On linearly ordered $H$-closed topological semilattices

ON LINEARL Y ORDERED H -CLOSED TOPOLOGI CAL SEMILA TTICES OLEG GUTIK AND DU ˇ SAN REPO V ˇ S Abstra ct. W e giv e a criterium when a li nearly o rdered topological semila ttice is H -closed. W e also prov e that any linearly ordered H -closed topological semilattice is absolutely H -closed and we sho w that every linearly ordered semilattice is a dense subsemilattice of an H -closed top ological semilattice. In this pap er all top ologica l spaces will b e assumed to b e Hausdorff. W e shall follo w the terminology of [2 ]–[5]. If A is a subs et of a top ological sp ace X , then we shall denote the closure of the s et A in X b y cl X ( A ). W e shall denote the first infinite cardinal by ω . A se milattic e is a semigroup with a commuta tiv e idemp oten t semigroup op eration. If S is a top ological space equipp ed with a cont in uous semigroup op eration, then S is called a top olo gic al semigr oup . A top olo gic al semilattic e is a top ological semigroup wh ich is algebraically a semilattice. If E is a semilattice , then the semilattice op eration on E d etermines the partial ord er 6 on E : e 6 f if an d only if ef = f e = e. This order is called natur al . An element e of a semilattice E is called minimal ( maximal ) if f 6 e ( e 6 f ) implies f = e f or f ∈ E . F or elemen ts e and f of a semilattice E w e write e < f if e 6 f and e 6 = f . A semilattice E is said to b e line arly or der e d or a chain if the natural order on E is linear. Let S b e a semilattice and e ∈ S . W e denote ↓ e = { f ∈ S | f 6 e } and ↑ e = { f ∈ S | e 6 f } . If S is a top ologica l semilattice then Prop ositions VI.1.6(ii ) and VI.1.14 of [5] imply that ↑ e and ↓ e are closed subsets in S for any e ∈ S . Let E b e a linearly ordered top ologica l s emilattice. Since ↓ e and ↑ e are closed for eac h e ∈ E , it follo w s that the top ology of E r efines the order top ology . Th us the principal ob jects wh ick we shall consider can b e alternativ ely view ed as linearly ord er ed sets equip p ed with the semilattice op eration of taking minim um and equ ipp ed with a top ology refinin g the order top ology for whic h the semilattice op eration is contin uou s . Let S b e some class of top ological semigroups. A semigroup S ∈ S is called H -close d in S , if S is a closed sub semigroup of an y top ologica l semigroup T ∈ S whic h conta ins S b oth as a s u bsemigroup and as a top ologica l space. If S co incides with th e class of all top ologica l semigroups, then the semigroup S is called H -close d . The H -closed top ological semigroups we re in trod u ced by Stepp [8 ], they w ere called maximal semigr oups . A top ologica l semigroup S ∈ S is called absolutely H -close d in the class S , if an y con tin u ous homomorphic image of S in to T ∈ S is H -closed in S . If S coincides with the class of all top ological semigroups, then the semigroup S is called absolutely H -close d . An algebraic semigroup S is called algebr aic al ly h -close d in S , if S equip p ed with discrete top ology d is abs olutely H -closed in S and ( S, d ) ∈ S . If S coincides w ith the class of all top ological semigroups, then the semigroup S is called algebr aic al ly h -close d . Absolutely H -closed top ological semigroups and algebraical ly h -closed semigroups w ere in tro duced by Stepp in [9], they w ere called absolutely maximal and algebr aic maximal , respective ly . G utik and Pa vlyk [6 ] observ ed that a top ological semilattice is [absolutely] H -closed if and only if it is [absolutely] H -closed in the class of top ological semilattices. Stepp [9] pro v ed that a semilattice E is algebraically h -closed if and only if any maximal chain in E is finite and he ask ed the follo wing question: Is every H -close d top olo gi c al semilattic e absolutely H -close d? In the presen t pap er w e giv e a criterium w h en a linearly ordered top ological semilattice is H -closed. W e also pr o v e that ev ery linearly ordered H -closed top ological semilatti ce is absolutely H -closed, w e sho w that ev ery linearly ordered semilattice is a dense sub semilattice of an H -closed top ological semilattice, Date : Octob er 30, 2018. 2000 Mathematics Subje ct Classific ation. 06A12, 06F30, 22A15, 22A26, 54H12. Key wor ds and phr ases. T op ological semi lattice, linearly ordered top ological semi lattice, H -closed top ological semilattice, absolutely H -closed topological semilattice. 1 2 OLEG GUTIK A ND DU ˇ SAN REPOV ˇ S and w e giv e an example of a linearly ordered H -closed lo cally compact top ological semilattice whic h do es not em b ed into a compact top ologica l semilattice. Let C b e a maximal chain of a top ological semilattice E . T hen C = \ e ∈ C ( ↓ e ∪ ↑ e ), and hence C is a closed subsemilattice of E . T herefore w e obtain the follo wing: Lemma 1. Let L b e a linearly ordered subsemilattice of a top ological semilattice E . Then cl E ( L ) is a linearly ordered subsemilattice of E . A linearly ordered top ological semilattice E is calle d c ompl ete if ev ery non-empt y subset of S has inf and sup. Theorem 2. A line arly or der e d top olo gic al semilattic e E is H -c lose d i f and only if the fol lowing c onditions hold: ( i ) E is c omplete; ( ii ) x = su p A for A = ↓ A \ { x } implies x ∈ cl E A , whenever A 6 = ∅ ; and ( iii ) x = inf B for B = ↑ B \ { x } implies x ∈ cl E B , whenever B 6 = ∅ Pr o of. ( ⇐ ) Supp ose to the con trary that there exists a linearly ordered non- H -closed top ological semilat- tice E which satisfies the conditions ( i ), ( ii ), and ( iii ). Then by Lemma 1 there exists a linearly ordered top ological semilattice T suc h that E is a d ense pr op er subsemilattice of T . Let x ∈ T \ E . Condition ( i ) implies that x 6 = sup E and x 6 = inf E , otherwise E is not complete. Therefore w e ha v e inf E < x < sup E . Let A ( x ) = ( T \ ↑ x ) ∩ E and B ( x ) = ( T \ ↓ x ) ∩ E . Since E is complete, we ha ve sup A ( x ) ∈ E and inf B ( x ) ∈ E . Let s = su p A ( x ) and i = inf B ( x ). W e obser ve that s < x < i , otherwise, if x < s , then A = ↓ x ∩ E = ( ↓ s \ { s } ) E is a closed subset in E , w hic h con tradicts the condition ( ii ), an d if i < x , then B = ↑ x ∩ E = ( ↑ i \ { i } ) E is a closed subset of E , which con tradicts the condition ( iii ). Then ↑ i and ↓ s are closed subs ets of T and T = ↓ s ∪ ↑ i ∪ { x } . Th erefore x is an isolated p oint of T . Th is con tr ad icts the assumption that x ∈ T \ E and E is a dense s u bspace of T . T his con tradiction imp lies that E is an H -closed top ological semilattice. ( ⇒ ) A t first we shall sho w that sup A ∈ E for every infin ite subs et S of E . Supp ose to the con tr ary , that th ere exists an infi nite subset A in E suc h that A h as no sup in E . Since S is a linearly ord ered semilattice , w e hav e that the set ↓ A also has no sup in E . W e consider t w o cases: ( a ) E \ ↓ A 6 = ∅ ; and ( b ) E \ ↓ A = ∅ . In case ( a ) the set B = E \ ↓ A has no inf , since E is a linearly ordered semilattice, otherwise inf B = sup A ∈ E . Let x / ∈ E . W e put E ∗ = E ∪ { x } . W e extend the semilattic e op eration from E on to E ∗ as follo ws : x · y = y · x =    x, if y ∈ E \ ↓ A ; x, if y = x ; y , if y ∈ ↓ A. Ob viously , the s emilattice op eration on E ∗ determines a linear order on E ∗ . W e defin e a top ology τ ∗ on E ∗ as f ollo ws. Let τ b e th e top ology on E . A t an y p oint a ∈ E = E ∗ \ { x } bases of topologies τ ∗ and τ coincide. W e pu t B ∗ ( x ) = { V a b ( x ) = E \ ( ↓ b ∪ ↑ a ) | a ∈ E \ ↓ A, b ∈ ↓ A } . Since the set E \ ↓ A has no in f and the set ↓ A has n o sup, the conditions (BP1)—(BP3) of [4] hold for the family B ∗ ( x ) and B ∗ ( x ) is a b ase of a Hausdorff top ology τ ∗ at the p oin t x ∈ E ∗ . Let c ∈ E \ ↓ A and d ∈ ↓ A . Th en there exist a ∈ E \ ↓ A and b ∈ ↓ A such that d < b < x < a < c . Then for any op en neighbou r ho o d s V ( c ) and V ( d ) of th e p oints c and d , resp ectiv ely , such that V ( c ) ⊆ E \ ↓ a = E ∗ ↓ a and V ( d ) ⊆ E \ ↑ b = E ∗ \ ↑ b we ha v e V a b ( x ) · V ( d ) ⊆ V ( d ) and V a b ( x ) · V ( c ) ⊆ V a b ( x ) . W e also hav e V a b ( x ) · V a b ( x ) ⊆ V a b ( x ) f or all a ∈ E \ ↓ A and b ∈ ↓ A . Therefore ( E ∗ , τ ∗ ) is a Hausdorff top ological semilattice which con tains E as a dense non-closed sub semilattice. This contradict s the assumption that E is an H -closed top ological semilattice . ON LINEARL Y ORDERED H -CLOSED TOPO LO GICAL SEMILA TTICES 3 Consider case ( b ). Let y / ∈ E . W e pu t E ⋆ = E ∪ { y } and extend the semilattice op eration from E ont o E ⋆ as follo ws : y · s = s · y =  y , if s = y ; s, if s 6 = y . Ob viously , the s emilattice op eration on E ⋆ determines a linear order on E ⋆ . W e defin e a top ology τ ⋆ on E ⋆ as f ollo ws. Let τ b e th e top ology on E . A t an y p oint a ∈ E = E ⋆ \ { x } bases of topologies τ ⋆ and τ coincide. W e pu t B ⋆ ( x ) = { V b ( x ) = E \ ↓ b | b ∈ ↓ A = E } . Since the set E = ↓ E has no s up, the conditions (BP1)—(BP3) of [4] h old for the family B ⋆ ( x ) and B ⋆ ( x ) is a b ase of a Hausdorff top ology τ ∗ at the p oint x ∈ E ∗ . Let c ∈ E . Then there exists b ∈ E such that c < b < y and for any op en neighbour ho o d V b ( y ) of y and any op en n eigh b ourho o d V ( c ) such that V ( c ) ⊆ E \ ↑ b we h a v e V b ( y ) · V ( c ) ⊆ V ( c ). W e also ha v e V b ( y ) · V b ( y ) ⊆ V b ( y ) for all b ∈ ↓ A = E . Th erefore ( E ⋆ , τ ⋆ ) is a Hausdorff top ological semilattice whic h con tains E as a dense non-closed subsemilattice. This con tradicts the assumption that E is an H -closed top ological semilattice. Th e obtained cont radictions imply that ev ery subset of the semilattice E h as sup. The pro of of the fact that ev ery subset of E has an inf is similar. Next we sho w that for eve ry H -closed linearly ordered top ological semilattice E cond ition ( ii ) h olds. Supp ose that there exists x ∈ E such that x = sup ( ↓ x \ { x } ) and x / ∈ cl E ( ↓ x \ { x } ). Since the top ological semilattice E is linearly ordered , L ◦ ( x ) = ↓ x \ { x } is a clop en subset of E . Let g / ∈ E . W e extend the semilattice op eration from E ont o E ◦ = E ∪ { g } as follo ws: g · s = s · g =  g , if s ∈ ↑ x ; s, i f s ∈ L ◦ ( x ) . Ob viously , the s emilattice op eration on E ◦ determines a linear order on E ◦ . W e defin e a top ology τ ◦ on E ◦ as f ollo ws. Let τ b e the top ology on E . A t an y p oint a ∈ E = E ◦ \ { g } bases of topologies τ ◦ and τ coincide. W e pu t B ◦ ( g ) = { U s ( g ) = { g } ∪ L ◦ ( x ) \ ↓ s | s ∈ L ◦ ( x ) } . Since sup L ◦ ( x ) = x , th e set U s ( g ) is n on-singleton for any s ∈ L ◦ ( x ). T h erefore the conditions (BP1)— (BP3) of [4] hold for the family B ◦ ( g ) and B ◦ ( g ) is a base of the top ology τ ◦ at the p oin t g ∈ E ◦ . Also since the set ↑ x is closed in ( S ◦ , τ ◦ ) and sup L ◦ ( x ) = x , w e h av e that the top ology τ ◦ is Hausdorff. The pro of of the cont in uit y of th e semilattice op eration in ( S ◦ , τ ◦ ) is similar as f or ( S ∗ , τ ∗ ) and ( S ⋆ , τ ⋆ ). Th us condition ( ii ) holds. The pro of of the assertion th at if E is a linearly ordered H -closed top ologica l semilatti ce, then the condition ( iii ) holds, is similar. Th erefore th e pro of of the theorem is complete.  Since the conditions ( i )—( iii ) of Theorem 2 are pr eserv ed b y con tinuous homomorph isms, we h a v e the follo win g: Theorem 3. Every line arly or der e d H -close d top olo gi c al semilattic e is absolutely H -close d. Theorem 2 also imp lies the follo wing: Corollary 4. Ev ery line arly or der e d H -close d top olo gic al semilattic e c ontains maximal and minimal idemp otents. Theorems 2 and 3 imply the follo w ing: Corollary 5. L et E b e a line arly or der e d H -close d top olo gic al semilattic e and e ∈ E . Then ↑ e and ↓ e ar e (absolutely) H -close d top olo gic al semilattic es. Theorem 6. Every line arly or der e d top olo g ic al semilattic e is a dense subsemilattic e of an H -close d lin- e arly or der e d top olo gic al semilattic e. Pr o of. Let E b e a linearly ord ered top ological semilattic e and let E a b e an algebraic cop y of E . Then E a with op eration inf and s up is a lattice. It is well known that ev ery lattic e embeds in to a complete lattice (cf. [2, Theorem V.2.1]). I n our construction we shall use the id ea of pro ofs of Theorem V.2.1 and Lemma V.2.1 in [2]. W e d enote the lattice of all ideals of E a b y e E a . Then p oin t wise op erations inf and 4 OLEG GUTIK A ND DU ˇ SAN REPOV ˇ S sup on e E a coincide with T and S on e E a , r esp ectiv ely . S ince E a is a linearly ordered semilattice, we can iden tify E a with the su bsemilattice of all principal ideals of E a in e E a . F or an ideal I ∈ e E a and a p rincipal ideal I e ∈ e E a generated by an idemp oten t e we p ut I e ρI if and only if ev ery op en neigh b ourh o o d of e in tersects I . Since E a and e E a are linearly order ed semilattices, f or principal ideals I e and I f generated by idemp oten ts e and f from E a , resp ectiv ely , we ha v e I e ρI f if an d only if I e = I f , i.e. e = f . W e put α = ∆ ∩ ρ ∪ ρ − 1 . Ob viously the relation α is an equ iv alence on e E a . Since e E a is a linearly ordered semilattice , α is a congruence on e E a and h ence e E = e E a /α is also a linearly ordered semilattice . W e observe that E a is a su bsemilattice of e E . W e d efi ne a top ology e τ on e E as follo ws. Let τ b e the top ology on E . A t any p oin t a ∈ E a ⊂ e E bases of top ologies e τ and τ coincide. F or x ∈ e E \ E a w e put e B ( x ) = { V b ( x ) = ↓ x \ ↓ b | b ∈ ↓ x ∩ E a } . Then conditions (BP1)—(BP3 ) of [4] hold for the family e B ( x ) and e B ( x ) is a base of a top ology e τ at the p oin t x and since τ is Hausdorff, s o is e τ . W e also observe that the definition of e τ implies that ↓ e and ↑ e are closed sub set of the top ologica l space ( e E , e τ ). Obviously the semilattice op eration on ( e E , e τ ) is con tin u ous, ( e E , e τ ) satisfies the conditions ( i ) and ( ii ) of Th eorem 2 and E is a dense subsemilattice of ( e E , e τ ). W e denote the lattice of all fi lters of e E by F ( E ). Then p oin t wise op erations in f and sup on F ( E ) coincide with S and T on F ( E ), resp ectiv ely . S ince e E is a lin early ordered semilattice , w e can identify e E with the sub semilattice in F ( E ) of all pr incipal filters of e E . By the dual theorem to Theorem V.2.1 of [2 ] the lattice F ( E ) is complete, and since e E is linearly ordered, so is F ( E ). F or a fi lter F ∈ e E a and a p rincipal filter F e ∈ F ( E ) generated by an id emp otent e ∈ e E we put F e e ρF if and only if ev ery op en n eigh b ourho o d of e intersect s F . Since e E and F ( E ) are linearly ordered semilattice s, for pr incipal filters F e and F f generated by idemp oten ts e and f from e E , resp ectiv ely , w e h av e F e e ρF f if and only if F e = F f , i.e. e = f . W e pu t e α = ∆ ∩ e ρ ∪ e ρ − 1 . Obviously e α is an equiv alence on F ( E ). Since F ( E ) is a linearly ordered semilattice, e α is a congruence on F ( E ) and hence f F ( E ) = F ( E ) / e α is a linearly ordered semilattice . W e define a top ology τ F on f F ( E ) as follo ws. At any p oin t a ∈ e E ⊆ f F ( E ) bases of top ologies τ F and e τ coincide. F or x ∈ f F ( E ) \ e E w e pu t e B F ( x ) = { W b ( x ) = ↑ x \ ↑ b | b ∈ ↑ x ∩ e E } . Then conditions (BP1)—(BP3 ) of [4] hold for the f amily e B F ( x ) and e B F ( x ) is a base of a topology τ F at the p oin t x and since e τ is Hausdorff, so is τ F . Also we observe that the d efi nition of τ F implies that ↓ e and ↑ e are closed s u bsets of the top ologica l s pace ( f F ( E ) , τ F ), and hence the semilattic e op eration on ( f F ( E ) , τ F ) is cont in uous, ( e E , e τ ) satisfies the conditions ( i ) and ( iii ) of Theorem 2 and E is a dense subsemilattice of ( f F ( E ) , τ F ). F u rther we shall sho w that the condition ( ii ) of Theorem 2 holds for the top ological semilattic e ( f F ( E ) , τ F ). S upp ose to the contrary that there exists a lo w er sub set A of F ( E ) suc h that sup A = x / ∈ A and x / ∈ cl F ( E ) A . Then A = F ( E ) \ ↑ x and A are clop en subsets of the top ological space ( F ( E ) , τ F ). Since x = sup A / ∈ A , there exists an increasing family of ideals I = { I α | α ∈ A} of th e semilattice E a suc h that I α ⊂ I β whenev er α < β , α, β ∈ A , sup S I = x , and S I ⊂ A . T he existence of the family I follo ws from the fact that the s emilattice e E is a d ense sub semilattice of ( f F ( E ) , τ F ). Ho wev er, S I is an ideal in E a and h en ce su p S I ∈ A , a con tradiction. The obtained con tradictio n implies that statemen t ( ii ) holds for the top ologica l semilattice ( f F ( E ) , τ F ).  Example 7. Let N b e the set of p ositiv e in tegers. Let { x n } b e an increasing s equ ence in N . Put N ∗ = { 0 } ∪ { 1 n | n ∈ N } . W e define the semilattice op eration on N ∗ as follo ws ab = min { a, b } , for a, b ∈ N ∗ . Ob viously , 0 is the zero elemen t of N ∗ . W e put U n (0) = { 0 } ∪ { 1 x k | k > n } , n ∈ N . A top ology τ on N ∗ is defin ed as follo ws: all nonzero element s of N ∗ are isolated p oints in N ∗ and B (0) = { U n (0) | n ∈ N } is the base of the top ology τ at the p oin t 0 ∈ N ∗ . It is easy to see that ( N ∗ , τ ) is a countable linearly ordered σ -compact 0-dimensional scattered locally compact metrizable top ological ON LINEARL Y ORDERED H -CLOSED TOPO LO GICAL SEMILA TTICES 5 semilattice and if x k +1 > x k + 1 for ev ery k ∈ N , then ( N ∗ , τ ) is a non-compact semilattice. W e also observ e that the family Hom( E , { 0 , 1 } ) of all h omomorphisms from a top ological semilattice E in to the discrete semilattice ( { 0 , 1 } , min ) separates p oint s for the top ological semilattice E . Theorem 2 implies the follo wing: Prop osition 8. ( N ∗ , τ ) is an H -close d top olo gic al semilattic e. Remark 9. Example 7 implies negativ e answe rs to the follo wing questions: (1) Is ev ery closed su bsemilattice of an H -closed top ological semilattice H -closed? (2) (I. Gu ran) Do es ev er y lo cally compact top ologica l semilattice em b ed into a compact semilattic e? (3) (cf. [1]) Does ev ery globally b ounded top ological inv erse Clifford semigroup emb ed into a compact semigroup? (4) Do es ev ery locally compact top ological semilattice ha v e a base with op en order con v ex subsets? Remark 10. Theorem 3 and E x amp le 7 im p ly that a closed subsemilattice of an absolutely H -closed top ological semilattice is not H -closed. Example 11 implies that th ere exist top ologically isomorphic linearly ord ered top ological semilattices E 1 and E 2 whic h is dense subsemilattice of linearly ord er ed top ologica l semilattice s S 1 and S 2 , resp ec- tiv ely , suc h that S 1 and S 2 are not algebraical ly isomorphic. Example 11. Let N b e the set of p ositiv e inte gers. Let S 1 = {− 1 n | n ∈ N } ∪ { 0 } ∪ { 1 n | n ∈ N } and S 2 = {− 1 − 1 n | n ∈ N } ∪ {− 1 } ∪ { 0 } ∪ { 1 n | n ∈ N } with usu al top ology and op eration min. Then E 1 = {− 1 n | n ∈ N } ∪ { 1 n | n ∈ N } and E 2 = {− 1 − 1 n | n ∈ N } ∪ { 1 n | n ∈ N } is discrete isomorphic semilattice , but th e semilattices S 1 and S 2 are not algebraical ly isomorphic. Theorem 12 giv es a metho d of constru cting new H -closed an d absolutely H -closed top ologica l semi- lattice s from old. Theorem 12. L et S = S α ∈ A S α b e a top olo gic al semilattic e such that: ( i ) S α is an (absolutely) H -close d top olo gic al semilattic e for any α ∈ A ; and ( ii ) ther e exists an (absolutely) H - close d top olo gic al semilattic e T such that T ⊆ S and S α S β ⊆ T for al l α 6 = β , α, β ∈ A . Then S is an (absolutely) H -close d top olo gic al semilattic e. Pr o of. W e shall consider only the case when S is an absolutely H -closed top ological semilattice. T h e pro of in the other case is similar. Let h : S → G b e a contin uou s homomorph ism from S in to a top ological semilattice G . Without loss of generalit y w e ma y assume that cl G ( h ( S )) = G . Supp ose that G \ h ( S ) 6 = ∅ . W e fix x ∈ G \ h ( S ). T h e absolute H -closedness of the top ological semilattice T implies that th er e exists an op en neigh b ourh o o d U ( x ) of the p oin t x in G s uc h that U ( x ) ∩ h ( T ) = ∅ . Since G is a top ological semilattice, there exists an op en neigh b ourho o d V ( x ) of x in G suc h that V ( x ) V ( x ) ⊆ U ( x ). Since the top ological semilattic e S α is absolutely H -closed, th e neigh b ourho o d V ( x ) in tersects in finitely many semilattices h ( S β ), β ∈ A . Th erefore V ( x ) V ( x ) ∩ h ( T ) 6 = ∅ . T h is is in disagreemen t with the c hoice of the n eigh b ourho o d U ( x ). T his con tradictio n implies the assertion of the th eorem.  A cknowledgements This researc h w as sup p orted by S RA gran ts P1-0292-0 101-0 4 and BI-UA/07- 08-001 . The authors thank the referee for imp ortan t r emarks and su ggestions. 6 OLEG GUTIK A ND DU ˇ SAN REPOV ˇ S Referen ces [1] Banakh, T., and O. Hry niv, On the structur e of c omp act top ol o gic al inverse semigr oups . Preprint. [2] Birkhoff, G., L attic e The ory , 3rd ed., Amer. Math. Soc. Coll. Pub l. 25, Providence, R.I., 1967. [3] Carruth, J. H., J. A. Hildebrant, and R. J. Ko ch, The The ory of T op olo gic al Semigr oups . V ol. I. Marcel Dek ker, Inc., New Y ork and Basel (1983). V ol. I I. Marcel Dekker, Inc., New Y ork an d Basel (1986). [4] Engelking, R ., Gener al T op olo gy , 2nd ed., Heldermann, Berlin, 1989. [5] Gierz, G., K. H. Hofmann, K. Keimel, J. D . Lawso n, M. W. Mis lo ve, and D. S. Scott, Continuous L attic es and Domains . Cam bridge Univ. Press, Cam bridge (2003). [6] Gutik, O. V., and K. P . Pa vly k , T op olo gic al Br andt λ -extensions of absolutely H -close d top olo gic al i nverse semigr oups . Visnyk Lv iv . U niv. Ser. Mekh.-Mat. 61 (2003), 98–105 . [7] Gutik, O . V ., and K. P . Pa v lyk, On top olo gic al semigr oups of m atrix units . Semigroup F orum 71 (2005), 389–400. [8] Step p, J. W., A note on maximal lo c al ly c omp act semigr oups . Proc. Amer. Math. So c. 20 (1969), 251–253. [9] Step p, J. W., A lgebr aic maximal semilattic es . Pacific J. Math. 58 (1975), 243–248. Dep ar tment of Ma thema tics, Iv an Frank o L viv Na tional Uni versity, Universytetska 1, L viv, 79000, Ukraine E-mail addr ess : o gutik@fran ko.lviv.u a Institute of Ma them a tics, Physics and M echanics, and F acul ty of Educa tion, Univ ersity of Ljubljana, P.O.B. 2964, Ljubljana, 1001, Slovenia E-mail addr ess : dusan. repovs@gue st.arnes. si