algebraic model of pre-tabular EM4 logic, The 19-th conference on applied and industrial mathematics, CAIM, September 22-25, 2011, Abstracts, Iaşi, Romania, 2011, p. 26-27.
137. M. Raţiu, A. Cașu, Gh. Ciocanu, V. Izbaș, P.Sârbu, Doctoral examen program on specialty 01.01.06- Mathematical Logic, Algebra and Number Theory (joint program with St.Univ. of Moldova and Tiraspol St. Univ.), Chişinău, 2011 (in Romanian).
138. M. Raţiu, Method of the formula realization of algebras and its application in mathematical logic, Proceedings of The 36th Anual Congress of the American Romanian Academy of Arts and Sciences (ARA). Learing Without Frontiers. Giola del Colle - Bari, Italia, May 30th - June 2nd, 2012. Presses Iternationales Polytechnique, Montréal, Quebec, Canada, 2012, pp. 137-140.
139. M. Raţiu, Expressibility of implication in intuitionistic logic with the method of the formula realization of algebras, In: The 20th Conference on Applied and Industrial
Mathematics, CAIM- 2012 (dedicated to academician Mitrofan M. Ciobanu), Chişinău, Republic of Moldova, August 22-25, 2012, Communications, p. 195-196.
140. M. Raţiu, Applying of the method of the series of matrixes to establish the cardinal of the set of iterative pseudo-boolean algebras, Proceedings of The 37th Anual Congress of the American Romanian Academy of Arts and Sciences (ARA). The university of European Political and Economic Studies "Constantin Stere",
June 04-09, 2013: Presses Internationales Polytechnique, Montreal, Quebec, Canada, Proceedings.-Chişinau, 2013, pp. 460-462.
141. I. Cucu, M. Raţiu, The implicit bases for chain pseudo-Boolean algebras, MITRE- 2013, Abstracts, Chişinau, p.36-37.
xx
chain super-intuitionistic logics, In the 21-thConf. on Applied and Industrial Mathematics, CAIM-2013, Book of Abstracts, Bucharest, Romania, p. 81-82, ISSN 1841-5512.
INTRODUCTION TO NEUTROSOPHIC HYPERGROUPS
Adesina Abdul Akeem Agboola1, Bijan Davvaz2
1Department of Mathematics, Federal University of Agriculture, Abeokuta, Nigeria 2Department of Mathematics, Yazd University, Yazd, Iran
[email protected], [email protected]
Abstract The objective of this paper is to introduce the concept ofneutrosophichypergroup and present some of its elementary properties.
Keywords: neutrosophic group, neutrosophic sub-group, hypergroup, sub-hypergroup, neutrosophic hypergroup, neutrosophic sub-hypergroup.
2010 MSC:03B60, 20A05, 20N20, 97H40.
1. INTRODUCTION
In 1995, Florentin Smarandache introduced the notion of Neutrosophy as a new branch of philosophy. Neutrosophy is the base of neutrosophic logic which is an ex- tension of the fuzzy logic in which indeterminancy is included. In the neutrosophic logic, each proposition is estimated to have the percentage of truth in a subset T, the percentage of indeterminancy in a subset I, and the percentage of falsity in a subsetF. Since the world is full of indeterminancy, several real world problems in- volving indeterminancy arising from law, medicine, sociology, psychology, politics, engineering, industry, economics, management and decision making, finance, stocks and share, meteorology, artificial intelligence, IT, communication etc can be solved by neutrosophic logic.
2. NEUTROSOPHIC ALGEBRAIC STRUCTURES
Using Neutrosophic theory, Vasantha Kandasamy and Florentin Smarandache in- troduced the concept of neutrosophic algebraic structures in [12]. Some of the neutro- sophic algebraic structures introduced and studied included neutrosophic fields, neu- trosophic vector spaces, neutrosophic groups, neutrosophic bigroups, neutrosophic N-groups, neutrosophic semigroups, neutrosophic bisemigroups, neutrosophic N- semigroup, neutrosophic loops, neutrosophic biloops, neutrosophic N-loop, neutro- sophic groupoids, neutrosophic bigroupoids and so on. In [13], Vasantha Kandasamy introduced and studiedneutrosophicrings. In [1], Agboola et al. studied the struc- ture of neutrosophic polynomial rings and in [2], Agboola et al. studied neutrosophic ideals andneutrosophicquotient rings. In [3], Agboola et al. studied neutrosophic groups and subgroups.
1
3. NEUTROSOPHIC GROUPS
Definition 3.1. [12] Let (G, ⋆) be any group and let N(G) = ⟨G∪I⟩. The couple (N(G), ⋆) is called a neutrosophic group generated by G and I under the binary operation⋆.
I is called the neutrosophic element with the property I⋆I =I. I−1, the inverse of I is not defined and hence does not exist.
N(G)is said to be commutative if a⋆b=b⋆a for all a,b∈N(G).
Theorem 3.1. [12]Let N(G)be a neutrosophic group. Then, (1) N(G)in general is not a group;
(2) N(G)always contain a group.
Definition 3.2. Let N(G)be a neutrosophic group. Then,
(1) A proper subset N(H) of N(G), where H ⊂ G, is said to be a neutrosophic subgroup of N(G) if N(H) is a neutrosophic group, that is, N(H) contains a proper subset which is a group;
(2) N(H) is said to be a pseudo neutrosophic subgroup if it does not contain a proper subset which is a group.
Example 3.1. (1) (N(Z),+), (N(Q),+) (N(R),+)and(N(C),+) are neutrosophic groups of integer, rational, real and complex numbers, respectively.
(2) (⟨{Q− {0}} ∪I⟩,·),(⟨{R− {0}} ∪I⟩,·)and(⟨{C− {0}} ∪I⟩,·)are neutrosophic groups of rational, real and complex numbers, respectively.
Example 3.2. [3] Let N(G) = {e, a, b, c, I, aI, bI, cI} be a set, where a2 = b2 = c2 = e, bc = cb = a,ac = ca = b,ab = ba = c, then N(G)is a commuta- tive neutrosophic group under multiplication since{e, a, b, c}is the Klein 4-group.
N(H)={e, a, I, aI}, N(K)={e, b, I, bI}and N(P)={e, c, I, cI}are neutrosophic subgroups of N(G).
Theorem 3.2. [3]Let N(H) be a non-empty proper subset of a neutrosophic group (N(G), ⋆). Then, N(H) is a neutrosophic subgroup of N(G) if and only if the follow- ing conditions hold:
(1) a,b∈N(H)implies that a⋆b∈N(H);
(2) there exists a proper subset A of N(H) such that(A, ⋆)is a group.
Theorem 3.3. [3]Let N(H) be a non-empty proper subset of a neutrosophic group (N(G), ⋆). Then, N(H) is a pseudo neutrosophic subgroup of N(G) if and only if the following conditions hold:
(1) a,b∈N(H)implies that a⋆b∈N(H);
(2) N(H)does not contain a proper subset A such that(A, ⋆)is a group.
4. HYPERGROUPS
The theory of hyperstructures was introduced in 1934 by Marty [9] at the 8th Congress of Scandinavian Mathematicians. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. Several books have been written on this topic, see [4, 5, 5, 8, 10]. Hyperstructure theory both extends some well-known group results and introduce new topics leading us to a wide variety of applications, as well as to a broadening of the investigation fields. In this part, we present the notion of hypergroup and some well-known related concepts. These concepts will be used in the building of neutrosophic hypergroups, for more details we refer the readers to see [4, 5, 5, 8, 9, 10].
Let H be a non-empty set and ◦ : H×H → P⋆(H) be a hyperoperation. The couple (H,◦) is called ahypergroupoid. For any two non-empty subsetsAandBof Handx∈H, we define
A◦B= ∪
a∈A,b∈Ba◦b, A◦x=A◦ {x} and x◦B={x} ◦B.
Definition 4.1. A hypergroupoid(H,◦)is called a semihypergroup if for all a,b,c of H we have(a◦b)◦c=a◦(b◦c), which means that
∪
u∈a◦bu◦c= ∪
v∈b◦ca◦v.
A hypergroupoid(H,◦)is called a quasihypergroup if for all a of H we have a◦H= H◦a= H. This condition is also called the reproduction axiom.
Definition 4.2. A hypergroupoid(H,◦)which is both a semihypergroup and a quasi- hypergroup is called a hypergroup.
Definition 4.3. Let(H,◦)and(H′,◦′)be two hypergroupoids. A mapϕ :H → H′, is called
(1) an inclusion homomorphism if for all x,y of H, we haveϕ(x◦y)⊆ϕ(x)◦′ϕ(y);
(2) a good homomorphism if for all x,y of H, we haveϕ(x◦y)=ϕ(x)◦′ϕ(y).
Let (H,◦) be a semihypergroup andRbe an equivalence relation onH. IfAandB are non-empty subsets ofH, then
ARBmeans that∀a∈A,∃b∈Bsuch thataRband
∀b′∈B,∃a′∈Asuch thata′Rb′; ARBmeans that∀a∈A,∀b∈B, we haveaRb.
Definition 4.4. The equivalence relationρis called
(1) regular on the right (on the left) if for all x of H, from aρb, it follows that (a◦x)ρ(b◦x)((x◦a)ρ(x◦b)respectively);
(2) strongly regular on the right (on the left) if for all x of H, from aρb, it follows that(a◦x)ρ(b◦x)((x◦a)ρ(x◦b)respectively);
(3) ρ is called regular (strongly regular) if it is regular (strongly regular) on the right and on the left.
Theorem 4.1. . Let(H,◦)be a semihypergroup andρbe an equivalence relation on H.
(1) If ρ is regular, then H/ρ is a semihypergroup, with respect to the following hyperoperation: x⊗y={z|z∈x◦y};
(2) If the above hyperoperation is well defined on H/ρ, thenρis regular.
Corollary 4.1. If(H,◦)is a hypergroup andρis an equivalence relation on H, then R is regular if and only if(H/ρ,⊗)is a hypergroup.
Theorem 4.2. Let(H,◦) be a semihypergroup andρ be an equivalence relation on H.
(1) Ifρis strongly regular, then H/ρis a semigroup, with respect to the following operation: x⊗y={z|z∈ x◦y};
(2) If the above operation is well defined on H/ρ, thenρis strongly regular.
Corollary 4.2. If(H,◦)is a hypergroup andρis an equivalence relation on H, then ρis strongly regular if and only if(H/ρ,⊗)is a group.
Definition 4.5. Let (H,◦) is a semihypergroup and A be a non-empty subset of H.
We say that A is a complete part of H if for any nonzero natural number n and for all a1, . . . ,anof H, the following implication holds:
A∩∏n
i=1ai ,∅ =⇒ ∏n
i=1ai ⊆A.
Theorem 4.3. If(H,◦)is a semihypergroup and R is a strongly regular relation on H, then for all z of H, the equivalence class of z is a complete part of H.
5. NEUTROSOPHIC HYPERGROUPS
Definition 5.1. Let(H, ⋆)be any hypergroup and let< H∪I >={x=(a,bI) :a,b∈ H}. The couple N(H)=(< H∪I >, ⋆)is called a neutrosophic hypergroup generated by H and I under the hyperoperation⋆. The part a is called the non-neutrosophic part of x and the part b is called the neutrosophic part of x.
If x=(a,bI)and y=(c,dI)are any two elements of N(H), where a,b,c,d∈H, we define x⋆y=(a,bI)⋆(c,dI)={(u,vI)|u∈a⋆c, v∈a⋆d∪b⋆c∪b⋆d}= (a⋆c,(a⋆d∪b⋆c∪b⋆d)I). Note that a⋆c⊆H and(a⋆d∪b⋆c∪b⋆d)⊆H.
Definition 5.2. Let N(H) be a neutrosophic hypergroup and let N(K) be a proper subset of N(H). Then,
(1) N(K) is said to be a neutrosophic sub-hypergroup of N(H)if N(K)is a neu- trosophic hypergroup, that is, N(K)must contain a proper subset which is a hypergroup;
(2) N(K)is said to be a pseudo neutrosophic sub-hypergroup of N(H) if N(K)is a neutrosophic hypergroup which contains no proper subset which is a hyper- group.
Theorem 5.1. Let N(H)be a neutrosophic hypergroup. Then, N(H)is a semihyper- group.
Proof. Letx= (a,bI), y=(c,dI), z= (e, f I) be arbitrary elements of N(H), where a,b,c,d,e,f ∈H. Then,
x⋆y = (a,bI)⋆(c,dI)
= {(u,vI)|u∈a⋆c, v∈a⋆d∪b⋆c∪b⋆d}
= (a⋆c,(a⋆d∪b⋆c∪b⋆d)I)
⊆ N(H). Hence, (N(H), ⋆) is a hypergroupoid.
Next,
x⋆(y⋆z) = (a,bI)⋆((c,dI)⋆(e, f I))
= (a,bI)⋆(c⋆e,(c⋆ f ∪d⋆e∪d⋆ f)I))
= (a⋆(c⋆e),((a⋆(c⋆ f))∪(a⋆(d⋆e))∪(a⋆(d⋆ f))∪(b⋆(c⋆e))
∪(b⋆(c⋆ f))∪(b⋆(d⋆e))∪(b⋆(d⋆ f)))I)
= ((a⋆c)⋆e,(((a⋆c)⋆ f)∪((a⋆d)⋆e)∪((a⋆d)⋆ f)∪((b⋆c)⋆e)
∪((b⋆c)⋆ f)∪((b⋆d)⋆e)∪((b⋆d)⋆ f))I)
= ((a,bI)⋆(c,dI))⋆(e,f I)
= (x⋆y)⋆z.
Accordingly, (N(H), ⋆) is a semihypergroup.
Lemma 5.1. Let N(H)be a neutrosophic hypergroup. Then, x⋆N(H)=N(H)⋆x⊂ N(H)for all x=(a,bI)∈N(H).
Proof. We have
x⋆N(H) = (a,bI)⋆N(H)
= (a,bI)⋆{(h1,h2I) :h1,h2∈H}
= {(a⋆h1,(a⋆h2∪b⋆h1∪b⋆h2)I) :a,b,h1,h2∈H}
= {(u,vI) :u∈a⋆h1,v∈(a⋆h2∪b⋆h1∪b⋆h2)}
⊂ N(H)
Similarly,N(H)⋆x⊂ N(H) and therefore,x⋆N(H)=N(H)⋆x⊂N(H).
Theorem 5.2. If N(H)is a neutrosophic hypergroup, then (1) N(H)in general is not a hypergroup;
(2) N(H)always contain a hypergroup.
Proof. (1) Follows directly from Theorem 5.3 and Lemma 5.4.
(2) Follows from the definition of aneutrosophichypergroup.
Example 5.1. Let H = {a,b,(a,aI),(a,bI),(b,aI),(b,bI)} be a set and let ⋆ be a hyperoperation on H defined in the table below.
⋆ a b (a,aI) (a,bI) (b,aI) (b,bI)
a a b (a,aI) (a,bI) (b,aI) (b,bI)
b b a
b (b,bI) (b,aI) (b,bI)
(a,bI) (b,bI)
(a,aI) (a,bI) (b,aI) (b,bI) (a,aI) (a,aI) (b,bI) (a,aI) (a,aI)
(a,bI)
(b,aI)
(b,bI) (b,bI) (a,bI) (a,bI) (b,aI)
(b,bI)
(a,aI) (a,bI)
(a,aI) (a,bI)
(b,aI) (b,bI)
(b,aI) (b,bI)
(b,aI) (b,aI) (b,bI) (a,bI)
(b,aI) (b,bI)
(b,aI) (b,bI)
(a,aI) (a,bI) (b,aI) (b,bI)
(a,aI) (a,bI) (b,aI) (b,bI)
(b,bI) (b,bI)
(a,aI) (a,bI) (b,aI) (b,bI)
(b,bI) (b,aI) (b,bI)
(a,aI) (a,bI) (b,aI) (b,bI)
(a,aI) (a,bI) (b,aI) (b,bI)
It is clear from the table that(H, ⋆)is a neutrosophic hypergroup since it contains a proper subset{a,b}which is a hypergroup under⋆.
Theorem 5.3. Let(N(H), ⋆1)and(N(K), ⋆2)be any two neutrosophic hypergroups.
Then,(N(H)×N(K), ⋆)is a neutrosophic hypergroup, where
(x1,x2)⋆(y1,y2)={(x,y) :x∈ x1⋆1y1,y∈x2⋆2y2, ∀(x1,x2),(y1,y2)∈N(H)×N(K)}.
