On Linearly Ordered H-Closed Topological Semilattices
by
Oleg Gutik
Ivan Franko National University of Lviv
Coauthors: Dusan Repovs

In our report all topological spaces will be assumed to be Hausdorff. We shall follow the terminology of [1], [2].

A topological semigroup S is called H-closed, if S is a closed subsemigroup of any topological semigroup T which contains S both as a subsemigroup and as a topological space [3].

A linearly ordered topological semilattice E is called complete if every non-empty subset of S has inf and sup.

We give a criterium when a linearly ordered topological semilattice is H-closed.

Theorem 1 A linearly ordered topological semilattice E is H-closed if and only if the following conditions hold:

(i) E is complete;
(ii) x=supA for A= ↓ A\{ x} implies x ∈ cl_EA, whenever A ≠ ∅; and
(iii) x=infB for B=↑B\{ x} implies x ∈ cl_EB, whenever B ≠ ∅.

A topological semigroup S is called absolutely H-closed, if any continuous homomorphic image of S into a topological semigroup T is H-closed [4].

Every linearly ordered H-closed topological semilattice is absolutely H-closed.

Theorem 2 Every linearly ordered topological semilattice is a dense subsemilattice of an H-closed linearly ordered topological semilattice.

References:

[1] G. Birkhoff, Lattice Theory, 3rd ed., Amer. Math. Soc. Coll. Publ. 25, Providence, R.I., 1967.

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

[3] J. W. Stepp, A note on maximal locally compact semigroups. Proc. Amer. Math. Soc. 20 (1969), 251-253.

[4] J. W. Stepp, Algebraic maximal semilattices. Pacific J. Math. 58 (1975), 243-248.

Date received: May 8, 2008