Isomorphism Theorems for Basic Constructive Algebraic Structures with Special Emphasize On Constructive Semigroups with Apartness—An Overview

  • Melanija MitrovićEmail author
  • Sergei Silvestrov
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 317)


This overview is an introduction to the basic constructive algebraic structures with apartness with special emphasises on a set and semigroup with apartness. The main purpose of this paper, inspired by Bauer [2], is to make some sort of understanding of constructive algebra in Bishop’s style position for those (classical) algebraists as well as for the ones who apply algebraic knowledge who might wonder what is constructive algebra all about. Every effort has been made to produce a reasonably prepared text with such definite need. In the context of basic constructive algebraic structures constructive analogous of isomorphism theorems will be given. Following their development, two points of view on a given subject: classical and constructive will be considered. This overview is not, of course, a comprehensive one.


Sets with apartness Groups and rings with tight apartness Semigroups with apartness Apartness isomorphism theorems 

MSC 2010 Classification

20M15 03F65 03D45 



Melanija Mitrović is financially supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia, Grant 174026, and by the Faculty of Mechanical Engineering, University of Niš, Serbia, Grant “Research and development of new generation machine systems in the function of the technological development of Serbia”. Melanija Mitrovicć is grateful to Mathematics and Applied Mathematics Research Environment MAM, Division of Applied Mathematics at the School of Education, Culture and Communication at Mälardalen University for cordial hospitality and excellent environment for research and cooperation during her visit.


  1. 1.
    Baroni, M., Bridges, D.S.: Continuity properties of preference relations. Math. Log. Q. 54(5), 454–459 (2008)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bauer, A.: Five stages of accepting constructive mathematics. Bull. (New Ser.) Am. Math. Soc. vol. 54(3), 481–498 (2017)Google Scholar
  3. 3.
    Beeson, M.J.: Foundations of Constructive Mathematics. Springer, Berlin (1985)CrossRefGoogle Scholar
  4. 4.
    Bishop, E.: Foundations of Constructive Analysis. McGraw-Hill, New York (1967)zbMATHGoogle Scholar
  5. 5.
    Bishop, E., Bridges, D.S.: Constructive Analysis, Grundlehren der mathematischen Wissenschaften 279. Springer, Berlin (1985)Google Scholar
  6. 6.
    Bridges, D.S., Richman, F.: Varieties of Constructive Mathematics. London Mathematical Society Lecture Notes, vol. 97. Cambridge University Press, Cambridge (1987)Google Scholar
  7. 7.
    Bridges, D.S., Reeves, S.: Constructive mathematics in theory and programming practice. Philos. Math. 7(3), 63–104 (1999)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bridges, D.S., Havea, R.: A constructive version of the spectral mapping theorem. Math. Log. Q. 47(3), 299–304 (2001)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bridges, D.S., Vîta, L.S.: Techniques in Constructive Analysis. Universitext. Springer, Berlin (2006)zbMATHGoogle Scholar
  10. 10.
    Bridges, D.S., Vîta, L.S.: Apartness and Uniformity - A Constructive Development. CiE Series on Theory and Applications of Computability. Springer, Berlin (2011)CrossRefGoogle Scholar
  11. 11.
    Calderón, G.: Formalizing constructive projective geometry in Agda. Electronic Notes in Theoretical Computer Science, 338, 61–77 (2018)Google Scholar
  12. 12.
    Clark, A.: Elements of Abstract Algebra, p. 1984. Dover Publications, Inc., New York (1974)Google Scholar
  13. 13.
    Crvenković, S., Mitrović, M., Romano, D.A.: Semigroups with apartness. Math. Log. Q. 59(6), 407–414 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Crvenković, S., Mitrović, M., Romano, D.A.: Basic otions of (constructive) semigroups with apartness. Semigroup Forum 92(3), 659–674 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Geuvers, H., Pollack, R., Wiedijk, F., Zwanenburg, J.: A constructive algebraic hierarchy in coq. J. Symb. Comput. 34, 271–286 (2002)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Gunther, E., Gadea, A., Pagano, M.: Formalization of universal algebra in Agda. Electronic Notes in Theoretical Computer Science, 338, 147–166 (2018)Google Scholar
  17. 17.
    Herstein, I.N.: Topics in Algebra, 2nd edn. Wiley, New Jersey (1975)zbMATHGoogle Scholar
  18. 18.
    Heyting, A.: Intuitionistische axiomatick der projectieve meetkunde. Thesis, P. Noordhoof (1925)Google Scholar
  19. 19.
    Heyting, A.: Zur intuitionistischen Axiomatik der projektiven Geometrie. Math. Ann. 98, 491–538 (1927)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Heyting, A.: Untersuchungen über intuitionistische Algebra. Nederl. Akad. Wetensch. Verh. Tweede Afd. Nat. 18(2) (1941)Google Scholar
  21. 21.
    Heyting, A.: Intuitionism, an Introduction. North-Holland (1956)Google Scholar
  22. 22.
    Hollings, C.: Mathematics Across the Iron Curtain: A History of the Algebraic Theory of Semigroups. American Mathematical Society, Providence (2014)CrossRefGoogle Scholar
  23. 23.
    Howie, J.M.: Why Study Semigroups? Lecture given to the New Zealand Mathematical Colloquium (1986)Google Scholar
  24. 24.
    Howie, J.M.: Fundamentals of Semigroup Theory. London Mathematical Society Monographs, New Series. Clarendon Press, Oxford (1995)Google Scholar
  25. 25.
    Hungerford, T.W.: Algebra. Springer, Berlin (2003). (Twelfth printing)Google Scholar
  26. 26.
    Ishihara, H., Palmgren, E.: Quotient topologies in constructive set theory and type theory. Ann. Pure Appl. Log. 141, 257–265 (2006)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Jacobs, B.: Bisimulation and Apartness in Coalgebraic Specification; 1995 - Notes of Lectures Given in January 1995 at the Joint TYPES/CLICS Workshop in Gothenburg and BRICS Seminar in AarhusGoogle Scholar
  28. 28.
    Martin-Löf, P.: Constructive mathematics and computer programming. In: Kohen, L.J., Los, J., Pfeiffer, H., Podewski, K.-P. (eds.) The Proceedings of the Sixth International Congress of Logic. Methodology and Philosophy of Science, pp. 153–175. North-Holland Publishing Company, Amsterdam (1982)Google Scholar
  29. 29.
    Mines, R., Richman, F., Ruitenburg, W.: A Course of Constructive Algebra. Springer, New York (1988)CrossRefGoogle Scholar
  30. 30.
    Mitrović, M.: Semilattices of Archimedean Semigroups. University of Niš - Faculty of Mechanical Engineering, Niš (2003)zbMATHGoogle Scholar
  31. 31.
    Mitrović, M., Darpö, E.: Apartness complements of Subsets of Constructive Sets and Semigroups with Apartness, (in preparation)Google Scholar
  32. 32.
    Moshier, M.A.: A rational reconstruction of the domain of feature structures. J. Log., Lang. Inf. 4(2), 111–143 (1995)Google Scholar
  33. 33.
    Richman, F.: Interview with a constructive mathematician. Mod. Log. 6, 247–271 (1996)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Romano, D.A.: Rings and fields, a constructive view. Math. Log. Q. (Formerly: Z. Math. Logik Grundl. Math.) 34(1) 25–40 (1988)Google Scholar
  35. 35.
    Ruitenburg, W.B.G.: Intuitionistic Algebra, Theory and Sheaf Models, Ph.D. (1982)Google Scholar
  36. 36.
    Ruitenburg, W.: Inequality in constructive mathematics. Notre Dame J. Form. Log. 32, 533–553 (1991)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Troelstra, A.S., van Dalen, D.: Constructivism in Mathematics, An Introduction, (Two volumes). North-Holland, Amsterdam (1988)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Mathematics and InformaticsFaculty of Mechanical Engineering, University of NišNišSerbia
  2. 2.Division of Applied Mathematics, School of Education, Culture and CommunicationMälardalen UniversityVästeråsSweden

Personalised recommendations