THE FIRST ISOMORPHISM THEOREM FOR (CO-ORDERED) $\Gamma$-SEMIGROUPS WITH APARTNESS

Year 2023, , 239 - 253, 31.07.2023

Daniel A. Romano

https://doi.org/10.33773/jum.1195108

Abstract

The notion of $\Gamma$-semigroups has been introduced by Sen and Saha in 1986. This author introduced the concept of $\Gamma$-semigroups with apartness and analyzed
their properties within the Bishop's constructive orientation. Many classical notions and processes of semigroups and $\Gamma$-semigroups have been extended to $\Gamma$-semigroups with apartness. Co-ordered $\Gamma$-semigroups with apartness have been studied by the author also. In this paper, as a continuation of previous research, the author investigates the specificity of two forms of the first isomorphism theorem for (co-ordered) $\Gamma$-semigroups with apartness which one of them has no a counterpart in the Classical case. In addition,
specific techniques used in proofs within algebraic Bishop's constructive orientation are exposed.

Keywords

Bishop, Intuitionistic logic, $\Gamma$-semigroup with apartness, ordered $\Gamma$-semigroup under co-order, co-congruence in $\Gamma$-semigroup, the isomorphism theorem.}

References

• E. Bishop, Foundations of Constructive Analysis, New York: McGraw-Hill, (1967).
• D. S. Bridges and F. Richman, Varieties of Constructive Mathematics, Cambridge: London Mathematical Society Lecture Notes, No. 97, Cambridge University Press, (1987). A. Cherubini and A. Frigeri, Inverse semigroups with apartness, Semigroup Forum, 98(3), 571--588 (2019).
• R. Chinram and K. Tinpun, Isomorphism theorems for $\Gamma$-semigroups and ordered $\Gamma$-semigroups, Thai Journal of Mathematics, 7(2), 231-–241 (2009).
• S. Crvenkovi\'c, M. Mitrovi\'c and D. A. Romano, Semigroups with apartness, Math. Logic Quarterly, 59(6), 407--414 (2013).
• S. Crvenkovi\'c, M. Mitrovi\'c and D. A. Romano, Basic notions of (Constructive) semigroups with apartness, Semigroup Forum, 92(3), 659--674 (2016).
• H. Hedayati, Isomorphisms via congruences on $\Gamma$-semigroups and $\Gamma$-ideals, Thai J. Math., 11(3), 563--575 (2013).
• N. Kehayopulu, On ordered $\Gamma$-semigroups. Sci. Math. Japonicae Online, e-2010, 37-–43 (2013).
• Y. I. Kwon amd S. K. Li, Some special elements in ordered $\Gamma$-semigroups, Kyungpook Math. J., 35(3), 679--685 (1996).
• R. Mines, F. Richman and W. Ruitenburg, A course of constructive algebra, Springer-Verlag, New York (1988).
• D. A. Romano, Some relations and subsets of semigroup with apartness generated by the principal consistent subset, Univ. Beograd, Publ. Elektroteh. Fak. Ser. Math, {13}, 7--25 (2002).
• D. A. Romano, A note on quasi-antiorder in semigroup, Novi Sad J. Math., 37(1), 3--8 (2007).
• D. A. Romano, An isomorphism theorem for anti-ordered sets, Filomat, 22(1), 145--160 (2008).
• D. A. Romano, On quasi-antiorder relation on semigroups, Mat. Vesn., 64(3), 190--199 (2012).
• D. A. Romano, $\Gamma$-semigroups with apartness, Bull. Allahabad Math. Soc., 34(1), 71--83 (2019).
• D. A. Romano, Some algebraic structures with apartness, A review, J. Int. Math. Virtual Inst., 9(2), 361--395 (2019).
• D. A. Romano, Semilattice co-congruence in $\Gamma$-semigroups, Turkish Journal of Mathematics and Computer Science, 12(1), 1--7 (2020).
• D. A. Romano, Co-filters in $\Gamma$-semigroups ordered under co-order, An. \c{S}tiin\c{t}. Univ. Al. I. Cuza Ia\c{s}i, Ser. Nou\u{a}, Mat., 67(1), 11--18 (2021).
• D. A. Romano. On co-filters in semigroup with apartness. \emph{Kragujevac J. Math.}, \textbf{45}(4)(2021), 607-–613.
• M. K Sen, On $\Gamma$-semigroups. In Proceeding of International Conference on 'Algebra and its Applications, (New Delhi, 1981)', (pp. 301--308). Lecture Notes in Pure and Appl. Math. 9, New York: Decker Publication, (1984).
• M. K. Sen and N. K. Saha, On $\Gamma$-semigroup, I. Bull. Calcutta Math. Soc., 78, 181--186 (1986).
• A. Seth, $\Gamma$-group congruence on regular $\Gamma$-semigroups, Inter. J. Math. Math. Sci., 15(1), 103--106 (1992).
• M. Siripitukdet and A. Iampan, On the least (ordered) semilattice congruence in ordered $\Gamma$-semigroups, Thai J. Math., 4(2), 403--415 (2006).
• A. S. Troelstra and D. van Dalen, Constructivism in Mathematics: An Introduction, Amsterdam: North-Holland, (1988).