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Algebras of multiplace functions

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References

  1. Belousov, V.D.,On properties of binary operations, Učenye Zap. Bel'c Gos. Ped. Inst. 1960, Vyp. 5, 9–28 (Russian).

    Google Scholar 

  2. ——,Balanced identities in algebras of quasigroups, Aeq.Math.8 (1974), 1–73.

    Google Scholar 

  3. —,Balanced identities in algebras of quasigroup operations. II, Latem. Issled. (Kišinev) No. 1, 3–40. (Russian).

  4. Berman, G.; Silverman, R.J.,Embedding of algebraic systems, Pacific J. Math.10 1960, 777–786.

    Google Scholar 

  5. Bredihin, D.A.; Schein, B.K.,Representations of ordered semigrous and lattices by binary relations, lloq. math.39(1978).

  6. Calabrese, P.,The menger algebras of 2-Place Functions in the 2-valued Logic, Notr Dame J. Formal Logic7 (1966), 333–340.

    Google Scholar 

  7. Čupona, G.,On finitary operations Fac.Sci.Univ. Skopje Annuaire11 (1959), 7–49 (1361)(Macedonian).

    Google Scholar 

  8. Dicker, R.M.,The substitutive law Proc. London Math. Soc.13 (1963), 493–510.

    Google Scholar 

  9. Garvackii, V.S.,Projection semigroups of reversible transformations, Studies in Algebra, Saratov, 1969, 11–22 (Russian).

  10. ——, ∩-semigroups of transformations, Teorija Polugrupp i Priloz., Saratov2 1971, 3–13 (Russian).

    Google Scholar 

  11. Garvackii, V.S.; Trohimenko, V.S.,The relation of semi-adjacency of certain semigroups of transformations, Izv. Vysš. Učebn. zaved., Mat., 1971, No. 2, 23–32 (Russian).

  12. Gluskin, L.M.,Algebras of multiplace functions, Summaries of Talks. Interuniv. Sci. Symp. General Algebra, Tartu, 1966, 32–37 (Russian).

  13. ——,Superposition of multiplace functions, Publ. Math. Debrecen17 (1970), 349–378 (Russian).

    Google Scholar 

  14. Gluskin, L.M.,On the algebras of functions, Dopov. Akad. Nauk URSR A 1971, 394–396 (Ukrainian).

  15. —,Dense embeddings in algebras of functions, Dopov.Akad.Nauk URSR A 1971, 680–683 (Ukrainian).

  16. —,Multiplicative mappins of algebras of functions, Dopov.Akad.Nauk URSR A1971, 771–773 (Ukrainian).

  17. Henno, Ja.,Group-like Menger systems, Trudy Tallinn.Polytehn.Inst. A, no. 312, 1971, 95–110 (Russian).

  18. ——,Green equivalences in Menger systems, Učen. Zap. Tartu.Gos.Univ.277 (1971), 37–46 (Russian).

    Google Scholar 

  19. ——,Densely embedded right ideals of Menger systems, Izv.Akad.Nauk Est.SSR21 (1972), 131–141 (Russian).

    Google Scholar 

  20. ——,Dense embeddings in Menger systems, Izv. Akad.Nauk Est.SST21 (1972), 231–238 (Russian).

    Google Scholar 

  21. ——,Densely embedded left ideals of Menger systems, I, Izv.Akad.Nauk Est.SSR22 (1973), 123–130 (Russian).

    Google Scholar 

  22. ——,Densely embedded left ideals of Menger systems, II, Izv.Akad.Nauk Est.SSR22 (1973), 365–372 (Russian).

    Google Scholar 

  23. —,Free Ω-systems, Trudy Tallinn.Polytehn. Inst. A, no. 345, 1973, 29–40 (Russian).

  24. ——,Embedding Ω-systems in Ω-systems with a minimal number of generators, Izv.Akad.Nauk Est. SSR23 (1974), 326–335 (Russian).

    Google Scholar 

  25. ——, Free Γ-commutative Menger systems, Trudy Tallinn.Polytenn.Inst.,8 (1975), no. 373, 19–26 (Russian).

    Google Scholar 

  26. ——,On the completeness of idempotent functions of infinite sets, Ann.Univ.TurkuA170 (1976), 1–9.

    Google Scholar 

  27. —,On idempotents and Green relations in the algebras of many-placed functions, Ann.Acad.Sci. Fennicae, Math. A 1977

  28. Hion, Ja.V., Ω-systems, Summaries of talks, Interuniv.Sci.Symp.General algebra, Tartu, 1966, 123–129 (Russian).

  29. ——, m-ary Ω-ringoids, Sibir.mat.Žurn.8 (1967), 174–194 (Russian).

    Google Scholar 

  30. —,Polyringoids, Summaries of talks, IX-th All-Union Alg.Colloq., Gomel', 1968, 198–199 (Russian).

  31. —,On generalized Menger systems, Summaries of talks, 1st All-Union Symp.Theory Semigroups, Sverdlovsk, 1969, 78–79 (Russian).

  32. Hoehnke, H.-J.,Superposition partieller Funktionen, Stud.zur Algebra und Anw., Berlin-DDR, 1972, Heft 16, 7–26.

  33. Kafka, V.,Axiomatics for systems of multiplace functions, M.S. Thesis, Ill.Inst.Technology, 1365.

  34. Knoebel, R.A.,A simplification of the functional completeness proofs of Quackenbush and Sierpinski, Alg.Univ.4 (1974), 133–134.

    Google Scholar 

  35. Länger, H.,Verallgemeinerung eines Satzes von Nöbauer und Philipp, Arch.der Matn.27 (1976), 1–2.

    Google Scholar 

  36. Lausch, H.; Nöbauer, W.,Algebra of polynomials, Amsterdam-London-New York, North-Holland & American Elsevier, 1973.

    Google Scholar 

  37. Mal'cev, A.I.,Iterative algebras and Poat varieties, Algebra i Logika, Novosibirsk5 (1966), No. 2, 5–24 (Russian).

    Google Scholar 

  38. Menger, K.,General algebra of analysis, Repts. Math.Colloq., Notre Dame Univ.7 (1946), 49–60.

    Google Scholar 

  39. —,On axiomatic theory of functions and fluents, Symp.Axiom.Method, Amsterdam, North-Holland, 1359, 454–473.

  40. Menger, K.,The algebra of functions:past, present, future, Rend.Mat.20 (1961), 409–430.

    Google Scholar 

  41. Menger, K.,Function algebra and propositional calculus, Self-Organizing Syst., Spartan Books, Wash., D.C., 1962, 525–532.

    Google Scholar 

  42. Menger, K.,A group of the substitutive algebra of the calculus of propositions, Arch.der Math.13 (1962), 471–478.

    Google Scholar 

  43. Menger, K.; Schultz, M.,Postulates for the substitutive algebra of the 2-place functors on the 2-valued calculus of propositions, Notre Dame J. Form. Logic4 (1963), 128–192.

    Google Scholar 

  44. Menger, K.; Whitlock H.,Two theorems on the generations of systems of functions, Fund.Math.58 (1966), 229–240.

    Google Scholar 

  45. Nöbauer, N.,Über die Darstellung von universellen Algebren durch Funktionenalgebren, Publ.Math.Lebrecen10 (1963), 151–154.

    Google Scholar 

  46. Nöbauer, N.,Über die Automorphismen von Kompositions-algebren, Acta Math.Acad.Sci.Hungar.26 (1975), 275–278.

    Google Scholar 

  47. Nöbauer, W.; Philipp, W.,Über die Einfachheit von Funktionenalgebren, Monatsh.Math.66 (1962), 441–452.

    Google Scholar 

  48. Nöbauer, W.; Philipp, W.,Die Einfachheit der mehrdimensionalen Funktionenalgebren, Arch.der Math.15 (1964), 1–5.

    Google Scholar 

  49. Pavlovskii, E.V.,The semi-adjacency relation in semigroups of partial transformations, Izv.Vyšs. Učebn.Zaved., Math. 1970, No.3, 70–75 (Russian).

  50. Peters R.,Two remarks concerning Menger's and Schultz' postulates for the substitutive algebra of the 2-place functors in the 2-lvalued calculus of propositions, Notre Dame J.Form.Logic5 (1964), 125–128.

    Google Scholar 

  51. Quackenbush, R.W.,On the composition of idempotent functions, Alg.Univ.1 (1971), 7–12.

    Google Scholar 

  52. Redi, E.,Representation of Menger systems by multi-place endomorphisms, Učen.Zap.Tartu.Gos. Univ.277 (1971), 47–51 (Russian).

    Google Scholar 

  53. Redi, E.,One-sided ideals of symmetric polycategories, Učen.Zap.Tartu.Gos.Univ.336 (1974), 31–61 (Russian).

    Google Scholar 

  54. Redi, E.,On symmetric polyringoids, Učen.Zap.Tartu.Gos.Univ.336 (1974), 62–99 (Russian).

    Google Scholar 

  55. Redi, E.,On the polycategory of multiplace relations and polyringoid of partial multi-place functions, Učen.Zap.Tartu.Gos.Univ.366 (1975), 3–25 (Russian).

    Google Scholar 

  56. Redi, E.,On the representation of polyringoids, Učen.Zap.Tartu.Gos.Uni.390 (1976), 43–55 (Russian).

    Google Scholar 

  57. Sandik, M.D.,An algebra of multi-operations, Izv.Akad.Nauk Mold.SSR, Ser.Fiz.-Tehn., Mat.Nauk 1972, no.1, 31–38 (Russian).

  58. Schein, B.M.,Embedding semigroups in generalized groups, Matem. Sb.55 (1961), 373–400 (Russian).

    Google Scholar 

  59. Schein, B.M.,Representation of semigroups by means of binary relations, Dokl.Akad.Nauk SSSR142 (1962), 808–811 (Russian).

    Google Scholar 

  60. Schein, B.M.,Representation of semigroups by means of binary relations, Matem.Sb.60 (1963), 293–303 (Russian).

    Google Scholar 

  61. —,On the theory of semigroups of transformations, Works of Young Scientists, Vyp.Mat., Saratov, 1964, 120–122 (Russian).

  62. Schein, B.M.,Representation of ordered semigroups, Matem. Sb.65 (1964), 188–197 (Russian).

    Google Scholar 

  63. Schein, B.M.,On certain classes of semigroups of binary relations, Sibir.Mat.Zurn.6 (1965), 616–635 (Russian).

    Google Scholar 

  64. Schein, B.M.,On certain comitants of semigroups of binary relations, Matem.Zap.Ural.Gos.Univ.5 (1965), no. 1, 73–91 (Russian).

    Google Scholar 

  65. Schein, B.M.,Transformative semigroups of transformations, Matem.Sb.71 (1966), 65–92 (Russian).

    Google Scholar 

  66. —,Theory of semigroups as theory of superpositions of one-and multi-place functions, Summaries of Short Communs. Internat.Congr.Math., Moscow, 1966, Algebra (2), 56 (Russian).

  67. —,Theory of semigroups as theory of superpositions of multiplace functions, Summaries of Talks, Interuniv.Sci.Symp.General Algebra, Tartu, 1966, 169–192 (Russian).

  68. Schein, B.M.,Restrictive-multiplicative algebras of transformations, Izv.Vysš.Učeon.Zaved., Mat. 1970, No.4, 91–102 (Russian).

  69. Schein, B.M.,Relation algebras and function semigroups, Semigroup Forum1 (1970), 1–62.

    Google Scholar 

  70. —,Stationary subsets,stabilizers and transitive,representations of semigroups. Dissertationes Math.77(1970).

  71. Schein, B.M.,The co-definability relation on semigroups of functions, Ordered Sets and Lattices, Saratov1 (1971), 86–89 (Russian).

    Google Scholar 

  72. ——,Compatible function semigroups, Colloq. Math.28 (1973), 19–26.

    Google Scholar 

  73. ——,Lattice-ordered semigroups of functions, Ordered Sets and Lattices, Saratov2 (1974), 109–113 (Russian).

    Google Scholar 

  74. ——,Function U-semigroups Colloq.Math.31 (1974), 11–20.

    Google Scholar 

  75. ——,A representation theorem for lattices, Alg. Univ.2 (1972), 177–178.

    Google Scholar 

  76. Schweizer, B.; Sklar, A.,The algebra of multiplace vector-valued functions, Bull. Amer. Math. Soc.73 (1957), 510–515.

    Google Scholar 

  77. ——; ——,A grammar of functions, I, Aeq. Math.2 (1968), 62–85.

    Google Scholar 

  78. ——; ——,A grammar of functions, II, Aeq. Math.3 (1969), 15–43.

    Google Scholar 

  79. Sierpiński, W.,Sur les fonctions de plusieurs variables, Fund. Math.33 (1945), 169–173.

    Google Scholar 

  80. Skala, H.L.,The irreducible generating sets of 2-place functions in the 2-valued logic, Notre Dame J. Form. Logic7 (1966), 341–343.

    Google Scholar 

  81. ——,Irreducibly generated algebras, Fund. Math.67 (1970), 31–37.

    Google Scholar 

  82. ——,Grouplike Menger algebras, Fund. Math.79 (1973), 129–207.

    Google Scholar 

  83. Skordev, D.G.,On multi-valued functions of several variables, Dokl. Bolgar. Akad. Nauk28 (1975), 885–888 (Russian).

    Google Scholar 

  84. ——,Recursion theory on iterative combinatory spaces, Bull. Acad. Polon. Sci., Ser. Sci. Math., Astr., Phys.24 (1976), 23–31.

    Google Scholar 

  85. Strazdins, I.E.,Automorphism groups of the algebra of Boolean functions and Post classes, Trudy Rižsk. Algebr. Semin., Riga 1969, 253–266 (Russian).

  86. Tamari, D.,Sur quelques proglèmes d'associativité, Ann. Fac. Sci. Univ. Clermont, Math. 1964, No. 24, 91–107.

  87. Trohimenko, V.S.,On projectors of algebras of multi-place functions, Studies in Alg., Saratov 1969, 52–60 (Russian).

  88. —,On algebras of multi-place partial transformations, Works of Young Scientists, Saratov, Vyp. 2, 1969, 136–142 (Russian).

  89. —,Stabilizers of certain algebras of multi-place functions, Proc. 1st Sci.-Techn. Conf., VVChKU, Saratov, 1970, 68–72 (Russian).

  90. —,Ordered algebras of multi-place functions, Izv. Vyss. Učebn. Zaved., Mat. 1971, No. 1, 90–98 (Russian).

  91. —,The semi-adjacency relation of ordered algebras of multi-placefunctions, Summaries of Talks, 7-th Sci.-Techn. Conf. Ul'janovsk. Polytechn. Inst., Ul'janovsk, 1971, 120–122. (Russian).

  92. —,Abstract characterizations of certain algebras of multi-place functions, Izv. Vyss. Učbn. Zaved., Mat. 1971, No. 4, 87–95 (Russian).

  93. ——,Algebras of reversive multi-place functions, Theory of Semigroups and its Appl., Saratov2 (1971), 75–84 (Russian).

    Google Scholar 

  94. —,Restrictive algebras of multi-place functions, Dopov. Akad. Nauk URSR A 1972, 340–342 (Ukrainian).

  95. ——,On algebras of binary operations, Matem. Issled., Kišinev7 (1972), No. 2, 253–261 (Russian).

    Google Scholar 

  96. —,Position systems of multi-place functions, Issled. Teor. Kvazigrupp Lup, Kišinev, 1973, 165–174 (Russian).

  97. —,A characterization of the semigroup of multi-place functions, Summaries of Talks, XII-th All-Union Alg. Colloq., Sverdlovsk 1973, 239 (Russian).

  98. —,Functions Menger systems, Izv. Vysš. Učebn. Zaved., Mat. 1973, No. 11, 71–73 (Russian).

  99. ——,The semi-adjacency relation in algebras of multi-place functions, Theory of Semigroups and its Appl., Saratov3 (1974), 108–118 (Russian).

    Google Scholar 

  100. ——,A characterization of P-algebras of multi-place functions, Sibir. Matem. Žurn.16 (1975), 599–611 (Russian).

    Google Scholar 

  101. —,On the co-definability relation on ordered Menger algebras of reversive functions. Summaries of Talks, All-Union Alg. Symp., Gomel' 1975, 453–454 (Russian).

  102. —,A characterization of the co-definability relation on ordered algebras of multi-place functions, Izv. Vysš. Učebn. Zaved., Mat. 1977, No. 9, 80–86 (Russian).

  103. Trohimenko, V.S.,On certain Menger algebras of relations, Izv. Vysš. Učebn. Zaved., Mat. 1978 (to appear in Russian).

  104. —,Algebras of multi-place functions, Ph.D. Thesis, Saratov Univ., 1970 (Russian).

  105. Trohimenko, V.S., Kolesnikov, C.V.,Representation of Menger operatives, Issled. po Sovrem. Problemam Summitov. i Približ. FUnkcii i in Prilož., Dnepropetrovsk5 (1974), 190–199 (Russian).

    Google Scholar 

  106. Trohimenko, V.S.; Schein, B.M.,Compatible algebras of multiplace functions, Theory of Semigroups and its Appl., Saratov4 (1978), 89–98 (Russian).

    Google Scholar 

  107. Trpenovski, B.; Čupona, G.,Finitary associative operations with neutral elements, Bull. Soc. Math. Phys. Macedoine12 (1961), 15–24 (1963) (Macedonian).

    Google Scholar 

  108. Wilde, C.; Raney, S.,Computation of the transformation semigroups on three letters, J. Austral. Math. Soc.14 (1972), 335.

    Google Scholar 

  109. Whitlock, H.,A composition algebra for multip-place functions, Math. Ann.157 (1964), 167–178.

    Google Scholar 

  110. —,On generality of singly generated Menger algebras, Notices Amer. Math. Soc.13 (1966), Abstract 636-42.

  111. Yakubov, T.,On the (2, n)-semigroup of n-ary operations, Izv. Akad. Nauk Moldav. SSR, Ser. Fiz.-Tehn., Mat. Nauk 1974, No. 1, 25–46 (Russian).

  112. Yaroker, Ja.N.,Completely simple Menger operatives, Dopov. Akad. Nauk URSR A 1972, 34–36 (Ukrainian).

  113. —,On the operatives of relations, Dopov. Akad. Nauk URSR A 1973, 49–51 (Ukrainian).

  114. Zareckii, K.A.,Representation of ordered semigroups by binary relations, Izv. Vysš. Učebn. Zaved., Mat. 1959, No. 6, 48–50 (Russian).

  115. ——,An abstract characterization of the bisemigroup of binary operations, Matem. Zametki1 (1967), 525–530 (Russian).

    Google Scholar 

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  1. 410 600 Saratov, U. S. S. R. ul. Vavilova 2, kv. 111, USSR

    Boris M. Schein

  2. 286 021 Vinnitsa, U. S. S. R. ul. Zavodskaya 19, kv. 60., USSR

    Valentin S. Trohimenko

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Schein, B.M., Trohimenko, V.S. Algebras of multiplace functions. Semigroup Forum 17, 1–64 (1979). https://doi.org/10.1007/BF02194309

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