Semigroup Closures of Finite Rank Symmetric Inverse Semigroups

SEMIGR OUP CLOSURES OF FINITE R ANK SYMMETRIC INVE RSE SEMIGR OUPS OLEG GUTIK, JIMMIE LA WSON, AND DU ˇ SAN REPOV ˇ S Abstract. W e int ro duce the notion of semigroup with a tight ideal series and investigate their closures in semitopologica l semigroups, particular ly inv erse semig roups with contin uous inv ersion. As a coro llary w e show that the symmetric inv erse semigro up of finite tr ansformatio ns I n λ of the rank 6 n is algebraically closed in the class of (semi)topolo g ical in verse s emigroups with cont inuous inv ersion. W e also derive r e la ted r esults ab out the no nexistence of (partial) compactifications of classes of semigroups that w e consider. Intr oduction A partia l one-to-o ne transformation on a set X is a one-to-one function with domain and range subsets of X (including the empty transformation with empt y domain). There is a natural asso ciativ e op eration of comp o sition on these transformations, ab ( x ) = a ( b ( x )) wherev er defined, and the resulting semigroup is called the symm etric inverse semigr oup I X [9]. The symmetric inv erse semigroup w as in tro duced b y V. V. W agner [31] and it plays a ma jor role in the theory of semigroups. If the domain is finite and has car dina lity n , which is then a lso the cardinalit y of the ra nge, the transformatio n is said to b e of r ank n . F or each n ≥ 0, the mem b ers of I X of rank less than or equal to n form an ideal of I X , denoted I n X . (Recall that a nonempty subset I of a semigroup S is an ide al if S I ∪ I S ⊆ I .) If X and Y hav e the same cardinality , then these resp ectiv e semigroups are isomorphic, a nd th us we may restrict our atten tion to a canonical one from that class, whic h w e take to b e the one arising b y taking X to b e the cardinal λ . W e thus lab el the corresp onding semigroups as I λ and I n λ resp ectiv ely . The semigroups I n λ for λ infinite form a motiv ating example for the considerations and dev elopmen ts of t his pap er. Man y topolo gists hav e studied topolo gical prop erties of top o logical spaces of part ia l con- tin uous maps P C ( X , Y ) f r o m a top ological space X in to a top ological space Y with v arious top ologies suc h as the Vietoris top ology , generalized compact-op en top olog y , graph top olo g y , τ -top ology , and others (see [1, 7, 10, 12, 17, 18, 19, 20]). Since the set of all partial con tin uous self-transformations P C T ( X ) of the space X with the op eration comp osition is a semigroup, man y semigroup theorists hav e considered the semigroup of con tin uous transformations (see surv eys [21] and [14]), or the semigroup of part ia l homeomorphisms of an arbitr a ry top olo g ical space (see [2, 3 , 4, 5, 13, 22, 27, 32]). Be ˘ ıda [6], Orlov [23, 24], and Subbiah [30] hav e consid- ered semigroup and inv erse semigroup top ologies of semigroups of par t ia l homeomorphisms of some classes of top olo gical spaces. In this con text the results of our pap er yield some notable results ab out the top o logical b eha vior of the finite rank symmetric inv erse semigroups sitting inside larger function space semigroups, or larg er semigroups in general. F or example, under reasonably general conditions, the in v erse semigroup of partial finite bijections I n λ of rank 6 n is a closed subsemigroup of a top ological semigroup whic h contains I n λ as a subsemigroup. Date : Octob er 24, 20 18. 2000 Mathematics Subje ct Classific ation. Primary 22A15, 20M20, 54H15. Sec o ndary 20M12, 20M17, 20M18, 54C25, 5 4D40. Key wor ds and phr ases. T opolo gical semigroup, semitop olog ical semigroup, top o lo gical inv erse semigroup, symmetric in verse semigr oup of finite transformations, algebr aically closed semigro up, ω -unstable set, semig roup with a tight idea l s eries. 1 2 OLEG GUTIK, JIMMIE LA WSON, AN D D U ˇ SAN REPOV ˇ S The class of semigroups that w e consider in this pap er is a general class o f semigroups that is mo deled o n the semigroup I n λ and includes it as a sp ecial case. In Section 2 we take a closer lo ok at this class including some categorical prop erties and a more general example than I n λ . A question o f in terest ov er the ye ars has b een to identify classes of semigroups that can b e em b edded in compact semigroups and classes t ha t resist suc h em b eddings. In Section 3 w e consider this question in the con text of the class of semigroups w e a re considering. In this pap er all top ological spaces will b e assumed to b e Hausdorff. W e shall follo w the terminology of [8, 9, 11, 25, 2 6]. If A is a subset of a top olo g ical space X , then we denote the closure of t he set A in X b y cl X ( A ), o r simply A if X is ob vious fr o m contex t. By  x 1 x 2 · · · x n y 1 y 2 · · · y n  w e denote a partial o ne-to-one transfor ma t io n whic h maps x 1 on to y 1 , x 2 on to y 2 , . . . , and x n on to y n , and b y 0 the empt y tr a nsformation. 1. Semigroup closures Definition 1. A subset D of a semigroup S is said t o b e ω - unstable if D is infinite and for any a ∈ D a nd infinite subset B ⊆ D , w e ha v e aB ∪ B a 6⊆ D . The next lemma giv es a basic example of ω -unstable sets. Lemma 2. F or λ infi n ite, D := I n λ \ I n − 1 λ is an ω -unstable s ubse t of I n λ . Inde e d for a ∈ D and B ⊆ D of c ar dinality at le ast n ! + 1 , aB ∪ B a 6⊆ D . Pr o of. In order for aB to b e con tained in D , it m ust b e the case that t he rang e of eac h mem b er of B equals the domain of a and in order for B a to b e con tained in D , it m ust b e the case the ev ery mem b er o f B ha v e the same domain as the range a . Thus there are only n ! p ossibilities for mem b ers o f B .  Recall that a semigroup S is a sem i top olo gic al semigr oup if it is equipp ed with a Hausdorff top ology for whic h all left translation maps λ s and all righ t translation maps ρ s are contin u- ous [26]. In this case w e say equiv alen tly that the multiplication is sep ar ately c ontinuous . W e come no w to the crucial lemma for a ll tha t fo llo ws. Lemma 3. L et S b e a semitop o l o gic al semigr oup, let T b e a subsemigr o up, let I b e an ide al o f T , and assume D := T \ I is ω -unstable. If s ∈ S is a lim it p oin t o f T , then for e ach t ∈ T , either st ∈ I or ts ∈ I . Pr o of. W e ha v e s ∈ T = D ∪ I . If s ∈ I , then b y separate con tinuit y of m ultiplication for eac h t ∈ T , st ∈ I t ⊆ I t ⊆ I . Th us sT ⊆ I , and by con tinuit y of left translation b y s , sT ⊆ I . Similarly T s ⊆ I . F or the case that s ∈ D , but s / ∈ I , supp ose for some t ∈ T that st, ts ∈ W := S \ I . Then t / ∈ I , for otherwise st, ts ∈ I , a nd therefore t ∈ D . Using the contin uit y of left a nd righ t translation b y t , w e find an op en set U con taining s suc h that U t ∪ tU ⊆ W . Sinc e s is a limit p oin t o f T and S is Hausdorff, the set B := T ∩ ( U \ I ) is infinite. Since B ⊆ T \ I ⊆ D and D is ω - unstable, either B t or tB meets I . But tB ∪ B t ⊆ tU ∪ U t ⊆ W , and W misses I , a con tradiction. W e conclude that for all t ∈ T , either st ∈ I or ts ∈ I . Th us the closed set ( λ s ) − 1 ( I ) ∪ ( ρ s ) − 1 ( I ) con tains T and hence T , whic h completes the pro of.  Corollary 4. L et S b e a semitop olo gi c al semigr oup, let T b e a subsem i g r oup, let I b e an ide al of T , and assume D := T \ I is ω -unstable. I f x, y ∈ T and xy x = x , then either x ∈ I or x ∈ T and x is an isol a te d p oint of T . SEMIGROUP CLOSU RES OF FINITE RANK SYMMETRIC INVERS E S EMIGROUPS 3 Pr o of. Supp o se tha t x / ∈ I . If x / ∈ T , then x m ust b e a limit p oin t of T . By Lemma 3 either xy ∈ I or y x ∈ I . But then x = ( xy ) x = x ( y x ) ∈ I , since I is an ideal o f T b y separate con tin uity . This contradicts our assumption that x / ∈ I . Th us x ∈ T \ I . If x is a limit p o int of T , then Lemma 3 w ould again imply that x = ( xy ) x = x ( y x ) ∈ I , a con tradiction. Th us x m ust b e an isolat ed p oin t of T and hence also of T .  Definition 5. An ide al series (see, for example, [9]) fo r a semigroup S is a chain of ideals I 0 ⊆ I 1 ⊆ I 2 ⊆ . . . ⊆ I m = S. W e call the ideal series tight if I 0 is a finite set and D k := I k \ I k − 1 is an ω -unstable subset for eac h k = 1 , . . . , m . Example 6. It follows from Lemma 2 that for an infinite cardinal λ , {∅} ⊆ I 1 λ ⊆ . . . ⊆ I m λ is a tigh t ideal series for S := I m λ . Recall that an elemen t x of a semigroup S is regular if there exists y ∈ S suc h that xy x = x and that S is regular if ev ery elemen t is regular [9]. If xy x = x , then it is straightforw ard to v erify tha t x ′ = y xy is an in verse for x , i.e., xx ′ x = x and x ′ xx ′ = x ′ . If x b elongs to an ideal I , then x ′ = y xy also b elongs to the ideal, and it follow s that an ideal of a regular semigroup is regular. Prop osition 7. L et S b e a semitop olo gic al r e gular semigr oup that admits a tight ide al seri e s I 0 ⊆ . . . , ⊆ I m = S . Then e ach I k is close d in S and e ac h memb er of S \ I m − 1 is an isolate d p oint of S . Pr o of. W e first pro v e b y finite induction that I k is closed in S for eac h k . First note that I 0 is closed since it is finite and S is Hausdorff. Assume that I k − 1 is closed for some k , 1 ≤ k ≤ m . If s ∈ I k , then by regularit y of the ideal I k and Corollary 4 s ∈ I k − 1 = I k − 1 or s ∈ I k . In either case s ∈ I k , so I k is closed. By induction I k is closed fo r all k , 0 ≤ k ≤ m . The last a ssertion no w follo ws fr o m Corolla r y 4.  By Example 6 and Prop osition 7 w e hav e the follow ing Corollary 8. L et λ > ω and let n b e any p ositive inte ger. I f τ is a top olo gy on I n λ such that ( I n λ , τ ) is a semitop olo gic al semigr oup, then every element α ∈ I n λ \ I n − 1 λ is an isolate d p oint of the top olo gic al sp ac e ( I n λ , τ ) . The next prop osition establishes that semigroups with tigh t ideal series are restricted in regard to the type of semigroups in whic h they ma y b e dense ly em b edded. Prop osition 9. L et S b e a semitop ol o gic al s e migr oup, let T b e a subsemigr o up, and let T b e its closur e in S . If T admits a tight ide al series, then any r e gular e l e ment of T , in p a rticular any ide m p otent, must alr e ady b e in T . Pr o of. Let x b e a regular elemen t of T , and let I 0 ⊆ . . . ⊆ I m = T b e a tigh t ideal series for T . Let k b e the smallest index suc h that x ∈ I k . W e claim x ∈ I k . If k = 0, then we are done since I 0 is finite, hence closed. So w e assume k ≥ 1. By hypothesis there exists y ∈ T suc h that xy x = x . Then x ′ := y xy satisfies xx ′ x = x and x ′ = y xy ∈ I k , since I k is an ideal of T . W e apply Corollary 4 (with T = I k and I = I k − 1 ) a nd conclude that x ∈ I k ⊆ T .  Recall that a semigroup is an inve rse semigr o up if it is a regular semigroup in whic h each elemen t x has a unique in v erse x ′ [25]. The semigroups I λ and I n λ are in vers e semigroups. Prop osition 10. L et S b e a semitop olo gic al inverse se m igr oup for which the inversion map x 7→ x ′ is c ontinuous. If T is an inverse subsemigr oup that admits a tight ide al series , then T is cl o se d in S . 4 OLEG GUTIK, JIMMIE LA WSON, AN D D U ˇ SAN REPOV ˇ S Pr o of. By separate con tin uity T is a subsem igroup. Since T is closed under the in v ersion map and the in v ersion map is contin uous, one readily sees that T is closed under inv ersion, i.e., T is an inv erse, hence regular, subsemigroup. Prop osition 9 then yields that T ⊆ T , i.e., T is closed.  Prop osition 10 applies directly to the symmetric in v erse semigroup I n λ for λ infinite and n a p ositiv e in t eger and yields the follo wing corollary . Corollary 11. L et S b e a semitop olo gic al inverse semigr oup for which the in v ersion map x 7→ x ′ is c ontinuous. If ( an isomorphic c opy of ) I n λ is a subsemigr oup of S , then it is a close d subset of S . Definition 12 ([15, 28]) . Let S b e a class of top ological semigroups. A top ological semigroup S ∈ S is called H -close d in the class S if S is a closed subsemigroup of any top olog ical semigroup T ∈ S whic h con tains S as a subse migroup. If S coincides with the class of all top ological semigroups, then the semigroup S is called H -close d . W e remark that in [28] the H -closed semigroups are called maximal . Definition 13 ([15, 2 9]) . Let S b e a class of top ological semigroups. A semigroup S is called algebr aic al ly close d in the class S if for an y top ology τ on S suc h tha t ( S, τ ) ∈ S w e ha v e that ( S, τ ) is an H -closed to p ological semigroup in the class S . If S coincides with the class of all top ological semigroups, then the semigroup S is called alge b r aic al ly close d . W e ha v e immediately from Corollar y 11 the follo wing corollaries. Corollary 14. F or a ny infinite c ar dinal λ and p ositive inte g e r n , the semigr oup I n λ is al- gebr aic al ly close d in the class of t op olo gic al invers e semigr oups ( inverse semigr oups that ar e top olo gic al semigr oups with c ontinuous inversion ) . Corollary 15. L et n b e any p ositive inte ge r and let τ b e any inverse s e m igr oup top o lo gy on I n λ . Then ( I n λ , τ ) is an H -close d top olo gic al inverse semig r oup i n the class of top olo gic al inverse semigr oups. The follo wing example implies that for a ll λ > ω , the semigroup I k λ with the discrete top ology is not H -closed in the class of a ll lo cally compact top olog ical semigroups, for an y p ositiv e in t eger k . Example 16. W e fix an y p ositiv e in teger k . Let a / ∈ I k ω . Let S = I k ω ∪ { a } . W e put a · a = a · x = x · a = 0 for all x ∈ I k ω . W e further en umerate t he elemen ts of the set ω by nat ural num b ers. Let A m =  2 l − 1 2 l     l > m  for eac h p ositiv e inte ger m . A top o logy τ on S is now defined as follow s: 1) all p oin ts of I k ω are isolated in S ; and 2) B ( a ) = { U n ( a ) = { a } ∪ A n | n = 1 , 2 , 3 , . . . } is the base o f the top o logy τ at the p oint a ∈ S . Then a) for all  x 1 x 2 · · · x i y 1 y 2 · · · y i  ∈ I k ω and n > max { x 1 , x 2 , . . . , x i , y 1 , y 2 , . . . , y i } w e hav e  x 1 x 2 · · · x i y 1 y 2 · · · y i  · U n ( a ) = U n ( a ) ·  x 1 x 2 · · · x i y 1 y 2 · · · y i  = { 0 } ; SEMIGROUP CLOSU RES OF FINITE RANK SYMMETRIC INVERS E S EMIGROUPS 5 b) U n ( a ) · U n ( a ) = U n ( a ) · { 0 } = { 0 } · U n ( a ) = { 0 } for any p o sitiv e integer n ; and c) U n ( a ) is a compact subset of S f or eac h p ositiv e in teger n . Therefore ( S, τ ) is a lo cally compact top ological semigroup. Obviously I k ω is no t a closed subset of ( S, τ ). The following example sho ws that for all λ > ω , the semigroup I ∞ λ := S n I n λ with the discrete top olog y is not H - closed in the class of all top ological inv erse semigroups. Example 17. Let λ > ω and let τ d b e the discrete to p ology on the semigroup I ∞ λ . F or an y ε ∈ E ( I ∞ λ ) we define M ( ε ) = { χ ∈ I ∞ λ | εχ = χε = ε } . Let S b e the semigroup I ∞ λ with the adjoined iden tit y ι . W e now define a top ology τ S on the semigroup S a s follows: ( i ) χ is an isolated p oint in S for all χ ∈ I ∞ λ ; and ( ii ) the family B ( ι ) = { U ε ( ι ) = { ι } ∪ M ( ε ) | ε ∈ E ( I ∞ λ ) } is the base of the top ology τ S at the p oin t ι . The definition of the family B ( ι ) implies that ι is not an isolated p oin t of a top ological space ( S, τ S ) a nd the restriction of the top ology τ S on the set I ∞ λ coincides with the to p ology τ d . Ob viously , this is sufficien t to sho w that the semigroup op eration on ( S, τ S ) is con tinuous in the follo wing cases: ( i ) ιι = ι ; and ( ii ) ιχ = χι = χ for all χ ∈ I ∞ λ . In case ( i ) w e ha v e U ε ( ι ) · U ε ( ι ) ⊆ U ε ( ι ) . In case ( ii ) w e denote χ =  x 1 x 2 · · · x n y 1 y 2 · · · y n  . Then w e put K = { x 1 } ∪ { x 2 } ∪ · · · ∪ { x n } ∪ { y 1 } ∪ { y 2 } ∪ · · · ∪ { y n } . Let b e K = { a 1 , a 2 , . . . , a k } . Obviously k 6 n . W e define ε =  a 1 a 2 · · · a k a 1 a 2 · · · a k  . Then w e hav e χε = εχ = χ and hence U ε ( ι ) · χ = U ε ( ι ) · χ = { χ } . Since ( U ε ( ι )) − 1 = U ε ( ι ), w e hav e that ( S, τ S ) is a top olog ical inv erse semigroup whic h con tains I ∞ λ as dense inv erse subsemigroup. 2. Semigroups with tight ideal series W e ha v e see n in the previous section that semigroups admitting a tigh t ideal series ha ve in teresting closure prop erties in larger semigroups. In this section w e t ak e a brief closure lo ok at this class of semigroups, primarily to see that suc h semigroups extend signfican tly b ey ond the finite rank symm etric in v erse semigroups. Lemma 18. The class of semigr oups ad mitting a tight ide al series is close d under finite pr o d- ucts. 6 OLEG GUTIK, JIMMIE LA WSON, AN D D U ˇ SAN REPOV ˇ S Pr o of. It suffice s to c hec k for the case n = 2. Let S hav e a t ig h t ideal series I 0 ⊆ . . . ⊆ I m = S and T hav e a tig h t ideal series J 0 ⊆ . . . ⊆ J n = T . Set K i = I i × J 0 for 0 ≤ i ≤ m and K i = S × J i − m for m < i ≤ n + m . Then a n y infinite B ⊆ K i +1 \ K i has an infinite pro jection in to either S for i ≤ m and into T for i > m , a nd since m ultiplicatio n is co ordinat ewise, it directly follo ws that aB ∪ B a meets K i .  Lemma 19. L et h : S → T b e a surje ctive semigr oup homom o rphism such that e ach p oin t inverse h − 1 ( t ) is finite. If S has a tight ide a l series, then so do e s T . Pr o of. It is easy to see that h − 1 ( I 0 ) ⊆ . . . ⊆ h − 1 ( I m ) = S is a tight ideal series for S if I 0 ⊆ . . . ⊆ I m is one fo r T .  Example 20. Let { X i | i ∈ I } b e a c ollection o f finite, pairwise disjoin t s ets, and let X = S i ∈ I X i . W e consider the s emigroup P T ( X , I ) of partial functions on X defined in the follow ing w a y . First c ho ose a subset J ⊂ I and let α : J → K b e a bijection to anot her subset of I . A function f : S i ∈ J X i → S i ∈ K is b y definition in P T ( X , I ) if a nd o nly if x ∈ X i implies f ( x ) ∈ X α ( i ) . All such partial functions ranging o v er all suc h α and correspo nding f form the subsem igroup P T ( X , I ) under composition, a subse migroup o f t he semigroup of all partial functions on X . W e define suc h a function to ha v e rank n if the cardinalit y of the range (and hence domain) of α has cardinality n and P T ( X , I ) n to b e all f unctions of rank less than or equal to n . W e then ha v e an ideal series for the semigroup P T ( X , I ) n defined b y I k = P T ( X , I ) k for 0 ≤ k ≤ n . There is a surjectiv e homomorphism h from P T ( X , I ) n to I n I , whic h assigns to f ∈ P T ( X , I ) n the corresp onding α : J → K b etw een the index sets. One c hec ks directly that this homomorphism has finite p oint in vers es, a nd hence it follo ws from Lemma 19 that the ideal series I k = P T ( X , I ) k for 0 ≤ k ≤ n is tigh t. W e remark that the semigroup P T ( X , I ) n is r egula r , but not an in v erse semigroup. It is w ell known that the semigroup of all tra nsformations T ( X ) is regular [9], a nd essen t ia lly the same pro of yields t he regularity of P T ( X , I ) n . Th us the principal results of Section 1 may b e applied to the semigroup P T ( X , I ) n . 3. Comp a c t embeddings Sev eral a uthors ha v e considered the problem of sho wing that v arious sp ecific semigroups or classes o f semigroups do o r do not em b ed in to compact semigroups. F o r example one can use the Sw elling Lemma to sho w that the bicyclic semigroup do es not admit an embedding in to a compact top ological sem igroup [8]. Closer to our current in v estigat io ns, it w as sho wn b y Gutik and Pa vlyk in [16] that a n infinite top olo g ical semigroup of λ × λ -matrix units B λ do es not em b ed in to a compact top olog ical semigroup, ev ery non- zero elemen t of B λ is an isolated p oin t of B λ , a nd B λ is algebraically closed in the class of top o logical inv erse semigroups. (This is ess en tially a sp ecial case o f results of this pa p er for I 1 λ .) How ev er, w e add a new wrinkle t o earlier in v estigatio ns b y showing that certain pa rtially compact em b eddings do not exist, more precisely that the closure of certain em b edded D -classes cannot b e compact. Recall the G reen’s relatio ns on a semigroup S . Tw o elemen ts a re L - equiv alen t if they generate the same pr incipal left ideal, i.e., s L t if { s } ∪ S s = { t } ∪ S t , and R related if they generate the same principal rig h t ideal. ( In the case that s is regular the principal left ideal reduces to S s s ince s = ss ′ s ∈ sS .) The join of the equiv alence relations L and R is denoted D . It is a standard semigroup result that D is alternat iv ely given by the relatio nal comp ositions L ◦ R = R ◦ L [9, Section 2.1]. A D - equiv alence class D is called a r e gular D -class if it contains a r egula r elemen t. This is the case if and only if eac h L -class and eac h R -class contained in D con tains at least one idemp oten t if and only if eve ry elemen t of D is regular [9, Chapter 2.3]. F urthermore, each inv erse of a mem b er of D is bac k in D [9, Chapter 2.3] W e recall a useful fact ab out regular D -classes. SEMIGROUP CLOSU RES OF FINITE RANK SYMMETRIC INVERS E S EMIGROUPS 7 Lemma 21. L e t a, c ∈ D , a r e gular D -class in a semigr oup S . Then ther e exist s, t ∈ D such that c = sat . Pr o of. Since D = R ◦ L , w e ma y pic k b ∈ D suc h that a R b and b L c . Pic k an idemp otent e in the R -class of a and u ∈ S suc h that au = e . S ince eS = aS = bS , w e ha v e also ea = a and eb = b . F or t = ub , w e hav e at = aub = eb = b . F urthermore, t = u b ∈ S b and b = at ∈ S t , so t L b , and th us t ∈ D . In a similar fashion one finds s ∈ D suc h that c = sb . Then c = sb = sat and s, t ∈ D .  The next theorem is our main one o n the non-existence of compact em b eddings of certain D -classes. Theorem 22. L et S b e a top olo gic al semigr oup and let T b e a subsem igr oup having a tight ide al series I 0 ⊆ . . . ⊆ I m . If D := I k +1 \ I k is a r e gular D -c lass, then D = cl S ( D ) is not c omp act. Pr o of. Supp o se the con trary . Then the infinite set D has a limit p oint x in the compact set D . Let x α denote a net in D \ { x } conv erging to to x . F or eac h α , pick an inv erse x ′ α , whic h m ust again b e in D . By compactness of D , some subnet of x ′ α (whic h w e aga in lab el x ′ α ) m ust con ve rge to some y ∈ D . By con tinuit y of m ultiplication x α = x α x ′ α x α → xy x , and b y uniqueness of limits x = xy x . Th us x is a regular elemen t. Fix some a ∈ D . By Lemma 2 2 for each α , there exists s α , t α ∈ D suc h that s α x α t α = a . Again passing to con v ergen t subnets, we hav e s α → s ∈ D , t α → t ∈ D , and a = s α x α t α → sxt . Therefore a = sxt . In a similar fashion, w e can write x α = u α av α and conclude x = uav for u, v ∈ D . It follows from Corollary 4 that x ∈ I k . Let j b e the smallest index suc h that x ∈ I j . Then j 6 = 0, for o t herwise since I 0 is finite, hence closed, x ∈ I 0 , and hence a = sxt ∈ I 0 , con tradicting the fact t ha t a / ∈ I k , whic h con tains I 0 . Th us j ≥ 1. If x / ∈ I j , then x w ould b e a limit p oint of I j , and hence a limit p o in t of I j \ I j − 1 , since x / ∈ I j − 1 . Again b y Corollary 4 it w ould follow that x ∈ I j − 1 , a contradiction. W e conclude tha t x ∈ I j , and then that a = sxt ∈ I j , since I j is an ideal of T . Applying Corollary 4 to the regular elemen t a , we conclude either that a ∈ I j , an imp ossibilit y since I j ⊆ I k , or a ∈ I j − 1 . The latter w ould imply x = uav ∈ I j − 1 , whic h w e ha v e just seen is not t he case. Th us we hav e reac hed a contradiction to our assumption that D is compact.  Since it is w ell-kno wn and direct to v erify that t he sets D = I k λ \ I k − 1 λ are D -classes for 1 ≤ k ≤ n in the s emigroup I n λ , we ha ve the follo wing corollary , whic h generalizes the previously men tio ned result of Gutik and P a vlyk in [16]. Corollary 23. F or an infi nite c ar din a l λ and p os i tive inte ger n , if I n λ is a subsemigr oup of a top olo gic al s emigr oup S , it c anno t b e the c ase that that cl S ( I k λ \ I k − 1 λ ) is c omp ac t for 1 ≤ k ≤ n . W e close with a theorem on I ∞ λ . Theorem 24. F or any infinite c ar dinal λ ther e exists no top olo gy τ on I ∞ λ such that ( I ∞ λ , τ ) is a c o m p act semitop o lo gic al semig r oup. Pr o of. Supp o se to the con trary , t ha t there exists a top olog y τ on I ∞ λ suc h that ( I ∞ λ , τ ) is a compact semitop ological semigroup. The definition o f the semigroup I ∞ λ implies that for an y idempo t ent ε ∈ I ∞ λ there exists an idempo ten t φ ∈ I ∞ λ suc h tha t ε φ = φε = ε and φ 6 = ε , i. e. ε < φ . Therefore there exists a subsets of idemp oten ts A = { ε 1 , ε 2 , . . . , ε n , . . . } in φ ∈ I ∞ λ suc h that 0 < ε 1 < ε 2 < . . . < ε n < . . . . Without lo os of generalit y w e can assume that ε k ∈ I k λ \ I k − 1 λ , for a n y k = 2 , 3 , 4 , . . . and ε 1 ∈ I 1 λ \ { 0 } . Then Coro lla ry 8 implies that the idemp oten t ε k has an op en neighbourho o d 8 OLEG GUTIK, JIMMIE LA WSON, AN D D U ˇ SAN REPOV ˇ S U ( ε k ) such that U ( ε k ) ∩ I k λ = { ε k } for all k = 1 , 2 , 3 , . . . . Since the translations in ( I n λ , τ ) are con tin uous maps, the set U l ( ε k ) = { β ∈ I ∞ λ | β ε k = ε k } is clop en in the to p ological space ( I ∞ λ , τ ) for all k = 1 , 2 , 3 , . . . . W e define the family O = { O k | k = 1 , 2 , 3 , . . . } as follo ws: ( i ) O 1 = I ∞ λ \ U l ( ε 1 ); and ( ii ) O k = U l ( ε k − 1 ) \ U l ( ε k ) fo r a ll k = 2 , 3 , 4 , . . . . Ob viously , the family O is a clop en co v er of the top ological space ( I ∞ λ , τ ), whic h does not con tain a finite sub cov er, a con tradiction.  A ckno wledgements This researc h w as supp o r t ed by SRA gran ts P1-0292- 0101-04 , J1-964 3-0101 and BI-R U/08- 09-002. Reference s [1] A. Abd-Allah and R. Br own, A c omp act-op en top olo gy on p artial maps with op en domains, J. Londo n Math. So c. 21 (2 ) (1980 ), 48 0 —486. [2] B. B. Baird, Inverse semigr oups of home omorphisms b etwe en op en su bsets, J. Austr al. Math. So c. Ser. A 24 :1 (1 977), 92 –102 . [3] B. B. Ba ird, Emb e dding inverse semigr oups of home omorphisms on close d subsets , Glasgow Math. J. 1 8 :2 (1977), 19 9–20 7. [4] B. B. 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Dep ar tment of Mechanics and Ma thema tics, Iv an Franko L viv N a tional University, U niver- sytetska 1 , L viv, 7 9000, Ukraine E-mail addr ess : o gutik@ frank o.lvi v.ua Dep ar tment of Ma thema tics, Louisiana St a te University, B a ton R ouge, LA 7 0803, U SA E-mail addr ess : law son@ma th.ls u.edu Institute of Ma thema tics, P hysics and Mechanics, and F acul ty of Ma thema tics and Phy sics, University of L jubljana, Jadransk a 19, Ljubljana , 1000, S lovenia E-mail addr ess : dus an.rep ovs@g uest.arnes.si