On Co-Ideals of Implicative Semigroups with Apartness

Abstract

The setting of this research is the Bishop's constructive mathematics - a mathematics based on the Intuitionistic Logic and the principled-philosophical constructive orientation. Implicative semigroups with apartness were introduced and analyzed in 2016-17 in two published articles (\emph{An introduction to implicative semigroups with apartness}, Sarajevo J. Math., 12(25)(2)(2016),  155-165 and \emph{Strongly extensional homomorphism of implicative semigroups with apartness}.  Sarajevo J. Math., 13(2)(2017),  155-162). In this paper, as a continuation of the mentioned articles, the concept of co-ideals was introduced in the implicative semigroups with apartness. Some of the important properties of these substructures in such semigroups have been proven. In addition, it has been shown that the family of all co-ideals in these semigroups forms a complete lattice.

Keywords

• Constructive mathematics,semigroup with apartness,co-ordered semigroups,implicative semigroups with apartness,co-ideals

References

1. Bang, K., So, K.S., \emph{Extended special sets in implicative semigroups}, Commun Korean Math Soc, \textbf{22}(2007), 9--14.
2. Beeson, M.A., Foundations of Constructive Mathematics., Berlin, Springer, 1985.
3. Bishop, E., Foundations of Constructive Analysis, New York, McGraw-Hill, 1967.
4. Bishop, E., Bridges, D.S., Constructive Analysis, Grundlehren der Mathematischen Wissenschaften 279, Berlin, Springer, 1985.
5. Birkhoff, G., Lattice Theory, 3rd ed., Rhode Island: American Mathematical Society Colloquium Publications, vol. 25, 1967.
6. Blyth, T.S., \emph{Pseudo-residuals in semigroups}, J London Math Soc, \textbf{40}(1965), 441--454.
7. Bridges, D.S., Richman, F., Varieties of Constructive Mathematics. Cambridge: London Mathematical Society Lecture Notes, No. 97, Cambridge University Press, 1987.
8. Cherubini, A., Frigeri, A., \emph{Inverse semigroups with apartness}, Semigroup Forum, \textbf{98(3)}(2019), 571--588.

9. Chan, M.W., Shum, K.P., \emph{Homomorphisms of implicative semigroups}, Semigroup Forum, \textbf{46}(1993), 7--15.
10. Crvenkovi\'c, S., Mitrovi\'c, M., Romano, D.A., \emph{Semigroups with apartness}, Math Logic Quart, \textbf{59(6)}(2013), 407--414.
11. Crvenkovi\'c, S., Mitrovi\'c, M., Romano, D.A., {\em Basic notions of (constructive) semigroups with apartness}, Semigroup Forum, \textbf{92(3)}(2016), 659--674.
12. Greenleaf, N., Linear Order in Lattices: A Constructive Study, In: Rota G-C. (Editor), Advances in Mathematics Supplementary Studies, 1, (pp. 11-30). New York, Academic Press, 1978.
13. Jun, Y.B., \emph{A note on ordered filters of implicative semigroups}, Bull Korean Math Soc, \textbf{34}(1997), 185--191.
14. Jun, Y.B., \emph{Some results on ordered filters of implicative semigroups}, Int J Math Math Sci, \textbf{26}(2001), 731--735.
15. Jun, Y.B, Meng, J., Xin, X.L., \emph{On ordered filters of implicative semigroups}, Semigroup Forum, \textbf{54}(1997), 75--82.
16. Jun, Y.B., Kim, K.H., \emph{On ideals of implicative semigroups}, Int J Math Math Sci, \textbf{27}(2001), 77--82.
17. Lee, S.Y., Shum, K.P., Wu, C., \emph{Filters in fuzzy implicative semigroups}, Comm. Algebra, \textbf{32}(2004), 4633--4651.
18. Lee, S.Y., Shum, K.P., Wu, C., \emph{Strong semilattice of implicative semigroups}, Algebra Colloqium, \textbf{15}(2008), 57--62.
19. Mines, R., Richman, F., Ruitenburg, W., A Course of Constructive Algebra, New York, Springer-Verlag, 1988.
20. Nemitz, W.C., \emph{Implicative semi-lattices}, Trans Amer Math Soc, \textbf{117}(1965), 128--142.
21. Negri, S., \emph{Sequent calculus proof theory of Intuitionistic apartness and order relations}, Archive Math Logic, \textbf{38}(1999), 521--547.
22. Romano, D.A., \emph{A note on quasi-antiorder in semigroup}, Novi Sad J. Math, \textbf{37}(2007), 3--8.
23. Romano, D.A., \emph{An isomorphism theorem for anti-ordered sets}, Filomat, \textbf{22}(2008), 145--160.
24. Romano, D.A., \emph{On semilattice-ordered semigroups. A constructive point of view}, Scientific Studies and Research. Series Mathematics and Informatics, \textbf{21}(2011), 117--134.
25. Romano, D.A., \emph{On quasi-antiorder relation on semigroups}, Mat. Vesnik, \textbf{64}(2012), 190--199.
26. Romano, D.A., \emph{An introduction to implicative semigroups with apartness}, Sarajevo J. Math, \textbf{12(2)}(2016), 155--165.
27. Romano, D.A., \emph{Strongly extensional homomorphism of implicative semigroups with apartness}, Sarajevo J. Math., \textbf{13(2)}(2017), 155--162.
28. Romano, D.A., {\em Co-filters in semilattice-ordered semigroup with apartness}, J. Adv. Math. Stud, \textbf{11(1)}(2018), 124--131.
29. Romano, D.A., \emph{Co-ideals and co-filters in ordered set under co-quasiorder}, Bull. Int. Math. Virtual Inst., \textbf{8(1)}(2018), 177--188.
30. Romano, D.A., \emph{Some algebraic structures with apartness, A review}, J. Int. Math. Virtual Inst., \textbf{9(2)}(2019), 361--395.
31. Troelstra, A.S., van Dalen, D., Constructivism in Mathematics: An Introduction, Amsterdam,vNorth-Holland, 1988.
32. Von Plato, J., Positive Lattices, In: P.Schuster, U.Berger and H.Osswald (Eds.), Reuniting the Antipodes-Constructive and Nonstandard Views of the Continuum (pp. 185-197), Dordrecht: Kluwer Academic Publishers, 2001.