# On H-complete topological semilattices

Author S. Bardyla, O. Gutik
sbardyla@yahoo.com, o_gutik@franko.lviv.ua, ovgutik@yahoo.com
Ivan Franko National University of Lviv

Abstract In the paper we describe the structure of $\mathscr{A\!H}$-completions and $\mathscr{H}$-completions of the discrete semilattices $(\mathbb{N},\min)$ and $(\mathbb{N},\max)$. We give an example of an $\mathscr{H}$-complete topological semilattice which is not $\mathscr{A\!H}$-complete. Also for an arbitrary infinite cardinal $\lambda$ we construct an $\mathscr{H}$-complete topological semilattice of cardinality $\lambda$ which has $2^\lambda$ many open-and-closed continuous homomorphic images which are not $\mathscr{H}$-complete topological semilattices. The constructed examples give a negative answer to Question 17 in the paper J. W. Stepp, Algebraic maximal semilattices, Pacific J. Math., 58 (1975), no.1, 243--248.
Keywords topological semilattice; free filter; complete semigroup; chain
Reference 1. J.H. Carruth, J.A. Hildebrant, R.J. Koch, The theory of topological semigroups, Vol. I, Marcel Dekker, Inc., New York and Basel, 1983; Vol. II, Marcel Dekker, Inc., New York and Basel, 1986.

2. I. Chuchman, O. Gutik, On H-closed topological semigroups and semilattices, Algebra Discrete Math., (2007), ¹1, 13–23.

3. R. Engelking, General topology, 2nd ed., Heldermann, Berlin, 1989.

4. 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.

5. O. Gutik, D. Pagon, D. Repov.s, On chains in H-closed topological pospaces, Order, 27 (2010), ¹1, 69–81.

6. O. Gutik, K. Pavlyk, Topological Brandt lambda-extensions of absolutely H-closed topological inverse semigroups, Visnyk Lviv. Univ. Ser. Mekh.-Mat., 61 (2003), 98–105.

7. O. Gutik, D. Repovs, On linearly ordered H-closed topological semilattices, Semigroup Forum, 77 (2008), ¹3, 474–481.

8. D.A. Raikov, On a completion of topological groups, Izv. Akad. Nauk SSSR, 10 (1946), ¹6, 513–528. (in Russian)

9. J.W. Stepp, A note on maximal locally compact semigroups, Proc. Amer. Math. Soc., 20 (1969), 251–253.

10. J. W. Stepp, Algebraic maximal semilattices, Pacific J. Math., 58 (1975), ¹1, 243–248.

11. T. Yokoyama, On completeness of H-closed pospaces, arXiv:1004.3038v1.

Pages 118-123
Volume 38
Issue 2
Year 2012
Journal Matematychni Studii
Full text of paper PDF
Table of content of issue HTML