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Functional representation of preiterative/combinatory formalism. (English) Zbl 1097.03012
Summary: Both formalisms model systems of multi-argument selfmaps closed under composition as well as argument permutation and identification by using abstract algebraic operations for these transformations and substitutions. Developed are additional requirements for each system to be representable as a system of concrete selfmaps on some set in which these operations act in the expected way.
03B40 Combinatory logic and lambda calculus
03C05 Equational classes, universal algebra in model theory
08A02 Relational systems, laws of composition
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[1] CURRY H. B.-FEYS R.: Combinatory Logic. N. Holland, Amsterdam, 1968. · Zbl 0197.00601
[2] DICKER R. M.: The substitutive law. Proc. London Math. Soc. 13 (1963), 493-510. · Zbl 0122.25501
[3] FLEISCHER I.: Semigroup of not bijective finite selfmaps of an infinite set. Algebra Universalis 33 (1995), 186-190; Semigroup Forum 58 (1999), 468-470. · Zbl 0821.03030
[4] HALMOS P. R.: Algebraic Logic. Chelsea Publ. Comp., New York, 1962. · Zbl 0101.01101
[5] HINDLEY J. R.-SELDIN J.: Introduction to Combinators and X-Calculus. London Math. Soc. Stud. Texts 1, Cambridge Univ. Press, Cambridge, 1986. · Zbl 0614.03014
[6] HOWIE J. M.: An Introduction to Semigroup Theory. London Math. Soc. Monogr. 7, Academic Press, London-New York-San Francisco, 1976. · Zbl 0355.20056
[7] JÓNSSON B.: Defining relations for full semigroups of finite transformations. Michigan Math. J. 9 (1962), 77-85. · Zbl 0111.03803
[8] LAUSCH H.-NÖBAUER W.: Algebra of Polynomials. North-Holland Math. Library 5, North-Holland Publ. Comp./Amer. Elsevier Publ. Comp., Inc, Amsterdam-London/New York, 1973. · Zbl 0283.12101
[9] MAĽCEV A. I.: Iterative Algebras and Pos\?s Varieties (Russian). [
[10] MENGER K.: On substitutive algebra and its syntax. Z. Math. Logik Grundlag. Math. 10 (1964), 81-104. · Zbl 0132.24601
[11] ROSENBERG I. G.: Maľcev algebras for universal algebra terms. Algebraic Logic and Universal Algebra in Computer Science, Conference, Ames, Iowa, USA, June 1-4, 1988. Proceedings (C. H. Bergman et al., Lecture Notes in Comput. Sci. 425, Springer-Verlag, Berlin, 1990, pp. 195-208.
[12] SCHÖNFINKEL M.: Bausteine der Mathematischen Logik. Math. Ann. 92 (1924), 305-316. [ · JFM 50.0023.01
[13] STENLUND S.: Combinators, X-Terms and Proof Theory. D. Reidel, Dordrecht, 1972. · Zbl 0248.02032
[14] WHITLOCK H. I.: A composition algebra for multiplace functions. Math. Ann. 157 (1964), 167-178. · Zbl 0126.03501
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