We study algebraic and topological properties of topological semigroups containing a copy of the bicyclic semigroup C(p, q). We prove that a topological semigroup S with pseudocompact square contains no dense copy of C(p, q). On the other hand, we construct a (consistent) example of a pseudocompact (countably compact) Tychonoff semigroup containing a copy of C(p, q).2000 Mathematics Subject Classification. 22A15, 54C25, 54D35, 54H15.Although all non-trivial applications concern infinite D, we do not restrict ourselves by infinite spaces and formulate our results for any (not necessarily infinite) discrete space D.Let D be a discrete topological space. If D is infinite, then let αD = D ∪ {∞} be the Aleksandrov compactification of D. If D is finite, then let αD = D ∪ {∞} be the topological sum of D and the singleton {∞} for some point ∞ / ∈ D. Given a map π : D → M to a T 1 -topological space M , consider the closed subspaceof the product αD×M . We shall identify the space D with the open discrete subspace {(x, π(x)) : x ∈ D} and M with the closed subspace {∞} × M of D ∪ π M . Letπ = π ∪ id M : D ∪ π M → M denote the projection to the second factor. Observe that the topology of the space D ∪ π M is the weakest T 1topology that induces the original topologies on the subspaces D and M of D ∪ π M and makes the map π continuous.The following (almost trivial) propositions describe some elementary properties of the space D ∪ π M .Proposition 1.1. If for some i ≤ 3 1 2 the space M satisfies the separation axiom T i , then so does the space D ∪ π M . Proposition 1.2. If M is (separable) metrizable and D is countable, then the space D∪ π M is (separable) metrizable too.We recall that a topological space X is countably compact if each countable open cover of X has a finite subcover. This is equivalent to saying that the space X contains no infinite closed discrete subspace.Proposition 1.4. If some power M κ of the space M is countably compact, then the power (D ∪ π M ) κ is countably compact too.Proof. Since D ∪ π M is a closed subspace of αD × M , the power (D ∪ π M ) κ is a closed subspace of (αD × M ) κ . So, it suffices to check that the latter space is countably compact. Since the product of a countably compact space and a compact space is countably compact [10, 3.10.14], the product M κ ×(αD) κ is countably compact and so is its topological copy (αD × M ) κ .

We prove that the semigroup of matrix units is stable. Compact, countably compact and pseudocompact topologies τ on the infinite semigroup of matrix units B λ such that (B λ , τ) is a semitopological (inverse) semigroup are described. We prove the following properties of an infinite topological semigroup of matrix units. On the infinite semigroup of matrix units there exists no semigroup pseudocompact topology. Any continuous homomorphism from the infinite topological semigroup of matrix units into a compact topological semigroup is annihilating. The semigroup of matrix units is algebraically h-closed in the class of topological inverse semigroups. Some H-closed minimal semigroup topologies on the infinite semigroup of matrix units are considered.

We describe the structure of 0-simple countably compact topological inverse semigroups and the structure of congruence-free countably compact topological inverse semigroups.

We give sufficient conditions when a topological inverse λ-polycyclic monoid P λ is absolutely Hclosed in the class of topological inverse semigroups. For every infinite cardinal λ we construct the coarsest semigroup inverse topology τ mi on P λ and give an example of a topological inverse monoid S which contains the polycyclic monoid P 2 as a dense discrete subsemigroup.Key words and phrases: inverse semigroup, bicyclic monoid, polycyclic monoid, free monoid, semigroup of matrix units, topological semigroup, topological inverse semigroup, minimal topology.Ivan Franko National University, 1 Universytetska str., 79000, Lviv, Ukraine E-mail: sbardyla@yahoo.com (Bardyla S.O.), o_gutik@franko.lviv.ua, ovgutik@yahoo.com (Gutik O.V.) In this paper all topological spaces will be assumed to be Hausdorff. We shall follow the terminology of [10,12,16,31]. If A is a subset of a topological space X, then we denote the closure of the set A in X by cl X (A). By N we denote the set of all positive integers and by ω the first infinite cardinal.A semigroup S is called an inverse semigroup if every a in S possesses a unique inverse, i.e. if there exists a unique element a −1 in S such thatA map that associates to any element of an inverse semigroup its inverse is called the inversion.A band is a semigroup of idempotents. If S is a semigroup, then we shall denote the subset of idempotents in S by E(S). If S is an inverse semigroup, then E(S) is closed under multiplication. The semigroup operation on S determines the following partial order on E(S): e f if and only if e f = f e = e. This order is called the natural partial order on E(S). A semilattice is a commutative semigroup of idempotents. A semilattice E is called linearly ordered or a chain if its natural order is a linear order. A maximal chain of a semilattice E is a chain which is properly contained in no other chain of E. The Axiom of Choice implies the existence of maximal chains in any partially ordered set. According to [35, Definition II.5.12] a chain L is called ω-chain if L is order isomorphic to {0, −1, −2, −3, . . .} with the usual order . Let E be a semilattice and e ∈ E. We denote ↓e = { f ∈ E | f e} andIf S is a semigroup, then we shall denote by R, L , D and H the Green relations on S (see [17] The R-class (resp., L -, H -, or D-class) of the semigroup S which contains an element a of S will be denoted by R a (resp., L a , H a , or D a ).