On feebly compact topologies on the semilattice $\exp_n\lambda$

O. Gutik; O. Sobol

doi:10.15330/ms.46.1.29-43

We study feebly compact topologies $\tau$ on the semilattice $\left(\exp_n\lambda,\cap\right)$ such that $\left(\exp_n\lambda,\tau\right)$ is a semitopological semilattice. All compact semilattice $T_1$-topologies on $\exp_n\lambda$ are described. Also we prove that for an arbitrary positive integer $n$ and an arbitrary infinite cardinal $\lambda$ for a $T_1$-topology $\tau$ on $\exp_n\lambda$ the following conditions are equivalent: $(i)$ $\left(\exp_n\lambda,\tau\right)$ is a compact topological semilattice; $(ii)$ $\left(\exp_n\lambda,\tau\right)$ is a countably compact topological semilattice; $(iii)$ $\left(\exp_n\lambda,\tau\right)$ is a feebly compact topological semilattice; $(iv)$ $\left(\exp_n\lambda,\tau\right)$ is a compact semitopological semilattice; $(v)$ $\left(\exp_n\lambda,\tau\right)$ is a countably compact semitopological semilattice. We construct a countably pracompact $H$-closed quasiregular non-semiregular topology $\tau_{\operatorname{\textsf{fc}}}^2$ such that $\left(\exp_2\lambda,\tau_{\operatorname{\textsf{fc}}}^2\right)$ is a semitopological semilattice with discontinuous semilattice operation and prove that for an arbitrary positive integer $n$ and an arbitrary infinite cardinal $\lambda$ every $T_1$-semiregular feebly compact semitopological semilattice $\exp_n\lambda$ is a compact topological semilattice.

1. A. V. Arkhangefskii, Topological Function Spaces, Kluwer Publ., Dordrecht, 1992.

2. J.H. Carruth, J.A. Hildebrant, 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.

3. A.H. Clifford, G.B. Preston, The Algebraic Theory of Semigroups, Vol. I., Amer. Math. Soc. Surveys 7, Providence, R.I., 1961; Vol. II., Amer. Math. Soc. Surveys 7, Providence, R.I., 1967.

4. R. Engelking, General Topology, 2nd ed., Heldermann, Berlin, 1989.

5. G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove, D.S. Scott, Continuous Lattices and Domains, Cambridge Univ. Press, Cambridge, 2003.

6. O. Gutik, On closures in semitopological inverse semigroups with continuous inversion, Algebra Discr. Math. 18 (2014), no.1, 59-85.

7. O. Gutik, J. Lawson, D. Repovs, Semigroup closures of finite rank symmetric inverse semigroups, Semigroup Forum, 78 (2009), no.2, 326-336.

8. O.V. Gutik, K.P. Pavlyk, On topological semigroups of matrix units, Semigroup Forum, 71 (2005), no.3, 389-400.

9. O.V. Gutik, K.P. Pavlyk, Topological semigroups of matrix units, Algebra Discrete Math., no.3 (2005), 1-17.

10. O. Gutik, K. Pavlyk, A. Reiter, Topological semigroups of matrix units and countably compact Brandt $\lambda^0$-extensions, Mat. Stud., 32 (2009), no.2, 115-131.

11. O. Gutik, O. Ravsky, Pseudocompactness, products and topological Brandt $\lambda^0$-extensions of semitopological monoids, Math. Methods and Phys.-Mech. Fields, 58 (2015), no.2, 20-37.

	On feebly compact topologies on the semilattice $\exp_n\lambda$
Author	O. Gutik, O. Sobol o_ gutik@franko.lviv.ua, ovgutik@yahoo.com; olesyasobol@mail.ru Ivan Franko National University of Lviv
Abstract	We study feebly compact topologies $\tau$ on the semilattice $\left(\exp_n\lambda,\cap\right)$ such that $\left(\exp_n\lambda,\tau\right)$ is a semitopological semilattice. All compact semilattice $T_1$-topologies on $\exp_n\lambda$ are described. Also we prove that for an arbitrary positive integer $n$ and an arbitrary infinite cardinal $\lambda$ for a $T_1$-topology $\tau$ on $\exp_n\lambda$ the following conditions are equivalent: $(i)$ $\left(\exp_n\lambda,\tau\right)$ is a compact topological semilattice; $(ii)$ $\left(\exp_n\lambda,\tau\right)$ is a countably compact topological semilattice; $(iii)$ $\left(\exp_n\lambda,\tau\right)$ is a feebly compact topological semilattice; $(iv)$ $\left(\exp_n\lambda,\tau\right)$ is a compact semitopological semilattice; $(v)$ $\left(\exp_n\lambda,\tau\right)$ is a countably compact semitopological semilattice. We construct a countably pracompact $H$-closed quasiregular non-semiregular topology $\tau_{\operatorname{\textsf{fc}}}^2$ such that $\left(\exp_2\lambda,\tau_{\operatorname{\textsf{fc}}}^2\right)$ is a semitopological semilattice with discontinuous semilattice operation and prove that for an arbitrary positive integer $n$ and an arbitrary infinite cardinal $\lambda$ every $T_1$-semiregular feebly compact semitopological semilattice $\exp_n\lambda$ is a compact topological semilattice.
Keywords	topological semilattice; semitopological semilattice; compact; countably compact; feebly compact; $H$-closed; semiregular space; regular space
DOI	doi:10.15330/ms.46.1.29-43
Reference	1. A. V. Arkhangefskii, Topological Function Spaces, Kluwer Publ., Dordrecht, 1992. 2. J.H. Carruth, J.A. Hildebrant, 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. 3. A.H. Clifford, G.B. Preston, The Algebraic Theory of Semigroups, Vol. I., Amer. Math. Soc. Surveys 7, Providence, R.I., 1961; Vol. II., Amer. Math. Soc. Surveys 7, Providence, R.I., 1967. 4. R. Engelking, General Topology, 2nd ed., Heldermann, Berlin, 1989. 5. G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove, D.S. Scott, Continuous Lattices and Domains, Cambridge Univ. Press, Cambridge, 2003. 6. O. Gutik, On closures in semitopological inverse semigroups with continuous inversion, Algebra Discr. Math. 18 (2014), no.1, 59-85. 7. O. Gutik, J. Lawson, D. Repovs, Semigroup closures of finite rank symmetric inverse semigroups, Semigroup Forum, 78 (2009), no.2, 326-336. 8. O.V. Gutik, K.P. Pavlyk, On topological semigroups of matrix units, Semigroup Forum, 71 (2005), no.3, 389-400. 9. O.V. Gutik, K.P. Pavlyk, Topological semigroups of matrix units, Algebra Discrete Math., no.3 (2005), 1-17. 10. O. Gutik, K. Pavlyk, A. Reiter, Topological semigroups of matrix units and countably compact Brandt $\lambda^0$-extensions, Mat. Stud., 32 (2009), no.2, 115-131. 11. O. Gutik, O. Ravsky, Pseudocompactness, products and topological Brandt $\lambda^0$-extensions of semitopological monoids, Math. Methods and Phys.-Mech. Fields, 58 (2015), no.2, 20-37.
Pages	29-43
Volume	46
Issue	1
Year	2016
Journal	Matematychni Studii
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