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 ).

In the paper we present various characterizations of chain-compact and chain-finite topological semilattices. A topological semilattice X is called chain-compact (resp. chain-finite) if each closed chain in X is compact (finite). In particular, we prove that a (Hausdorff) T1-topological semilattice X is chain-finite (chain-compact) if and only if for any closed subsemilattice Z ⊂ X and any continuous homomorphism h :1991 Mathematics Subject Classification. 22A26; 54D30; 54D35; 54H12.

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We find (completeness type) conditions on topological semilattices X, Y guaranteeing that each continuous homomorphism h : X → Y has closed image h(X) in Y .1991 Mathematics Subject Classification. 22A26; 54D30; 54D35; 54H12.

An H-closed quasitopological group is a Hausdorff quasitopological group which is contained in each Hausdorff quasitopological group as a closed subspace. We obtained a sufficient condition for a quasitopological group to be H-closed, which allowed us to solve a problem by Arhangel'skii and Choban and to show that a topological group G is H-closed in the class of quasitopological groups if and only if G is Raǐkov-complete. Also we present examples of non-compact quasitopological groups whose topological spaces are H-closed.

In this paper we investigate locally compact semitopological graph inverse semigroups. Our main result is the following: if a directed graph E is strongly connected and has finitely many vertices, then any Hausdorff shift-continuous locally compact topology on the graph inverse semigroup G(E) is either compact or discrete. This result generalizes results of Gutik and Bardyla who proved the above dichotomy for Hausdorff locally compact shift-continuous topologies on polycyclic monoids P 1 and P λ , respectively.

We study the interplay between three weak topologies on a topological semilattice X: the weak • topology W • X (generated by the base consiting of open subsemilattices of X), the weak • topology W • X (generated by the subbase consisting of complements to closed subsemilattices), and the I-weak topology WX (which is the weakest topology in which all continuous homomorphisms h : X → [0, 1] remain continuous). Also we study the interplay between the weak topologies W • X , W • X , WX of a topological semilattice X and the Scott and Lawson topologies SX and LX, which are determined by the order structure of the semilattice. We prove that the weak • topology W • X on a Hausdorff semitopological semilattice X is compact if and only if X is chain-compact in the sense that each closed chain in X is compact. This result implies that the Lawson topology LX on a semilattice X is compact if and only if X is a continuous semilattice if and only if X complete in the sense that each non-empty chain C in X has inf(C) and sup(C) in X.For a chain-compact Hausdorff topological semilattice X with topology TX we prove the inclusions WX ⊂ LX ⊂ W • X ⊂ TX. For a compact topological semilattice X we prove that TX = W • X if and only if TX = W • X if and only if TX = LX . 1991 Mathematics Subject Classification. 22A15, 22A26.

A topologized semilattice X is called complete if each non-empty chain C ⊂ X has inf C ∈C and sup C ∈C. We prove that for any continuous homomorphism h : X → Y from a complete topologized semilattice X to a sequential Hausdorff semitopological semilattice Y the image h(X) is closed in Y .1991 Mathematics Subject Classification. 06B30, 06B35, 54D55.