On monoids of monotone injective partial selfmaps of $L_n\times_{\operatorname{lex}}\mathbb{Z}$ with co-finite domains and images

Oleg Gutik ; Inna Pozdnyakova

Algebra and discrete mathematics, Tome 17 (2014) no. 2, pp. 256-279

Résumé

We study the semigroup $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ of monotone injective partial selfmaps of the set of $L_n\times_{\operatorname{lex}}\mathbb{Z}$ having co-finite domain and image, where $L_n\times_{\operatorname{lex}}\mathbb{Z}$ is the lexicographic product of $n$-elements chain and the set of integers with the usual order. We show that $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ is bisimple and establish its projective congruences. We prove that $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ is finitely generated, and for $n=1$ every automorphism of $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ is inner and show that in the case $n\geqslant 2$ the semigroup $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ has non-inner automorphisms. Also we show that every Baire topology $\tau$ on $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ such that $(\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}}),\tau)$ is a Hausdorff semitopological semigroup is discrete, construct a non-discrete Hausdorff semigroup inverse topology on $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$, and prove that the discrete semigroup $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ cannot be embedded into some classes of compact-like topological semigroups and that its remainder under the closure in a topological semigroup $S$ is an ideal in $S$.

Détail

Keywords: topological semigroup, semitopological semigroup, semigroup of bijective partial transformations, symmetric inverse semigroup, congruence, ideal, semigroup topologization, embedding.
Mots-clés : automorphism, homomorphism, Baire space

Bibliographie

[1] L. W. Anderson, R. P. Hunter and R. J. Koch, “Some results on stability in semigroups”, Trans. Amer. Math. Soc., 117 (1965), 521–529 | DOI | MR | Zbl

[2] T. Banakh, S. Dimitrova and O. Gutik, “The Rees-Suschkiewitsch Theorem for simple topological semigroups”, Mat. Stud., 31:2 (2009), 211–218 | MR | Zbl

[3] T. Banakh, S. Dimitrova and O. Gutik, “Embedding the bicyclic semigroup into countably compact topological semigroups”, Topology Appl., 157:18 (2010), 2803–2814 | DOI | MR | Zbl

[4] M. O. Bertman and T. T. West, “Conditionally compact bicyclic semitopological semigroups”, Proc. Roy. Irish Acad., A76:21–23 (1976), 219–226 | MR | Zbl

[5] J. H. Carruth, J. A. Hildebrant and R. J. Koch, The Theory of Topological Semigroups, v. I, Marcel Dekker, Inc., New York and Basel, 1983 ; v. II, Marcel Dekker, Inc., New York and Basel, 1986 | MR | Zbl | Zbl

[6] A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups, v. I, Amer. Math. Soc. Surveys, 7, Providence, R.I., 1961 ; v. II, Amer. Math. Soc. Surveys, 7, Providence, R.I., 1967 | MR | Zbl | MR

[7] W. W. Comfort, “Topological groups”, Handbook of set-theoretic topology, eds. Kunen K., Vaughan J., Elsevier, 1984, 1143–1263 | DOI | MR

[8] V. Doroshenko, “Generators and relations for the semigroups of increasing functions on $\mathbb{N}$ and $\mathbb{Z}$”, Algebra Discr. Math., 2005, no. 4, 1–15 | MR | Zbl

[9] Ukr. Math. J., 61:6 (2009), 859–872 | DOI | MR | Zbl

[10] C. Eberhart and J. Selden, “On the closure of the bicyclic semigroup”, Trans. Amer. Math. Soc., 144 (1969), 115–126 | DOI | MR | Zbl

[11] R. Engelking, General Topology, 2nd ed., Heldermann, Berlin, 1989 | MR | Zbl

[12] O. V. Gutik and K. P. Pavlyk, “On topological semigroups of matrix units”, Semigroup Forum, 71:3 (2005), 389–400 | DOI | MR | Zbl

[13] O. Gutik, K. Pavlyk and A. Reiter, “Topological semigroups of matrix units and countably compact Brandt $\lambda^0$-extensions”, Mat. Stud., 32:2 (2009), 115–131 | MR | Zbl

[14] O. V. Gutik and A. R. Reiter, “Symmetric inverse topological semigroups of finite rank $\leqslant n$”, Mat. Metody Phis.-Mech. Polya, 53:3 (2009), 7–14 | MR

[15] O. Gutik and A. Reiter, “On semitopological symmetric inverse semigroups of a bounded finite rank”, Visnyk Lviv Univ. Ser. Mech.-Math., 72 (2010), 94–106 (in Ukrainian) | Zbl

[16] O. Gutik and D. Repovš, “On countably compact $0$-simple topological inverse semigroups”, Semigroup Forum, 75:2 (2007), 464–469 | DOI | MR | Zbl

[17] O. Gutik and D. Repovš, “Topological monoids of monotone, injective partial selfmaps of $\mathbb{N}$ having cofinite domain and image”, Stud. Sci. Math. Hungar., 48:3 (2011), 342–353 | MR | Zbl

[18] O. Gutik and D. Repovš, “On monoids of injective partial selfmaps of integers with cofinite domains and images”, Georgian Math. J., 19:3 (2012), 511–532 | DOI | MR | Zbl

[19] E. Hewitt and K. A. Roos, Abstract Harmonic Analysis, v. 1, Springer, Berlin, 1963 | Zbl

[20] J. A. Hildebrant and R. J. Koch, “Swelling actions of $\Gamma$-compact semigroups”, Semigroup Forum, 33 (1988), 65–85 | DOI | MR

[21] J. M. Howie, Fundamentals of Semigroup Theory, London Math. Monographs, New Ser., 12, Clarendon Press, Oxford, 1995 | MR | Zbl

[22] M. Lawson, Inverse Semigroups. The Theory of Partial Symmetries, World Scientific, Singapore, 1998 | MR

[23] Transl. Amer. Math. Soc. (1), 8 (1962), 195–272 | MR

[24] A. Yu. Ol'shanskiy, “Remark on counatable non-topologized groups”, Vestnik Moscow Univ. Ser. Mech. Math., 39 (1980), 1034 (in Russian)

[25] M. Petrich, Inverse Semigroups, John Wiley $\$ Sons, New York, 1984 | MR | Zbl

[26] L. S. Pontryagin, Topological Groups, Gordon $\$ Breach, New York, 1966 | MR

[27] A. D. Taimanov, “The topologization of commutative semigroups”, Mat. Zametki, 17:5 (1975), 745–748 (in Russian) | MR

[28] V. V. Vagner, “Generalized groups”, Dokl. Akad. Nauk SSSR, 84 (1952), 1119–1122 (in Russian) | MR | Zbl