Embedding the bicyclic semigroup into countably compact topological semigroups

EMBEDDING THE BICYCLI C SEMIGR OUP INTO COUNT ABL Y COMP A CT TOPOLOGIC AL SEMIGR OUPS T ARA S BANAKH, SVETLAN A DIMITROV A, AND OLEG GUTIK Abstra ct. W e study algebraic and t op ological p roperties of top ological semigroups containing a cop y of the bicyclic semigroup C ( p, q ). W e prov e that a top ological semigroup S with pseudo compact square con- tains no dense copy of C ( p, q ). On the other hand, we construct a (consisten t) example of a pseudocompact (countably compact) Tyc honoff semigroup contai ning a copy of C ( p, q ). In this pap er we study the structural prop erties of top ological semigroups that con tain a copy of the bicyclic semigroup C ( p, q ) and present a (consisten t) example of a Tyc honoff pseud o compact (coun tably compact) semigroup S that con tains C ( p, q ). This example s ho ws that the theorem of K o c h and W allace [17] sa ying th at compact top ological semigroups do not con tain bicyclic subs emigroups cannot b e gen- eralized to the class of pseudo compact or count ably compact top ological semigroups . Also this example sho ws that the pr esence of an inv ersion is essentia l in a result of Gutik and Rep ov ˇ s [14] who pr o v ed that the bicyclic s emigroup do es n ot embed in to a coun tably compact top ological inv erse semigroup. The presence or absence of a bicyclic su bsemigroup in a give n (top ologica l) s emigroup S h as imp ortan t implications f or u nderstanding the algebraic (and top olo gical) structure of S . F or example, the well- kno wn Andersen Theorem [2 ], [6 , 2.54] s a ys th at a simp le semigroup with an idemp otent but without a cop y of C ( p, q ) is completely simple and h ence b y th e Rees-Susc hk ewitsc h Theorem [21], has the str u cture of a s andwic h pro duct [ X, H , Y ] σ of t w o sets X , Y and a group H connected by a suitable sandwich function σ : Y × X → H . The R ees-Su sc hk ewitsc h Theorem has also a top ological version, see [4]. Ha ving in mind the men tioned result of Koch a nd W allace [17], I.I. Guran ask ed if the b icyclic semigroup can b e em b edd ed into a countably compact top ological semigroup. In this pap er we shall find man y conditions on a top ological semigroup S whic h forbid S to conta in a b icyclic subs emigroup . One of the simplest conditions is the coun table compactness of the squ are S × S . On th e other hand , we constru ct a Tyc honoff pseudo compact semigroup that con tains a b icyclic semigroup. Moreo v er, assuming the existence of a counta bly compact ab elian torsion-free top ological group without con v ergen t sequences w e shall construct an example of a Tyc honoff counta bly compact top ological semigroup that con tains a cop y of the b icyclic semigroup. T o construct suc h pathological semigroups , we shall stud y the op eration of attac hing a discrete semi- group D to a top ological semigroup X along a homomorphism π : D → X . This construction has tw o ingredien ts: top ological and algebraic, discussed in th e next four sections. In section 5 w e establish some structure prop erties of top ological semigroups that conta in a cop y of the bicyclic subsemigroup and in Section 6 w e constr u ct our main coun terexample. Our metho d of constructing this coun terexample is rather standard and exploits the ideas of D. Robbie, S. Sv etlic hny [23] (who co nstructed a counta bly com- pact cancellativ e semigroup under CH) and A. T omita [26] (who weak ened the Con tin uum Hyp othesis in their r esu lt to a we ak er version of Martin’s Axiom). All top ological spaces app earing in this pap er are assumed to b e Hausdorff. 1. A tt aching a discre te s p ace t o a topological sp ace In this section we describ e a s imple construction of attac hing a discr ete s p ace D to a top ological space M along a map π : D → M and w ill inv estigate top ologica l prop erties of the obtained s pace D ∪ π M . 2000 Mathematics Subje ct Classific ation. 22A15, 54C25, 54D35, 54H15. Key wor ds and phr ases. T op ological semigroup, semitop ological semigroup, bicyclic semigroup, em b edding, ex tension, semigroup compactification. 1 2 T ARAS BANAKH, SVETLANA DIMITR OV A, AND OLEG GUTIK Although all non-trivial applicatio ns concern in fi nite D , w e do not restrict ourselve s by infi nite spaces and formulate ou r results f or an y (n ot necessarily in fi nite) discrete space D . Let D b e a discrete top ological sp ace. If D is infi nite, then let αD = D ∪ {∞} b e the Aleksandrov compactificatio n of D . If D is fin ite, then let αD = D ∪ {∞} b e the top olog ical sum of D and the singleton {∞} for some p oin t ∞ / ∈ D . Giv en a map π : D → M to a T 1 -top ologica l sp ace M , consider the closed subsp ace D ∪ π M = { ( x, π ( x )) : x ∈ D } ∪ ( {∞} × M ) of the pr o duct αD × M . W e shall iden tify the space D with the op en discrete subspace { ( x, π ( x )) : x ∈ D } and M w ith the closed subspace {∞} × M of D ∪ π M . Let ¯ π = π ∪ id M : D ∪ π M → M d enote th e pro jection to the seco nd factor. Observe that the top ology of the space D ∪ π M is the we ak est T 1 - top ology that indu ces the original top ologies on the su bspaces D and M of D ∪ π M and makes th e map ¯ π con tin uous. The follo wing (almost trivial) prop ositio ns describ e some elemen tary prop erties of the sp ace D ∪ π M . Prop osition 1.1. If for some i ≤ 3 1 2 the sp ac e M satisfies the sep ar ation axiom T i , then so do e s the sp ac e D ∪ π M . Prop osition 1.2. If M is (sep ar able) metrizable and D is c ountable, then the sp ac e D ∪ π M is (sep ar able) metrizable to o. Prop osition 1.3. If the sp ac e M is c omp act, then so is the sp ac e D ∪ π M . W e recall that a top ological space X is c ountably c omp act if eac h coun table op en co v er of X h as a finite sub cov er. Th is is equiv alent to sa ying that the space X con tains no infi n ite closed discrete subspace. Prop osition 1.4. If some p ower M κ of the sp ac e M is c ountably c omp act, then the p ower ( D ∪ π M ) κ is c ountably c omp act to o. Pr o of. Since D ∪ π M is a closed su bspace of αD × M , the p o w er ( D ∪ π M ) κ is a closed sub space of ( αD × M ) κ . So, it suffices to chec k that th e latter space is count ably compact. Since th e pro d uct of a coun tably compact space and a co mpact space is countably compact [10, 3.10. 14], th e p ro duct M κ × ( αD ) κ is coun tably compact and so is its top ological cop y ( αD × M ) κ .  If the space M is T y chonoff, then D ∪ π M is a subsp ace of the compact Hausd orff sp ace D ∪ π β M where β M is the Stone- ˇ Cec h compactification of M . Assuming that M is countably compact at π ( D ) w e sh all sh o w that D ∪ π β M coincides with the Stone- ˇ Cec h compactification of D ∪ π M . W e shall sa y that a top ological space X is c ountably c omp act at a subset A ⊂ X if eac h infi nite subs et B ⊂ A h as an accum ulation p oint x in X . The latter means that eac h neigh b orh o o d O ( x ) of x conta ins infinitely many p oints of the set B . Prop osition 1.5. If the sp ac e M is T ychonoff and is c ountably c omp act at the subset π ( D ) , then D ∪ π β M is the Stone- ˇ Ce ch c omp actific ation of D ∪ π M . Pr o of. By Prop osition 1.5, the sp ace D ∪ π M is T yc honoff and h en ce has the Stone- ˇ Cec h compactificatio n β ( D ∪ π M ). Since the space M is a retract of D ∪ π M , the compactification β M is a retract of β ( D ∪ π M ). Let β i : β ( D ∪ π M ) → D ∪ π β M b e th e Stone- ˇ Cec h extension of the id en tit y inclusion i : D ∪ π M → D ∪ π β M . W e claim that β i is a h omeomorphism. First w e sho w that the subset D ∪ β M ⊂ β ( D ∪ π M ) is compact. In deed, giv en an op en co v er U of D ∪ β M we can find a fin ite sub co ve r V ⊂ U of β M and then consider the set D ′ = D \ S V . W e claim that this s et D ′ is finite. Assu ming the con verse and u sing the countable compactness of M at π ( D ) we could fi nd a p oint a ∈ M suc h th at for ev ery neigh b orho o d O ( a ) ⊂ M the set { x ∈ D ′ : π ( x ) ∈ O ( a ) } is infinite. T ak e an y op en set V ∈ V con taining the p oint a . By the definition of th e top olog y on D ∪ π M there is a neigh b orh o od O ( a ) ⊂ M ∩ V of a in M and a fi nite su bset F ⊂ D suc h that ¯ π − 1 ( O ( a )) \ F ⊂ V . Then the set { x ∈ D ′ : π ( x ) ∈ O ( a ) } lies in F and hence is finite, whic h is a con tradiction. Hence the set D ′ is fin ite and w e can find a fin ite subf amily W ⊂ U with D ′ ⊂ S W . Then V ∪ W ⊂ U is a fi nite BICYCLIC SEMIGROUP IN COUNT ABL Y COMP ACT SEM IGR OUPS 3 sub cov er of D ∪ β M . No w w e see that the subs et D ∪ β M , b eing compact and dense in β ( D ∪ π M ), coincides with β ( D ∪ π M ). It f ollo ws that the con tinuous map β i = β i | D ∪ β M is bijectiv e and h ence is a homeomorphism .  F ollo wing A.V. Arkhan gel’skii [1, I I I . § 4], we say that a top olo gical space X is c ountably pr ac omp act if X is countably compact at a dense subset of X . It is clear that eac h countably compact space is coun tably pr acompact. Prop osition 1.6. The sp ac e D ∪ π M is c ountably pr ac omp act if and only if M is c ountably c omp act at a dense sub set A ⊃ π ( D ) of M . Pr o of. If the space D ∪ π M is coun tably pracompact, then it is countably compact at some den s e subset A ⊂ D ∪ π M . Th e set A , b eing dense, con tains the op en discrete sub space D of D ∪ π M . Th e con tin uit y of the retraction ¯ π : D ∪ π M → M imp lies that the space M is coun tably compact at the dens e subset ¯ π ( A ) ⊃ π ( D ) of M , so M is coun tably p racompact. No w assu me conv ersely that the space M is countably compact at a d en se su b set A ⊃ π ( D ). W e claim th at D ∪ π M is coun tably compact at the dense sub set D ∪ A . W e n eed to chec k that eac h infinite subset B ⊂ D ∪ A h as a cluster p oint in D ∪ π M . If B ∩ A is infinite, then the s et B ∩ A ⊂ B has an accum u lation p oin t in M b ecause M is counta bly compact at A . If π ( B \ A ) is infinite, then π ( B \ A ) has an accum ulation p oint x in M b ecause of th e counta ble compactness of M at π ( D ) ⊂ A . B y the definition of the top ology on D ∪ π M , the p oin t x is an accum ulation p oint of the set B \ A . It remains to consider the case when the sets A ∩ B and π ( B \ A ) are finite. In this case for some p oint c ∈ π ( B \ A ) the set C = B ∩ π − 1 ( c ) is infin ite and then c is an accum ulation p oint of the set C ⊂ B by th e d efi nition of the top olog y of D ∪ π M .  A top ologica l space X is d efined to b e pseudo c omp act if eac h lo cally finite op en cov er of X is fin ite. According to [10, 3.10.22] a T yc honoff space X is pseudo compact if and only if eac h con tin uous real-v alued function on X is b ound ed. F or eac h top ologica l sp ace w e h a ve the follo wing implications: coun tably compact ⇒ counta bly pracompact ⇒ pseud o compact. Prop osition 1.7. The sp ac e D ∪ π M is pseudo c omp act if and only if M is pseudo c omp act and M c ountably c omp act at the subset π ( D ) ⊂ M . Pr o of. Assume that the space D ∪ π M is pseudo compact. Then the sp ace M is ps eu do compact, b eing a con tinuous image of the pseudo compact space D ∪ π M . Next, we pr o ve that M is coun tably compact at π ( D ). Assuming the con v erse, we could find a sequen ce D ′ = { x n : n ∈ ω } ⊂ D s u c h that π ( x n ) 6 = π ( x m ) for n 6 = m and the image π ( D ′ ) is closed and d iscrete in M . Define an u n b oun ded function f : D ∪ π M → R letting f ( x ) = ( n if x = x n for some n ∈ ω 0 otherwise, and c hec k that f is con tin uous, whic h cont radicts the pseudo compactness of D ∪ π M . T o prov e the “if ” p art, assume that the space M is p s eudo compact and is coun tably compact at the subset π ( D ). T o pro v e th at the space D ∪ π M is ps eu do compact, fix a locally finite op en co ve r U of D ∪ π M and consider the lo cally finite op en sub co v er V = { U ∈ U : U ∩ M 6 = ∅} of M . The pseud o compactness of M guaran tees that the co ver V is finite. Rep eating the argumen t of the pro of of Prop osition 1.5, w e can chec k that the set D ′ = D \ S V is finite. The lo cal finiteness of th e family U implies th at the f amily W = { U ∈ U : U ∩ D ′ 6 = ∅} is finite. Since U = V ∪ W , th e co ver U of D ∪ π M is fin ite.  F ollo wing [3], we d efi ne a top ological space X to b e op enly factorizable if ev ery cont inuous map f : X → Y to a metrizable separable space Y can b e written as the comp ositio n g ◦ p of an op en con tinuous map p : X → K on to a metrizable s ep arable space K and a contin uous map g : K → Y . Prop osition 1.8. If the set D is c ountable and M is op enly factorizable, then the sp ac e D ∪ π M is op enly factorizable to o. 4 T ARAS BANAKH, SVETLANA DIMITR OV A, AND OLEG GUTIK Pr o of. Fix any contin uous m ap f : D ∪ π M → Y to a metrizable separable space Y . Since M is op enly factorizable, there are an op en contin uous map p : M → K on to a separable metrizable sp ace K and a con tinuous map g : K → Y suc h that f | M = g ◦ p . Consider the map pπ = p ◦ π : D → K and th e corresp ondin g space D ∪ pπ K that is sep arable and metrizable by Prop osition 1.2. Let ¯ p = id ∪ p : D ∪ π M → D ∪ pπ K b e the map that is iden tit y on D and coincides w ith the map p on M . It follo w s from the op enn ess of the map p th at the map ¯ p is op en (and con tinuous). No w extend the map g : K → Y to a map ¯ g : D ∪ pπ K → Y letting ¯ g | D = f | D . It is easy to see that f = ¯ g ◦ ¯ p . It r emains to c hec k that the map ¯ g is con tin uous. T ak e an y op en set U ⊂ Y an d observe that ¯ g − 1 ( U ) = ¯ p ( f − 1 ( U )) b ecause f is con tin u ous an d ¯ p is op en.  2. Comp act e xtensions of topological semigroups In this section we surv ey some kno w n results on compact extensions of semitop ological semigroups. By a semitop olo gic al semigr oup we un derstand a top olo gical space S en d o wed with a separately con tin uous semigroup op eration ∗ : S × S → S . I f the op eration is jointly contin uous, then S is called a top olo gic al semigr oup . Let C b e a class of compact Hausd orff semitop ological semigroups. By a C -c omp actific ation of a semitop ologica l semigroup S w e understand a pair ( C ( S ) , η ) consisting of a compact semitop ological semigroup C ( S ) ∈ C and a conti nuous homomorphism η : S → C ( S ) (called the c anonic homomo rphism ) suc h that f or eac h con tin uous h omomorphism h : S → K to a semitop ologica l semigroup K ∈ C there is a un ique contin uous homomorphism ¯ h : C ( S ) → K such that h = ¯ h ◦ η . It follo ws that any tw o C -compactifica tions of S are top ologically isomorp hic. W e shall b e in terested in C -compacti fications for the follo wing classes of semigroups: • W AP of compact semitop ological semigroup s; • AP of compact top olo gical semigroups; • SAP of compact top ologica l group s. The corresp ondin g C -compactifications of a semitop ological semigroup S will b e denoted by W AP( S ), AP( S ), and SAP( S ). The notatio n came fr om the abbreviations for w eakly almost p erio dic, almost p erio dic, and strongly almost p eriod ic function rings that determine those compactifications, see [5, Ch.IV], [24, Ch.I I I], [16, § 21]. The inclusions SAP ⊂ AP ⊂ W AP ind uce canonic homomorphisms η : S → W AP ( S ) → AP( S ) → SAP( S ) for any semitop ological semigroup S . It should b e m entioned that th e canonic homomorph ism η : S → W AP( S ) need not b e injectiv e. F or example, for the group H + [0 , 1] of orien tation-preserving homeomorphisms of the in terv al its W AP-compact ification is a singleton, see [20]. Ho w ev er, for coun tably compact semitop ological semigroups the situation is more optimistic. The f ollo wing t wo r esults are d ue to E. Reznic h enk o [22]. Theorem 2.1 (Reznic henko) . F or any T ychonoff c ountably c omp act semitop olo gi c al semigr oup S the semigr oup op er ation of S extends to a sep ar ately c ontinuous semigr oup op er ation on β S , which implies that β S c oincides with the W AP -c omp actific ation of S . The same conclusion holds for T yc honoff p seudo compact top ological semigroups. Theorem 2.2 (Reznic hen ko) . F or any T ychonoff pseudo c omp act top olo gic al semigr oup S the semig r oup op er ation of S extends to a sep ar ately c ontinuous semigr oup op er ation β S , which implies that β S c oincides with the W AP -c omp actific ation of S . This theorem combined with the Glic ksb erg Theorem [10, 3.12.20( c)] on the Stone- ˇ Cec h compactifica- tions of p ro ducts of pseudo compact spaces, implies the f ollo wing imp ortan t result, see [3, 1.3]. Theorem 2.3. F or any T ychonoff top olo gic al semigr oup S with pseudo c omp act squar e S × S the semigr oup op er ation of S extends to a c ontinuous semigr oup op er ation on β S , which implies that β S c oincides with the AP -c omp actific ation of S . BICYCLIC SEMIGROUP IN COUNT ABL Y COMP ACT SEM IGR OUPS 5 Another r esult of the same s pirit inv olve s op enly f actoriza ble spaces w ith w eakly Lindel¨ of squares. W e recall that a top ological space X is we akly Lindel¨ of if eac h op en co ver U of X con tains a countable sub collection V ⊂ U whose un ion S V is d ense in X . The f ollo wing extension theorem is pro v ed in [3]. Theorem 2.4. F or any T yc honoff op enly factorizable top olo gic al semigr oup S with we akly Lindel¨ of squar e S × S the semigr oup op er ation of S extends to a c ontinuous semigr oup op er ation on β S , which implies that β S is an AP - c omp actific ation of S . The follo wing theorem also is p ro v ed in [3]. It giv es conditions on a p seudo compact top ological semi- group S under whic h its Stone- ˇ Cec h compact ification β S coincides with the SAP-compactification SAP( S ) of S . Theorem 2.5. F or a T ychonoff pseudo c omp act top olo gic al semigr oup S the Stone- ˇ Ce ch c omp actific ation β S is a c omp act top olo gic al gr oup pr ovide d that one of the fol lowing c onditions holds: (1) S c ontains a total ly b ounde d top olo gic al gr oup as a dense su b gr oup; (2) S c ontains a dense sub gr oup and S × S is pseudo c omp act. 3. A tt aching a discrete semigroup to a se mitopological semigroup In this section w e extend the construction of the s p ace D ∪ π M to the category of semitop ological semigroups and their con tin uous homomorphisms . Giv en a homomorphism π : D → M fr om a discrete semigroup D in to a semitop ological semigroup M let us extend the s emigroup op erations from ( D , · ) and ( M , · ) to D ∪ π M b y letting xy =      x · y if x, y ∈ D or x, y ∈ M , π ( x ) · y if x ∈ D and y ∈ M , x · π ( y ) if x ∈ M , y ∈ D . Endo we d with th e so-extended op eration, the space S = D ∪ π M b ecomes a semitop ological semigroup con taining D as a su bsemigroups and M as a t wo -sided ideal. Moreo v er, the map ¯ π = π ∪ id M : D ∪ π M → M is a con tin u ous semigroup homomorph ism. No w we will find some conditions guaranteei ng that S = D ∪ π M is a top ological semigroup. Definition 3.1. A homomorphism π : D → M is called finitely r esolvable if for ev ery a, b ∈ M and c ∈ D the set { ( x, y ) ∈ D × D : π ( x ) = a, π ( y ) = b, xy = c } is finite. Observe that eac h one-to-one homomorphism is fi n itely resolv able. Theorem 3.2. L et π : D → M b e a homomor phism fr om a discr ete semig r oup D to a top olo g i c al semigr oup M . F or the semitop olo gic al semigr oup S = D ∪ π M the fol lowing c onditions ar e e qui v alent: (1) S i s a top olo g ic al semigr oup; (2) f or e ach c ∈ D the set D c = { ( x, y ) ∈ D × D : xy = c } is close d in S × S . If the homomo rphism π is finitely r esolvable, then the c onditions (1),(2) ar e e q u ivalent to (3) and fol low fr om (4): (3) F or e ach c ∈ D the set π 2 ( D c ) = { ( π ( x ) , π ( y )) : ( x, y ) ∈ D c } is close d and discr ete in M × M . (4) the subsp ac e π ( D ) is discr ete in M and the c omplement M \ π ( D ) is a two-side d ide al in M . Pr o of. (1) ⇒ (2) Assumin g that S is a top ological semigroup, w e need to chec k that for every c ∈ D the set D c = { ( x, y ) ∈ D × D : xy = c } is closed in S × S . Assum ing the con ve rse, we could find an accum u lation p oint ( a, b ) ∈ S × S for the s et D c . Since D c ⊂ D × D is discrete, either a ∈ M or b ∈ M and hence ab ∈ M . Ho wev er ab = c by the con tin uit y of the semigroup op eration on S , w hic h is a con tr ad iction as c / ∈ M . (2) ⇒ (1) Assum e that for eac h c ∈ D the set D c is closed in S × S . W e n eed to chec k the con tin u it y of the m ultiplication at eac h pair ( x, y ) ∈ S × S . If x or y b elongs to D , then this follo ws from the con tinuit y of left and right s h ifts on S . So, w e can assume that x, y ∈ M . Let z = xy and O ( z ) ⊂ S b e an op en neigh b orh o o d of z . It follo ws f r om the d efinition of the top olog y of S = D ∪ π M that there 6 T ARAS BANAKH, SVETLANA DIMITR OV A, AND OLEG GUTIK are a neigh b orho o d U ( z ) ⊂ M and a finite su bset F ⊂ D su c h that ¯ π − 1 ( U ( z )) \ F ⊂ O ( z ). By th e con tinuit y of the semigroup op eratio n on M the p oin ts x, y ha v e neigh b orho o ds U ( x ) , U ( y ) ⊂ M such that U ( x ) · U ( y ) ⊂ U ( z ). Since the set D F = S c ∈ F D c is closed in S × S and ( x, y ) / ∈ D F (b ecause ( x, y ) ∈ M × M ), we can fi nd neigh b orho o d s O ( x ) ⊂ ¯ π − 1 ( U ( x )) and O ( y ) ⊂ ¯ π − 1 ( U ( y )) of th e p oin ts x, y suc h that the set O ( x ) × O ( y ) is d isjoin t from th e set D F . In this case O ( x ) · O ( y ) ⊂ S \ F and O ( x ) · O ( y ) ⊂ ¯ π − 1 ( U ( x )) · ¯ π − 1 ( U ( y )) ⊂ ¯ π − 1 ( U ( z )), whic h implies O ( x ) · O ( y ) ⊂ ¯ π − 1 ( U ( z )) \ F ⊂ O ( z ). (2) ⇒ (3) Assu m e th at for some c ∈ D the set D c is closed in S × S . W e shall show that its image π 2 ( D c ) = { ( π ( x ) , π ( y )) : ( x, y ) ∈ D c } is closed and discrete in M × M . Assuming the conv erse, we could find an accum ulation p oin t ( a, b ) ∈ M × M of π 2 ( D c ). W e claim that ( a, b ) is an accum ulation p oin t of the set D c . Fix an y neigh b orh o o ds O ( a ) and O ( b ) of the p oints a and b in S , resp ectiv ely . By the definition of the top ology of D ∪ π M , th er e are neighborh o ods U ( a ) and U ( b ) of those p oin ts in M and a finite subset F ⊂ D su c h that O ( a ) ⊃ ¯ π − 1 ( U ( a )) \ F and O ( b ) ⊃ ¯ π − 1 ( U ( b )) \ F . Since ( a, b ) is an accum u lation p oint of th e set π 2 ( D c ), th ere is a pair ( x, y ) ∈ D c \ F 2 suc h that ( π ( x ) , π ( y )) ∈ U ( a ) × U ( b ). This pair ( x, y ) b elongs to the n eighb orh o o d O ( a ) × O ( b ), witn essin g th at ( a, b ) is an accum ulation p oin t of the set D c . Since ( a, b ) / ∈ D c , the set D c is not closed in S × S . F rom no w on w e assu m e that the homomorphism π is finitely resolv able. (3) ⇒ (2) Assume that for some c ∈ D the s et π 2 ( D c ) is clo sed and discrete in M × M . W e need to c hec k that the set D c is closed in S × S . In the opp osite case th is s et has an accum ulation p oin t ( a, b ) in S × S . The con tinuit y of the retraction ¯ π implies that the pair ( π ( a ) , π ( b )) lies in the closure of the set π 2 ( D c ) and hence is an isolated p oint of π 2 ( D c ). Then the set W = { ( x, y ) ∈ D c : π ( x ) = π ( a ) , π ( y ) = π ( b ) } is infinite b ecause ( a, b ) is an accumulat ion p oin t of D c . On the other hand, the set W is finite by the finite resolv abilit y of the homomorph ism π . (4) ⇒ (3) Assume that π ( D ) is discrete and S \ π ( D ) is a tw o-sided ideal in M . Giv en an y c ∈ D , we need to c heck that the subsp ace π 2 ( D c ) is closed and discrete in M × M . The space π 2 ( D c ) is discrete b ecause so is the space π ( D ) × π ( D ) ⊃ π 2 ( D c ). If the set π 2 ( D c ) is not closed in M × M , then it has an accumulatio n p oint ( a, b ) ∈ M × M , whic h do es not b elongs to π ( D ) × π ( D ) as the latte r space is discrete. Since M \ π ( D ) is a t w o-sided ideal in M , ab / ∈ π ( D ). On the other h and, by the contin uit y of the homomorp hism ¯ π , we get ¯ π ( ab ) = π ( c ) ∈ π ( D c ) and this is a required con trad iction.  Corollary 3.3. L et D b e a semigr oup such that for some c ∈ D the set D c = { ( x, y ) ∈ D × D : x y = c } is infinite and let π : D → M b e a homomorph ism into a top olo gic al se mig r oup M . If S = D ∪ π M is a top olo gic al semigr oup, then D c is an op en-and-close d discr ete subsp ac e of S × S and henc e S × S i s not pseudo c omp act. 4. A tt aching the b icyclic semigroup to a topo logical s emigroup In this section we study the structur e of the semigroups D ∪ π M in the case D = C ( p, q ) is the bicyclic semigroup. The bicyclic group pla ys an imp ortan t r ole in the structure theory of semigroups , see [6]. A remark able p r op ert y of this semigroup is that it is n on-top ologiza ble in the sense that any Hausd orff top ology tur ning C ( p, q ) into a top ological semigroup is d iscrete [9]. The bicyclic s emigroup C ( p, q ) is generate d by tw o elemen t p, q and one relatio n q p = 1, see [6]. It follo ws that eac h elemen t of C ( p, q ) can b e uniquely written as the p ro duct p n q m for some n, m ∈ ω . The elemen t 1 = p 0 q 0 is a t wo -sided unit for C ( p, q ). The pro d uct p m q n · p i q j of t w o element s of the bicyclic semigroup C ( p, q ) is equal to p m q n − i + j if n ≥ i and to p m + i − n q j if n ≤ i . The semigroup E C = { p n q n : n ∈ ω } of the idemp oten ts of C ( p, q ) is isomorp hic to the semigroup ω of finite ordinals endo w ed w ith the op eration of m aximum. If π : C ( p, q ) → H is any homomorphism of C ( p, q ) into a group, then π (1) is the iden tit y elemen t e of the group H and the relation q p = 1 implies that π ( q ) and π ( p ) are m utually inv erse elemen ts of H , generating a cyclic sub group of H , s ee [6, 1.32]. If the image π ( C ( p, q )) is infinite, then it is easy to c h ec k that the homomorphism π : C ( p, q ) → H is fin itely resolv able. BICYCLIC SEMIGROUP IN COUNT ABL Y COMP ACT SEM IGR OUPS 7 Theorem 4.1. L et π : C ( p, q ) → M b e a homo morphism of the bicyc lic semigr oup into a top olo gic al semigr oup M such that Z = π ( C ( p, q )) is a dense i nfinite cyclic sub gr oup of M and M is c ountably c omp act at Z . The semitop olo gic al semigr oup S = C ( p, q ) ∪ π M has the fol lowing pr op e rties: (1) i f M is c ountably c omp act, then so is C ( p, q ) ∪ π M . (2) S is a top olo gi c al semigr oup iff for every c ∈ C ( p, q ) the set D c = { ( x, y ) ∈ C ( p, q ) 2 : xy = c } is close d in S × S iff the subsemigr oup π 2 ( D 1 ) = { ( π ( q n ) , π ( p n )) : n ∈ ω } is close d and discr ete in M × M . (3) S is a top olo gic al semigr oup pr ovide d the sub g r oup Z is discr ete in M and M \ Z is an ide al in M . (4) If S is a top olo gic al semig r oup, then the squar e S × S is not pseudo c omp act. (5) If S is a top olo gic al semig r oup and the sp ac e M is T ychonoff, then (a) M is not op enly factorizable, (b) M × M is not pseudo c omp act, (c) M c ontains no dense total ly b ounde d top olo gi c al sub gr oup. Pr o of. 1. The fir s t item follo ws from Prop ositio n 1.4. 2. Th e second item will follo w from Theorem 3.2 as so on as we pro v e that for ev ery c ∈ C ( p, q ) the set π 2 ( D c ) = { ( π ( x ) , π ( y )) : x , y ∈ C ( p, q ) , xy = c } is closed and d iscrete in M × M pro vided th at the subsemigroup π 2 ( D 1 ) = { ( π ( q n ) , π ( p n )) : n ∈ ω } is closed an d discrete in M × M . So we assume that π 2 ( D 1 ) is closed and discrete in M × M . T aking in to accoun t that th e cyclic subgroup Z is den se in the top ologica l semigroup M , w e conclude that the semigroup M is commutat iv e and e = π (1) is a tw o-sided unit of M . Moreo v er, for eac h z ∈ Z th e left shift l z : M → M , l z : x 7→ z x , is a homeomorphism of M with in v erse l z − 1 . Th is imp lies that for ev ery a, b ∈ Z the set ( a, b ) · π 2 ( D 1 ) = { ( ax, by ) : ( x, y ) ∈ π 2 ( D 1 ) } is closed and discrete in M × M . It is easy to chec k that f or ev ery c = p i q j ∈ C ( p, q ), D c = { ( p k q n , p i − k + n q j ) : 0 ≤ k ≤ i, n ∈ ω } ∪ { ( p i q j − k + n , p n q k ) : 0 ≤ k ≤ j, n ∈ ω } and then π 2 ( D c ) =  [ 0 ≤ k ≤ i ( π ( p k ) , π ( p i − k q j )) · π 2 ( D 1 )  ∪  [ 0 ≤ k ≤ j ( π ( p i q j − k ) , π ( q k )) · π 2 ( D 1 )  is closed and discrete in M × M (b eing th e union of fin itely m an y sh if ts of the closed discrete sub space π 2 ( D 1 ) of M × M ). 3. The third item follo ws from the implication (4) ⇒ (1) of T heorem 3.2. 4. The four th item follo w s from Corollary 3.3. 5. No w assume th at S is a top ological semigroup and the space M is Tyc honoff. Being count ably compact at the dense su bset Z , the sp ace M is pseud o compact. 5a. If the space M is op enly f actoriza ble, then so is the space S according to Prop osition 1.8. By Prop osition 1.7, the space S is pseudo compact. Being s eparable, the square S × S is wea kly Lind el¨ of. By Theorem 2.4 , the S tone- ˇ Cec h compactificatio n β S is a top ological semigroup that contai ns the bicyclic semigroup C ( p, q ), which is forbid d en b y the theorem of K o c h and W allace [17]. 5b. Assum e that the square M × M is pseu do compact. Since M con tains a dense cyclic subgroup Z , by Theorem 2.5(2), the Stone- ˇ Cec h compactification β M is a compact topological group . Compact top ologica l groups, b eing Dugun d ji compact, are op enly factorizable, whic h implies th at β M is op enly factorizable. By P rop ositions 2.3 of [3], the op en f actoriza bilit y of the Stone- ˇ Cec h compactificatio n β M implies the op en factorizabilit y of the space M , whic h con tradicts the preceding item. 5c. Assume that the semigroup M contai ns a dense totally b ounded top ological sub group. By Th e- orem 2.5(1), the Stone- ˇ Cec h compactification β M of M is a compact top ological group. F urther we con tinue as in the preceding item.  8 T ARAS BANAKH, SVETLANA DIMITR OV A, AND OLEG GUTIK 5. The structure of top ological semigroups tha t cont ain bicyc lic semigroups In fact, man y p rop erties of the top ological semigroups C ( p, q ) ∪ π M , established in T h eorem 4.1 h old for any top olog ical semigroup conta ining a (dense) cop y of the bicyclic semigroup C ( p, q ). Theorem 5.1. If a top olo gic al semigr oup S c ontains the bicyclic semigr oup C ( p, q ) as a dense subsemi- gr oup, then (1) the c omplement S \ C ( p, q ) is a two-side d ide al in S ; (2) f or every c ∈ C ( p, q ) the set D c = { ( x, y ) : x, y ∈ C ( p, q ) , xy = c } is a close d-and-op e n discr ete subsp ac e of S × S ; (3) the squar e S × S is not pseudo c omp act; (4) β S is not op enly factorizable; (5) the almost p erio dic c omp actific ation AP( S ) of S i s a c omp act ab elian top olo gic al gr oup and henc e the c anonic homomorphism η : S → AP( S ) is not inje ctive. Pr o of. 1. The fact th at S \ C ( p, q ) is a tw o-sided ideal in S was prov ed b y Eb erhard and Selden in [9]. 2. Giv en any p oin t c ∈ C ( p, q ) w e should c hec k that the set D c = { ( x, y ) ∈ C ( p, q ) 2 : xy = c } is an op en-and -closed d iscrete subspace of S × S . By [9], the top ology on C ( p, q ) induced from S is discrete. Consequentl y , the subs p ace C ( p, q ), b eing discrete and dense in S , is op en in S . Then the square C ( p, q ) × C ( p, q ) is op en and discrete in S × S and s o is its su bspace D c . It remains to c hec k that the set D c is closed in S × S . Assuming th e opp osite, fi nd an accum ulation p oint ( a, b ) ∈ S × S of the subset D c . The conti nuit y of the semigroup op eration implies that ab = c . On the other han d , since the space C ( p, q ) × C ( p, q ) is discrete, one of the p oint s a, b b elong to the ideal S \ C ( p, q ) and hence ab ∈ S \ C ( p, q ) cannot b e equ al to c . 3. The space S × S fails to b e p s eudo compact b ecause it conta ins the infinite closed-and-op en discrete subspace D 1 = { ( x, y ) : x, y ∈ C ( p, q ) , xy = 1 } = { ( q n , p n ) : n ∈ ω } . 4. Assuming that the Stone- ˇ Cec h compactification β S of S is op enly factorizable, we ma y app ly Prop osition 2.3 of [3] to conclude that S is an op enly facto rizable p seudo compact space. S ince the space S has separable (hence w eakly Lindel¨ of ) square, w e can apply Theorem 2.4 to conclude that β S is a compact top olo gical semigroup th at con tains the b icyclic semigroup. But this is forbidden by the Hilden brandt-Ko ch T h eorem [15]. 5. Let η : S → AP( S ) b e the homomorph ism of S into its almost p eriod ic compactification. Th e restriction η |C ( p, q ) cann ot b e inj ectiv e b ecause compact top ological semigroups do not con tain b icyclic semigroups. Consequentl y , the image Z = η ( C ( p, q )) is a cyclic subgroup of AP( S ) b y Corollary 1.32 of [6]. Since C ( p, q ) is dens e in S , the subgroup Z is dens e in AP( S ). No w Theorem 2.5(2) guaran tees that AP( S ) is a compact ab elia n top olo gical group.  The follo wing th eorem extends (and corrects) T heorem 2.6 of [13]. Theorem 5.2. L e t S b e a top olo gi c al semigr oup c ontaining the bicyclic semigr oup C ( p, q ) as a dense subsemigr oup. If the sp ac e S is c ountably c omp act at the set E C = { p n q n : n ∈ ω } of the idemp otents of C ( p, q ) , then (1) the closur e ¯ E C of the set E C in S is c omp act and has a unique non-isolate d p oint e that c ommutes with al l elements of S ; (2) the map π : S → S , π : x 7→ x · e = e · x , is a c ontinuous homomorphism that r etr acts S onto the ide al M = S \ C ( p, q ) having the i demp otent e as a two-side d unit; (3) the element a = π ( p ) g ener ates a dense cyclic sub gr oup Z of M ; (4) π ( p n q m ) = a n − m for al l n, m ∈ ω ; (5) lim n →∞ p n + k q n = a k for every k ∈ Z ; (6) the sp ac e S is r e gular if and only if the sp ac e M = S \ C ( p, q ) is r e gular. (7) If the sp ac e S i s r e gular and c ountably c omp act at C ( p, q ) , then the semigr oup S is top olo gic al ly isomorph ic to C ( p, q ) ∪ π M and the subsemigr oup { ( q n e, p n e ) : n ∈ ω } is close d and discr ete in M × M . BICYCLIC SEMIGROUP IN COUNT ABL Y COMP ACT SEM IGR OUPS 9 (8) If the sp ac e S is T ychonoff and c ountably c omp act at C ( p, q ) , then the sp ac e M is not op enly factorizable, M × M i s not pseudo c omp act, and the semigr oup M c ontains no dense total ly b ounde d top olo gic al sub gr oup. Pr o of. 1. Th e set E C = { p n q n : n ∈ ω } of the idemp otent s of the bicyclic semigroup C ( p, q ) h as an accum u lation p oin t e ∈ ¯ E C b ecause S is counta bly compact at E C . W e claim that this accum ulation p oint e is u nique. Assume con versely that E C has another accumulati on p oint e ′ 6 = e . Then the pro d uct ee ′ differs from e or e ′ . W e lose no generalit y assum ing that ee ′ 6 = e ′ . S in ce S is Hausdorff, we ca n find t wo disjoin t op en sets O ( ee ′ ) ∋ ee ′ and O ′ ( e ′ ) ∋ e ′ . By the con tin uit y of th e semigroup op eration on S , there are t wo neighborh o o ds O ( e ) and O ( e ′ ) ⊂ O ′ ( e ′ ) of the p oin ts e , e ′ in S s u c h that O ( e ) · O ( e ′ ) ⊂ O ( ee ′ ). Since e is an accum ulation p oint of the set E C , we can find a num b er n ∈ ω suc h that p n q n ∈ O ( e ). By a similar r eason, there is a num b er m ≥ n such that p m q m ∈ O ( e ′ ). Then O ( e ′ ) ∋ p m q m = p n q n · p m q m ∈ O ( e ) · O ( e ′ ) ⊂ O ( ee ′ ) , whic h is n ot p ossible as O ′ ( e ′ ) and O ( ee ′ ) are disjoint . Therefore the set E C has a unique accumulati on p oint e . W e claim that the sequence { p n q n } ∞ n =0 con verges to the p oin t e . Otherwise, we would find a n eigh b orho o d O ( e ) su c h that the complement E C \ O ( e ) is infi nite and hence h as an accumula tion p oint e ′ ∈ S \ O ( e ) different fr om e , whic h is n ot p ossible. This prov es that th e sequence { p n q n } ∞ n =0 con verges to e and hence the set ¯ E C = E C ∪ { e } is compact and metrizable. Since the set E = { x ∈ S : xx = x } of idemp ote nts of S is closed, the accumulatio n p oint e of the set E C = E ∩ C ( p, q ) is an idemp oten t. Next, we sh o w that e comm utes w ith all the elemen ts of S . W e start with the elemen t p : p · e = p · lim k →∞ p k q k = lim k →∞ p k +1 q k = lim k →∞ p k +1 q k +1 p = e · p. By analogy w e can p ro v e that q · e = e · q . Moreo v er, pe · eq = peq = p · ( lim k →∞ p k q k ) · q = lim k →∞ pp k q k q = lim k →∞ p k +1 q k +1 = e, whic h means that the element s pe = ep and q e = eq are mutually in v erse. It follo ws that the element a = pe generates a cyclic sub group Z of S . W e claim th at for every n, m ∈ ω w e h a ve p n q m · e = e · p n q m = a n − m . Indeed, if n ≥ m , then p n q m · e = p n q m · lim k →∞ p k q k = lim k →∞ p n q m p k q k = lim k →∞ p n p k − m q k = = lim k →∞ p n − m p k q k = p n − m lim k →∞ p k q k = p n − m · e = ( pe ) n − m = a n − m . Similarly , e · p n q m = lim k →∞ p k q k p n q m = lim k →∞ p k q k − n q m = lim k →∞ p n − m p k − n + m q k − n + m = = p n − m lim k →∞ p k − n + m q k − n + m = p n − m · e = ( pe ) n − m = a n − m . By analogy w e can tr eat the case n ≤ m . Therefore, e comm utes w ith all elements of the bicyclic semigroup C ( p, q ). Consequent ly , the closed subset { x ∈ S : xe = ex } of S conta ins the den s e su bset C ( p, q ) of S and thus coincides with S , which means that th e idemp oten t e comm u tes with all elements of S . T aking into accoun t that the subs p ace C ( p, q ) is discrete in S [9], w e conclude that the idemp otent e , b eing an accumula tion p oin t of C ( p, q ), b elongs to the complement M = S \ C ( p, q ), w hic h is a tw o-sided ideal in S according to Theorem 5.1(1). C onsequen tly , xe = ex ∈ M for all x ∈ S . 2. It follo ws that the map π : S → M , π : x 7→ xe = ex , is a con tin uous homomorphism. Let us sho w that π ( x ) = x for ev er y x ∈ M . Assumin g the con v erse, fi nd x ∈ M with π ( x ) 6 = x . It is clear that x 6 = e . S ince S is Hausdorff, the p oints x, e and π ( x ) = xe = ex , hav e neighborh o ods O ( x ) , O ( e ) , O ( π ( x )) ⊂ S suc h th at O ( x ) · O ( e ) ∪ O ( e ) · O ( x ) ⊂ O ( π ( x )) and O ( x ) ∩ O ( π ( x )) = ∅ . T ak e 10 T ARAS BANAKH, SVETLANA DIMITR OV A, AND OLEG GUTIK an y id emp oten t p k q k ∈ O ( e ) ∩ C ( p, q ). The intersectio n O ( x ) ∩ C ( p, q ) is infi nite and hence conta ins a p oint p i q j ∈ O ( x ) ∩ C ( p, q ) such that i + j > 2 k . Then either i > k or j > k . If i > k , then p i q j = p k q k p i q j ∈ O ( x ) ∩ ( O ( e ) · O ( x )) ⊂ O ( x ) ∩ O ( π ( x )) = ∅ . If j > k , then p i q j = p i q j p k q k ∈ O ( x ) ∩ ( O ( x ) · O ( e )) ⊂ O ( x ) ∩ O ( π ( x )) = ∅ . Both cases lead to a con tradiction that completes the p r o of of the equalit y π ( x ) = x f or x ∈ M . This means that π r etracts S onto M . 3. As w e ha v e already prov ed, π ( p n q m ) = a n − m ∈ Z for every n , m ∈ ω . Sin ce C ( p, q ) is dense in S its image Z = π ( C ( p, q )) is dense in π ( S ) = M . 4–5. The statemen ts (4)–(5) h a ve b een p ro v ed in the first item. 6. If S is r egular, then so is its subspace M = S \ C ( p, q ). No w assume that M is regular. Giv en a p oint x ∈ S and an op en neighborho od U of x in S we need to fi nd a neighborh o o d V of x in S suc h that V ⊂ U . If th e p oint x is isolated, then we can pu t V = { x } . So, w e assum e th at x is non-isolated in S . In this case x ∈ M (b ecause C ( p, q ) is an op en discrete subspace of S b y [9]). By the regularit y of the space M the p oin t x has an op en neigh b orho o d W ⊂ M such that W ⊂ U . Th e con tin uit y of the retraction π : S → M implies that V = U ∩ π − 1 ( W ) is an op en neighb orh o o d of x in S . It is easy to c h ec k th at this neighb orh o o d has the required pr op ert y: V ⊂ U . 7. Assume that S is regular and coun tably compact at C ( p, q ). W e claim that the id en tit y map h : S → C ( p, q ) ∪ π M is a h omeomorph ism. The contin uity of this map follo ws from th e con tin uit y of the map π : S → M and the definition of the top ology of C ( p, q ) ∪ π M . Since eac h p oint of C ( p, q ) is isolated in C ( p, q ) ∪ π M , the inv erse identit y map h − 1 : C ( p, q ) ∪ π M → S is con tin uous at the set C ( p, q ). So, it remains to c hec k the con tin uit y of h − 1 at a p oint x ∈ M . Giv en any neigh b orho od U of x in S , w e need to find a neighborh o od V of x in C ( p, q ) ∪ π M suc h that V ⊂ U . By the r egularit y of M , the p oint x has an op en n eigh b orho o d W in M suc h that W ⊂ U . W e claim that the set F = π − 1 ( W ) \ U is finite. Otherw ise, by th e coun table compactness of S at C ( p, q ), w e can find an accum ulation p oint y of F . Since F ⊂ S \ U , the p oint y b elongs to th e closed su bset S \ U of S . Since C ( p, q ) is d iscrete in S , the p oint y , b eing non-isolated in S , b elongs to the complemen t M = S \ C ( p, q ). The contin uity of the retraction π : S → M implies th at y = π ( y ) is an accum ulation p oin t of the set π ( F ) ⊂ W and hence y ∈ π ( F ) ⊂ W ⊂ U , whic h contradicts y ∈ S \ U . Thus F is fin ite, and the set V = π − 1 ( W ) \ F is a required n eigh b orho o d of x in C ( p, q ) ∪ π M with V ⊂ U . Th us S is top ologically isomorphic to C ( p, q ) ∪ π M and hence C ( p, q ) ∪ π M is a top olo gical semigroup. By Th eorem 4.1(2), the su bsemigroup π 2 ( D 1 ) = { ( q n e, p n e ) : n ∈ ω } is closed and discrete in M × M . 8. If S is Tyc honoff and counta bly compact at C ( p, q ), then S is top olog ically isomorph ic to C ( p, q ) ∪ π M b y the pr eceding item. No w Theorem 4.1(5) implies that the space M is not op enly factorizable, M × M is not p s eudo compact, and the semigroup M con tains no dense totally b ounded top ological su bgroup.  6. A count abl y (pr a)comp act sem igr oup tha t cont ains C ( p, q ) In this section w e shall construct a coun tably (pra)compact top ological semigroup conta ining a bicyclic semigroup. Our main result is: Theorem 6.1. The bicyc lic su b gr oup is a subsemigr oup of some T ychonoff c ountably pr ac omp act top o- lo gi c al semigr oup. The pro of of this theorem relies on four lemmas. Lemma 6.2. A sub gr oup H of a top olo gic al gr oup G c ontains a non-trivial c onver ge nt se qu enc e if and only if H c ontains a non-trivial se quenc e that c onver ges in G . Pr o of. If a sequence { x n } n ∈ ω ⊂ H con verge s to a p oint x ∈ G \ H , then ( x − 1 n +1 x n ) n ∈ ω is a n on-trivial sequence in H that conv erges to the neutral elemen t e = x − 1 x .  The follo wing well-kno wn lemma can b e pr o ved by a standard argument inv olving bin ary trees. BICYCLIC SEMIGROUP IN COUNT ABL Y COMP ACT SEM IGR OUPS 11 Lemma 6.3. If a T ychonoff sp ac e X is c ountably c omp act at an i nfinite subset H ⊂ X that c ontains no non-trivial se que nc e that c onver ge s in X , then the closur e cl X ( A ) of any infinite subset A ⊂ H has c ar dinality ≥ c . A subset L of an ab elian group G is called line arly indep endent if for an y pairwise distinct p oin ts x 1 , . . . , x k ∈ L and an y intege r num b ers n 1 , . . . , n k the equ ality n 1 x 1 + · · · + n k x k = 0 implies n 1 = · · · = n k = 0. It is easy to see that L ⊂ G is linearly indep enden t if and only if for the free ab elian group F A( L ) generated by L the un ique h omomorphism h : F A( L ) → G suc h that h | L = id L is in jectiv e. F or a linearly indep enden t su b set L ⊂ G we sh all identify the free ab elian group F A( L ) with the subgroup of G generated by L . Lemma 6.4. L et an Ab elian torsion-fr e e top olo gic al gr oup G is c ountably c omp act at a sub gr oup H ⊂ G that c ontains no non-trivial c onver gent se quenc e. Each line arly i ndep endent subset L 0 ⊂ G of size | L 0 | < c c an b e e nlar ge d to a line arly indep endent subset L ⊂ G of size | L | = c such that the set L \ L 0 c ontains an ac cumulation p oint of e ach infinite sub set A ⊂ F A( L ) ∩ H ⊂ G . Pr o of. Without loss of generalit y , L 0 6 = ∅ . T ak e any faithfully indexed set X = { x α : α < c } of cardinalit y | X | = c suc h that X ∩ L 0 = ∅ and consider the free ab elian group F A( L 0 ∪ X ) generated b y the un ion L 0 ∪ X . F or ev ery ordinal α < c let X <α = L 0 ∪ { x β : β < α } and X ≤ α = L 0 ∪ { x β : β ≤ α } . So, L 0 ∪ X = X < c . Denote b y A th e set of all countable su bsets of the free ab elia n group F A( X < c ). Since | F A ( X < c ) | = c , the set A has size | A | = c ω = c . T o eac h set A ∈ A assign th e smallest ord in al ξ ( A ) ≤ c such that A ⊂ F A ( X <ξ ( A ) ) and observ e that ξ ( A ) < c b ecause c has uncount able cofinalit y . It follo ws that ξ ( A ) = 0 if and only if A ⊂ F A( L 0 ). W e claim that there is an en umeration A = { A α : α < c } of th e s et A such that ξ ( A α ) ≤ α for every ordinal α < c . T o construct su c h an en umeration, first fix any enumeration A = { A ′ α : α < c } suc h that A ′ 0 ⊂ F A( L 0 ) and f or ev ery A ∈ A the set { α < c : A ′ α = A } h as the size con tin uum. Next, f or ev ery α < c put A α = ( A ′ α if ξ ( A ′ α ) ≤ α A ′ 0 otherwise . The identit y inclusion X < 0 = L 0 ⊂ G extend s to a unique group homomorphism h < 0 : F A( X < 0 ) → G whic h is in jectiv e b ecause of th e linear ind ep endence of L 0 . Inductive ly , for eac h ordinal α < c w e sh all construct an inj ectiv e h omomorphism h α : F A( X ≤ α ) → G suc h that • h α | F A( X ≤ β ) = h β for all β < α ; • if h α ( A α ) ⊂ H , th en the p oin t ¯ x α = h α ( x α ) ∈ G is an accumulatio n p oin t of the set h α ( A α ). W e start with c ho osing a p oin t ¯ x α . Consider the injectiv e group homomorphism h <α : F A( X <α ) → G suc h that h <α | F A( X ≤ β ) = h β for all β < α . Th e image h <α (F A( X <α )) is a free ab elian subgroup of size < c in G . Cons ider the subgroup G <α =  x ∈ G : ∃ n > 0 n x ∈ h <α (F A( X <α ))  . Since G is torsion-free, | G <α | ≤ ℵ 0 · | F A( X <α ) | < c . Since the homomorphism h <α : F A ( X <α ) → G is injectiv e, the set B α = h <α ( A α ) is infinite. If B α ⊂ H , then by Lemmas 6.2 and 6.3 , the closure B α of B α in G has cardinalit y | B α | ≥ c . Consequently , w e can fin d a p oin t ¯ x α ∈ B α \ G <α . If B α 6⊂ H , th en tak e ¯ x α b e an y p oin t of the set H \ G <α . Such a p oint ¯ x α exists b eca use the closure H of H in G h as cardinalit y | H | ≥ c > | G <α | . The c hoice of the p oin t ¯ x α / ∈ G <α guaran tees that the in jectiv e homomorphism h <α extends to an injectiv e homomorph ism h α : F A( X ≤ α ) → G suc h that h α ( x α ) = ¯ x α . This completes th e in ductiv e step as well as the ind uctiv e construction. No w consider the injective h omomorp hism h = h < c : F A( X < c ) → G and observe that the image L = h ( X < c ) of X < c = L 0 ∪ X is a linearly in dep endent subset of G . By the choic e of the h omomorphism h < 0 , w e h a ve L 0 = h ( L 0 ) ⊂ L . W e claim th at the sub group F A( L ) = h (F A( X < c ) of G generated by the s et L is coun tably compact at th e s u bset H ∩ F A( L ). T ake any coun table infin ite sub set B ⊂ H ∩ F A( L ) and consider its preimage 12 T ARAS BANAKH, SVETLANA DIMITR OV A, AND OLEG GUTIK A = h − 1 ( B ) ⊂ F A( X < c ). It f ollo ws th at A = A α for some α < c . The c h oice of the p oin t ¯ x α ∈ L \ L 0 guaran tees that ¯ x α is an accum u lation p oin t of the set B = h ( A α ).  A (top ological) semigroup S is called a ( top olo gic al ) monoid if S has a tw o-sided u n it 1. The subgroup H 1 = { x ∈ S : ∃ y ∈ S xy = y x = 1 } is called the maximal sub gr oup of a monoid S . F or any subset B b y FM( B ) w e d enote the free ab elia n monoid generated by M . This is the subsemigroup of the f ree ab elian group F A( B ) generated b y the set B ∪ { 1 } , wh ere 1 is the neu tr al elemen t of F A( B ). Lemma 6.5. Assume that a torsion-fr e e Ab elian top olo gic al gr oup G i s c ountably c omp act at a dense infinite cyclic sub gr oup Z ⊂ G that c ontains no non-trivial c onver gent se q uenc e. Then ther e is a T ychonoff c ountably pr ac omp act top olo g i c al monoid M such that (1) M is algebr aic al ly isomorphic to the dir e ct sum Z ⊕ FM( c ) ; (2) the maximal sub gr oup H 1 of M is cyclic, discr ete, and dense i n M ; (3) M \ H 1 is an ide al in M ; (4) M admits a c ontinuous one-to-one homomor phism h : M → G such that h ( H 1 ) = Z ; (5) the semigr oup M is c ountably c omp act pr ovide d the gr oup G is c ountably c omp act and c ontains no non-trivial c onver gent se qu e nc e. Pr o of. Let H = Z if G is n ot coun tably compact and H = G if G is coun tably compact. L et a ∈ Z b e a generator of the cyclic group Z . By L emm a 6.4, the linearly indep enden t set L 0 = { a } can b e enlarged to a linearly in dep endent s u bset L ⊂ G of size | L | = c that generates the (fr ee ab elian) sub group F A( L ) in G such that for eac h infi nite subs et A ⊂ H ∩ F A( L ) ⊂ G the closure ¯ A meets the s et L \ L 0 . Let M b e the subsemigroup of G generated by the set {− a, a } ∪ L . Since eac h infinite subset of Z ⊂ H ∩ F A( L ) has an accum u lation p oin t (in L ⊂ M ), th e s pace M is coun tably compact at the subset Z . If G is countably compact, then H = G and th en M is counta bly compact b eca use eac h in fi nite subset of M ⊂ H ∩ F A( L ) has an accumulat ion p oint in L ⊂ M . It is clear that M is a monoid whose maximal subgroup H 1 coincides with Z and thus is dense in M . Also it is clear th at M is algebraical ly isomorp h ic to Z ⊕ FM( c ). No w we enlarge the top ology τ on M induced from G in order to mak e the maximal sub group Z = H 1 discrete. It is easy to see that th e top ology τ ′ = { U ∪ A : U ∈ τ , A ⊂ Z } on M has th e required p rop erty: Z b ecomes discrete but remains dense in this top olo gy . It is easy to c hec k that the space M end o wed with this stronger topology remains a top ologica l semigroup (this follo ws from the fact that M \ Z is an ideal in M ). Moreo v er, th e top ological s p ace ( M , τ ′ ) is T yc honoff, see [10, 5.1.22 ]. It r emains to chec k that the space ( M , τ ′ ) is coun tably compact at H 1 . T ak e an y infinite subset A ⊂ H 1 = Z . By Lemma 6.3, the closure ¯ A of A in the top ology τ has size | ¯ A | ≥ c and consequen tly , ¯ A con tains a p oin t a / ∈ Z . It follo ws f rom the d efinition of the top ology τ ′ that the p oin t a remains an accum u lation p oin t of the set A in the top olo gy τ ′ . If the group G is countably compact, then so is the semigroup M and the p receding argument ensures that M remains countably compact in the stronger top ology τ ′ .  No w we are ready to present the Pro of of Theorem 6.1 . Fix an Ab elian torsion-free top ologica l group G whic h is countably compact at a den s e infin ite cyclic subgroup Z ⊂ G con taining no non-trivial con v ergen t sequence. F or G w e can tak e the Bohr compactificat ion b Z of th e group of in tegers Z and for Z the im age Z ♯ of Z in b Z . It is w ell-kno wn that the Bohr compactification b Z is torsion-free and its subgroup Z ♯ con tains no n on-trivial con vergen t s equence, see [8] or [11]. By Lemmas 6.4 and 6.5 , there is a comm utativ e Tyc honoff top ological monoid M su ch that the maximal subgroup H 1 of M is cyclic, discrete, and d en se in M , M is countably compact at H 1 , and M \ H 1 is an ideal in M . Let h : Z → H 1 b e an y isomorphism. Define a homomorp hism π : C ( p, q ) → M letting π ( p n q m ) = h ( n − m ) for n, m ∈ ω . By T heorem 4.1(3), the semitop ological semigroup S = C ( p, q ) ∪ π M is a top ologica l semigroup. By Pr op ositions 1.1 and 1.6, the space S is T yc honoff and coun tably p racompact. BICYCLIC SEMIGROUP IN COUNT ABL Y COMP ACT SEM IGR OUPS 13 Moreo ve r, if th e group G is countably compact and con tains no n on -trivial con v ergen t sequ ence, then the semigroup M is co untably compact acco rding to Lemma 6.5(5), and then the semigroup S is co untably compact by Prop osition 1.4.  Let us remark that the ab o v e p ro of yields a bit more than requir ed in Th eorem 6.1, n amely: Theorem 6.6. If ther e is a torsion-fr e e Ab elian c ountably c omp act top olo gic al gr oup G without non-trivial c onver gent se quenc es, then ther e exists a T ychonoff c ountably c omp act semigr oup S c ontaining a bicyclic semigr oup. The first example of a grou p G with pr op erties r equired in Theorem 6.6 w as constructed by M. Tk a- c h enk o under the Contin uu m Hyp othesis [25]. Later, the Con tin uum Hyp othesis wa s we ak ened to Mar- tin’s Axiom for σ -cen tered p osets b y A. T omita in [26], for counta ble p osets in [18], and fi nally to the existence of con tinuum many incomparable selectiv e ultrafilters in [19]. Y et, the problem of th e existence of a countably compact group without conv ergen t sequences in ZFC seems to b e op en, see [7]. Those consistency r esults com bined with Th eorem 6.6 imply Corollary 6.7. Martin ’s Axi om implies the existenc e of a T ychonoff c ountably c omp act top olo gic al semi- gr oup S that c ontains a bicyclic semigr oup. Remark 6.8. By Th eorem 5.1(5), the almost p erio d ic compactification AP( S ) of the countably (pra)- compact semigroup S ⊃ C ( p, q ) constructed in Theorem 6.6 (or 6.1) is a compact top ologica l group. Consequent ly , the canonic homomorph ism η : S → AP( S ) is not in jectiv e in contrast to the canonic homomorphism η : S → W AP( S ) = β S w h ic h is a top ologica l emb edding by Theorem 2.2. In p articular, S is a coun tably (pra)compact top ologic al semigroup that d o es not em b ed in to a compact top ological semigroup. 7. Some Open Problems The consistency nature of Th eorem 6.6 and Corollary 6.7 suggests: Problem 7.1. Is ther e a Z F C-example of a c ountably c omp act top olo gic al semigr oup that c ontains the bicyclic semigr oup? Another op en problem w as s u ggested by the referee: Problem 7.2. Is ther e a pseudo c omp act top olo gic al semigr oup S that a c ontains the bicyclic semigr oup as a close d subsemigr oup? Theorem 6.1 giv es an example of a counta bly pracompact top olog ical s emigroup S for which the canonical homomorph ism η : S → AP( S ) is not injectiv e. Problem 7.3. Is ther e a non-trivial c ountably (pr a)c omp act top olo gic al semigr oup S whose almost p eri- o dic c omp actific ation AP( S ) is a singleton? 8. 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Instytut Ma tem a tyki, Un iwersytet Humani styczno-Przyro dniczy Jana Ko chanowsk iego w K ielcach, Poland, and De p ar tment of Ma thema tics, L viv Na tional Uni versity, Unive rsytetska 1, 79000, Ukraine E-mail addr ess : T.O.Banakh@gmail.c om Na tional Te chnical University “Kharkiv Pol ytechnical Institute”, Frunze 21, Kharkiv, 61002, Ukraine E-mail addr ess : s.dimitrova@mail.r u Dep ar tmen t of Mechanics and Ma thema ti cs, Iv an Frank o L viv Na tional University, Universytetska 1, L vi v, 79000, U kraine E-mail addr ess : ovgutik@yahoo.com