The Rees-Suschkewitsch Theorem for simple topological semigroups

THE REES-SUSCHKEWITSCH THEOREM F OR SIMPLE TOPOLOGICAL SEMIGROUPS T ARAS BANAKH, SVETLANA DIMITRO V A, AND OLEG GUTIK Abstract. W e detect topological semigr oups that are topological paragroups, i.e., are isomorphic to a Rees pro duct [ X × H × Y ] σ of a top ological group H ov er top ological spac es X, Y with a con tin uous sandwic h function σ : Y × X → H . W e prov e that a simple top ological semigroup S is a topological paragroup if one of the foll o wing conditions is satisfied: (1) S is completely simple and the maximal subgroups of S are topological groups, (2) S con tains an i dempotent and the square S × S i s coun tably compact or pseudocompact, (3) S is sequen tially compact or the pow er S 2 c is counta bly compact . The last item generalizes an old W allace’s result s a ying that eac h simple compact topological semigroup is a top ological paragroup. This pap er was motiv ated b y the cla ssical Rees-Suschk ewitsch Theorem that de- scrib es the algebraic str ucture of co mpletely simple semigro ups and the topo logical versions o f this theorem proved for compact top ologica l semigroups by W allace [28], for compact semitop olog ical semig roups by Rupper t [25], and fo r sequential count- ably co mpact top olog ical semigr o ups by Gutik, Pagon and Rep ov ˇ s [14]. All top o- logical se migroups co nsidered in this pap er are Hausdorff. W e recall tha t a semigr oup S is simple if S contains no pro pe r tw o -sided idea l. A simple semigr oup S is called c ompletely simple if the s et E = { e ∈ S : ee = e } of idempo tent s of S contains a primitive idemp otent , that is, a minimal idemp otent with resp ect to the partial order e ≤ f on E defined by ef = f e = e . In this case all the idempo tents ar e primitive and H e = eS e is a group for every e ∈ E , see [7, Section 2.7 , Ex . 6(b)] or [29]. Let us observe that each g r oup is a completely simple semigro up. Less triv ial examples of such semigro ups app ea r as minimal ideals in co mpact right-topolo gical semigroups, see [25, Theorem I.3.1 3], [16, Theorem 2.9] and [24]. A generic example of a c o mpletely s imple semigro up can b e construc ted as follows. T ak e any g roup H and a function σ : Y × X → H defined on the pro duct of t wo sets. This function σ induces the semig roup op eration ( x, h, y ) · ( x ′ , h ′ , y ′ ) = ( x, hσ ( y , x ′ ) h ′ , y ′ ) on the pr o duct X × H × Y turning it into a completely simple semigr oup, called the R e es pr o duct of H over X and Y r elative to the sandwich map σ [17] or a p ar agr oup [2 5] and denoted by [ X , H, Y ] σ . The Rees-Suschk ewitsch Structure Theorem [22] says that the conv erse is also true: each completely simple semig roup S is isomor phic to the parag roup [ X e , H e , Y e ] σ where e is any idempo tent of S , H e = eS e is the maximal subgroup of S containing Date : October 29, 2018. 2000 Mathematics Subje ct Classific ation. Primar y 22A15, 20M20. Secondary 20M18, 54H15. Key wor ds and phr ases. T opological semi group, semitopological semi gr oup, Rees- Susc hk ewitsc h Theorem, topological paragroup. 1 2 T ARAS BANAKH, S VETLANA DIMITRO V A, AND OLEG GUTIK e , X e = S e ∩ E , Y e = eS ∩ E , and the sandwich function σ : Y e × X e → H e is defined b y σ ( y , x ) = y x . In fact, the map R : [ X e , H e , Y e ] σ → S, R : ( x, h, y ) 7→ xhy , is an isomorphism ca lled the Re es isomorphism . Its in verse R − 1 : S → [ X e , H e , Y e ] σ is defined by the formula R − 1 ( s ) = ( s ( ese ) − 1 , ese, ( ese ) − 1 s ) . Now assume that S is a topo logical semig roup (i.e., a top ologica l space endowed with a co nt inuous s emigroup op eratio n). In this ca s e the spa ces X e = S e ∩ E , H e = eS e , and Y e = eS ∩ E ca rry the induced top ologie s while the par agro up [ X e , H e , Y e ] σ carries the pro duct top olog y making the semigro up op er ation contin- uous, i.e., [ X e , H e , Y e ] σ is a topo logical semigroup. Le t us observe that the maximal subgroup H e of S is a paratop olog ical gr oup, whic h mea ns that the gr oup mult i- plication on H e is jointly contin uous. If, in addition, the inv ersion map x 7→ x − 1 is contin uous on H e , then H e is a t op olo gic al gr oup . Lo oking at the Rees isomo rphism R : [ X e , H e , Y e ] σ → S, R : ( x, h, y ) 7→ xhy , we s e e that it is contin uous while its inv erse R − 1 : s 7→ ( s ( ese ) − 1 , ese, ( ese ) − 1 s ) is contin uous if the paratop olog ical gro up H e is a top ologica l g roup. In this case the to po logical s emigroup [ X e , H e , Y e ] σ is called a top olo gic al p ar agr oup [1 7]. More precisely , by a top olo gic al p ar agr oup we understand a top olo g ical se migroup that is topo logically isomorphic to the Rees pro duct [ X , H , Y ] σ where H is a top o - logical group and σ : X × Y → H is a contin uous function defined on the pro duct of tw o top olo gical spa c es. In such a wa y w e hav e obtained the following topo logical Rees-Suschk ewitsch structure theorem. Theorem 1. A t op olo gic al semigr oup S is a top olo gic al p ar agr oup if and only if S is c ompletely simple and e ach maximal sub gr oup H e of S is a t op olo gic al gr oup. There is a simple algebra ic ch arac terization of c ompletely s imple semig roups, see [7, Theo rem 2 .54] and [2]. Theorem 2 (Andersen) . A semigr oup S is simple if and only if S has an idemp o- tent but c ontains no isomorphic c opy of the bicyclic semigr oup C ( p, q ) . W e r ecall that C ( p, q ) is a se mig roup with a two-sided unit 1 , gener ated by tw o elements p , q and one relation q p = 1. Combining the Andersen Theorem 2 with Theorem 1 we obta in another charac- terization of top olo gical par agro ups. Theorem 3. A t op olo gic al semigr oup S is a top olo gic al p ar agr oup if and only if S is simple, c ont ains an idemp otent, c ontains n o c opy of the bicyclic gr oup, and e ach maximal sub gr oup H e of S is a t op olo gic al gr oup. F or co mpact topo logical semigroups the la st thr ee conditions alwa ys are satisfied: such semigro ups contain an idemp otent by the Iwassaw a-Numakura Theo rem (see [18, 1 1, 20, 27] or [6, V ol. 1, Theo rem 1 .8 ]), contain no copy of the bicyclic semigro up THE REES-SUSCHKEWITSCH THEOREM 3 by the Ko ch-W allace Theor em [1 9] and the maxima l subgroups H e = eS e co rre- sp onding to minimal idemp otents a re top olo gical g roups, b eing compact par atop o- logical g roups, see [9]. In such a wa y we hav e prov ed the fo llowing theore m due to W allace [28]. Theorem 4 (W allace) . Each simple c omp act t op olo gic al semigr oup S is top olo gi- c al ly isomorphi c t o a top olo gic al p ar agr oup. In [25, Theorem I.5.3] the W allace Theorem w as g eneralized to compa ct se mi- top ological semigro ups. By a semitop olo gic al semigr oup we unders ta nd a Hausdor ff top ological space S endow ed with a separa tely continous semigro up op eration. Theorem 5 (Rupp ert) . Ea ch simple c omp act semitop olo gic al semigr oup S is t op o- lo gic al ly isomorphic to a top olo gic al p ar agr oup. Another directio n of gener alization of the W allace Theorem consists in replacing the c o mpactness assumption by a weak er prop erty . T he fir st step in this directio n was made in [14]. Theorem 6 (Gutik-Pagon-Rep ovs) . Each simple se quential c ountably c omp act top olo gic al semigr oup is a top olo gic al p ar agr oup. In this pap er we ge ne r alize b oth the W allace and Gutik-Pagon-Rep ov ˇ s The o - rems proving that simple top o lo gical semigro ups satisfying certa in compactness- like pr o p erties ar e top ologica l paragro ups. All top ologica l spaces co nsidered in this pap er a re assumed to b e Hausdor ff. W e reca ll that a top ologic a l space X is • c ountably c omp act if ea ch clo sed discrete subspa ce of X is finite; • pseudo c omp act if X is Tychono v and ea ch co nt inuous rea l-v a lue d function on X is b ounded; • se quential ly c omp act if each s equence { x n } n ∈ ω ⊂ X has a co nv ergent sub- sequence; • p -c omp act for some free ultrafilter p on ω if each seq uence { x n } n ∈ ω ⊂ X has a p -limit x ∞ = lim n → p x n in X . Here the no ta tion x ∞ = lim n → p x n means that fo r each neig hborho o d O ( x ∞ ) ⊂ X of x ∞ the set { n ∈ ω : x n ∈ O ( x ∞ ) } b elo ngs to the ultrafilter p . It is clear that each sequentially compact and ea ch compact topo logical space is p -compact for every ultr afilter p . By [12], a to p o logical space X is p -compact for some fr ee ultrafilter p o n ω if and o nly if each p ower X κ of X is countably compact if and o nly if the p ow er X 2 c is countably compa ct. It is easy to s ee that each sequence ( x n ) n ∈ ω in a countably compact top olo gical s pace X has p - limit lim n → p x n for some free ultrafilter p on ω . W e shall say that for s o me free filter p on ω a double sequence { x m,n } m,n ∈ ω ⊂ X has a double p -limit lim n → p lim m → p x m,n if P = { n ∈ ω : ∃ lim m → p x m,n ∈ X } ∈ p and the sequence ( lim m → p x m,n ) n ∈ P has a p -limit in X . W e define a top olog ic al space X to b e doubly c ountably c omp act if ea ch double sequence ( x m,n ) m,n ∈ ω in X has a double p - limit lim n → p lim m → p x m,n ∈ X for so me free ultrafilter p on ω . Prop ositi on 1 . A top olo gic al sp ac e X is doubly c ountably c omp act if X is either se qu ential ly c omp act or p -c omp act for some fr e e ultr afilter p on ω . 4 T ARAS BANAKH, S VETLANA DIMITRO V A, AND OLEG GUTIK Pr o of. The double c o unt able compactness o f p -co mpact spaces is obvious. Now assume that X is sequentially compact a nd take any double s e quence ( x m,n ) m,n ∈ ω in X . By the sequential compactness o f X there is an infinite subset A 0 ⊂ ω such that the subsequence ( x m, 0 ) m ∈ A 0 conv erges to some p o in t x 0 ∈ X in the sense that for ea ch neighbo r ho o d O ( x 0 ) ⊂ X the set { n ∈ A 0 : x m, 0 / ∈ O ( x 0 ) } is finite. Now consider the sequence ( x m, 1 ) m ∈ A 0 and by the s e quential compac tnes s of X find an infinite subset A 1 ⊂ A 0 such that the subseq uence ( x m, 1 ) m ∈ A 1 conv erges to some po int x 1 . Next we pro ceed by induction and for e very n ∈ ω construct an infinite subset A n ⊂ A n − 1 such that the seq ue nc e ( x m,n ) m ∈ A n conv erges to so me p oint x n ∈ S . Now take any infinite subset A ⊂ ω such that for A ⊂ ∗ A n for every n ∈ ω . The la tter means that the complement A \ A n is finite. It follows that for every n ∈ ω the sequence ( x m,n ) m ∈ A conv erges to the p oint x n . By the sequential compactness of S for the sequence ( x n ) n ∈ A there is a n infinite subset B ⊂ A such that the sequence ( x n ) n ∈ B conv erges to some p oint x ∈ X . Finally , take any fre e ultrafilter p ∋ B and observe that x = lim n → p lim m → p x m,n .  Theorem 3 ensures that a simple top ologica l semigroup S is a top ologica l para- group provided (1) S has an idemp otent; (2) S contains no copy of the bicyclic semigr oup; (3) all maximal subgro ups of S a re top olog ical gro ups. T opolo gical semigroups containing an idempo tent ca n be characterized as follows. Theorem 7. A t op olo gic al semigr oup S c ontains an idemp otent if and only if for some x ∈ S the double se quenc e ( x m − n ) m ≥ n has a double p -limit lim n → p lim m → p x m − n ∈ S for some fr e e ultr afilter p on ω . Pr o of. The “only if ” par t is trivial: just take a ny idemp otent x o f X and observe that lim n → p lim m → p x m,n = x for any free ultrafilter p on ω . T o prove the “ if ” part, assume that for some x ∈ S the double sequence ( x m − n ) m ≥ n has a do uble p - limit e = lim n → p lim m → p x m − n for so me free ultrafilter p on ω . W e claim that e is a n idemp otent. Let P ∈ p b e the set of the num bers n for which ther e is a p -limit e − n = lim m → p x m − n in S . Then e = lim P ∋ n → p e − n . Assuming that e fails to b e an idemp otent, w e ca n find a neig hborho o d O ( e ) ⊂ S of e s uch that O ( e ) · O ( e ) is dis joint with O ( e ). Since e = lim P ∋ n → p e − n , the s e t P 1 = { n ∈ P : e − n ∈ O ( e ) } b e lo ngs to the ultrafilter p . T ak e any element n ∈ P 1 and observe that lim m → p x m − n = e − n ∈ O ( e ) implies P 2 = { m ∈ P 1 : m > n and x m − n ∈ O ( e ) } ∈ p . Pick any m > n in P 1 and observe that lim i → p x i − m = e − m ∈ O ( e ) a nd th us the set P 3 = { i ∈ P 2 : i > m and x i − m ∈ O ( e ) } belo ngs to p . Now ta ke any num ber i ∈ P 3 and observe that i ∈ P 3 ⊂ P 2 and m ∈ P 2 imply x i − n , x i − m , x m − n ∈ O ( e ). On the o ther hand, x i − n = x i − m x m − n ∈ O ( e ) · O ( e ) ⊂ S \ O ( e ), which is a desir ed contradiction.  This characterization will b e applied to o btain so me conv enient conditions o n a top ological semigroup X guara n teeing the existence of a n idempo tent e ∈ S . Theorem 8. A top olo gic al semigr oup S c ontains an idemp otent if S satisfies one of the fol lowing c onditions: THE REES-SUSCHKEWITSCH THEOREM 5 (1) S is doubly c ountably c omp act; (2) S is se quential ly c omp act; (3) S is p - c omp act for some fr e e ultr afilter p on ω ; (4) S 2 c is c ountably c omp act; (5) S κ ω is c oun t ably c omp act, wher e κ is the minimal c ar dinality of a close d subsemigr oup of S . Pr o of. The first item follows immedia tely fro m Theore m 7 and the definiton of a doubly se q uent ially countably compact space. The next tw o a ssertions follow fro m the fir st one and Prop o s ition 1. The fourth assertion follows from the third one and the characteriz ation of spaces with count- ably co mpact p ow er S 2 c as p -compact spaces for some fre e ultrafilter p , see [12]. It rema ins to prove the last ass ertion. Let κ b e the s mallest cardinality of a closed subsemigroup of S and assume that the p ower S κ ω is countably co mpact. Replacing S by a suita ble c losed subsemigr o up, we ca n assume that | S | = κ . Now it suffices to prov e that the spa c e S is p -co mpact for some free ultrafilter p . F or every n ∈ ω co ns ider the functional δ n : S ω → S assigning to e a ch function f ∈ S ω its v alue δ n ( f ) = f ( n ) at n . This functional is an element of the p ow er S S ω . The countable compactness of S S ω guarantees that the sequence ( δ n ) n ∈ ω has an accumulation po int δ ∞ ∈ S S ω and hence δ ∞ = lim n → p δ n for some free ultrafilter p on ω . Then every function f ∈ S ω has the p - limit lim n → p f ( n ) = lim n → p δ n ( f ) = δ ∞ ( f ) , which mea ns that the space S is p -compact.  Remark 1 . Theorem 8 g e ne r alizes many known results r elated to ide mp otents in top ological semigro ups. In particular, it gener alizes a r esult of A. T omita [26] on the existence of idemp otents in p -compact cancellative semigroups as well as the classical Iwassaw a-Numakura Theo rem [6, V ol.1, Theo rem 1.6] on the existence o f an idemp otent in compa ct top olog ical semigr oups. The following theo rem generaliz ing b oth the W allace and Gutik-Pagon-Rep ov ˇ s Theorems is the main result of this note. Theorem 9. A simple top olo gic al semigr oup S is a top olo gic al p ar agr oup if S is doubly c ountably c omp act and has c ountably c omp act squar e S × S . This theorem follows immediately from Theorem 7 a nd the following character- izing Theorem 10. A top olo gic al semigr oup S with c ountably c omp act squar e S × S is a top olo gic al p ar agr oup if and only if S is simple and c ontains an idemp otent . Pr o of. The “only if ” part is trivial. T o prove the “if ” par t, assume that S is simple and co ntains a n idemp otent. First we c heck that S contain no co py of C ( p, q ). Assume conv ersely that C ( p, q ) ⊂ S and c o nsider the sequence { ( q n , p n ) ∞ n =1 } in C ( p, q ) × C ( p, q ) ⊂ S × S . The countable compac tnes s of S × S guara ntees that this sequence has a n accumulation po int ( a, b ) ∈ S × S . Since q n p n = 1, the contin uity o f the semigroup op era tion on S guara ntees that ab = 1. By Co rollar y I.2 [8], the bicyclic semigro up C ( p, q ) endow ed with the top o lo gy induced fro m S is a discrete top olo g ical space. So, we can find a neighborho o d O (1 ) ⊂ S o f 1 ∈ C ( p, q ) containing no other p oints of the 6 T ARAS BANAKH, S VETLANA DIMITRO V A, AND OLEG GUTIK semigroup C ( p, q ). Since ab = 1, the po in ts a, b hav e neig h b orho o ds O ( a ) , O ( b ) ⊂ S such that O ( a ) · O ( b ) ⊂ O (1). Since a is an a c cum ulation p oint of the sequence q n , we can find n ∈ N with q n ∈ O ( a ). By the s ame reaso n, there is a num ber m > n such tha t p m ∈ O ( b ). Then q n p m = p m − n ∈ O ( a ) · O ( b ) ∩ C ( p, q ) = { 1 } , which is a contradiction. T his contradiction shows that the simple semigro up S contains no copy of C ( p, q ) and hence is co mpletely simple by the Andersen’s Theo r em 2. F or each idemp otent e ∈ E the ma ximal se mig roup H e = e S e is countably compact, b eing a contin uous image o f the coun tably co mpact space S . More over, the sq uare H e × H e is count ably c ompact, b eing a contin uous imag e of the countably compact space S × S . Then H e is a top ologica l gr o up, b eing a para top o logical g roup with countably compact square , see [21] or [1, 2 .2]. Now Theorem 3 assur es that S is a top olog ical para group.  F or Tyc honov top ologic al semigroups the countable compactness of the square S × S in the pre ceding theor em ca n b e replaced by its pseudo compactness . Theorem 11. A top olo gic al semigr oup S with pseudo c omp act squar e S × S is a top olo gic al p ar agr oup if and only if S is simple and c ontains an idemp otent. Pr o of. The “o nly if ” part of the theore m is trivial. T o prov e the “o nly if ” part, assume that the square S × S is pseudo co mpact. By [3, 1 .3], the Stone- ˇ Cech compactification β S o f S is a co mpact top o logical s e migroup. B y the Ko ch-W allace Theorem [19], the top olo gical semigr oup β S , be ing co mpact, contains no is o morphic copy of the bicyclic semigr oup and consequently , and so do es the subsemig r oup S of β S . By the Andersen Theorem [7, Theorem 2 .54], the simple semigroup S is completely simple. In order to a pply Theorem 1, it rema ins to prov e that e ach maximal subg roup H e of S is a top olo g ical g roup. Since the idemp otent e of S is primitive, the maximal group H e coincides with eS e and hence pseudo compact, being the contin uous imag e of the pseudo compact s pa ce S . Applying The o rem 2.6 of [23], we co nclude that H e , b eing a pseudo co mpact paratop olo gical gro up, is a top ological group.  F or completely simple semigro ups S the pseudo compac tness of the s q uare S × S in the preceding theorem can b e repla c ed by the pseudo compactness of S . Theorem 12. A pseudo c omp act top olo gic al semigr oup S is a top olo gic al p ar agr oup if and only if S is c omple tely simple. Pr o of. The “only if ” pa rt is tr ivial. T o prov e the “if ” par t, ass ume that S is a completely s imple pseudo co mpact top olo gical se mig roup. T a ke any primitive idem- po tent e of S and observe tha t the maxima l subgro up H e = eS e is pseudo compact, being the co n tinuous imag e of the pseudo compact spa c e S . By Theo rem 2.6 of [23], the parato po logical gr oup H e , b eing pseudo compa ct, is a to po logical gro up. Applying Theor em 1, we conclude that S is a top olo gical par agro up.  Our fina l r esult descr ibe s the structure of simple sequential countably compact top ological semigr oups. W e reca ll that a to p o logical spa ce X is called se quential if for ea ch non-clo sed subset A ⊂ X there is a s e quence { a n } n ∈ ω ⊂ A that conv erges to so me p oint x ∈ X \ A . Theorem 1 3. F or a simple top olo gic al semigr oup S the fol lowing c onditions ar e e quivalent: (1) S is a r e gular se quential c ountably c omp act top olo gic al sp ac e; THE REES-SUSCHKEWITSCH THEOREM 7 (2) S is top olo gic al ly isomorphic to a top olo gic al p ar agr oup [ X , G, Y ] σ for some r e gular se quential c ountably c omp act top olo gic al sp ac es X and Y and a se- quential c ountably c omp act top olo gic al gr oup G . Pr o of. (1) ⇒ (2). Assume that S is a regula r sequential co un tably co mpact top o- logical space. It follows that S is sequentially compact. By Theorem 9, S is top olog - ically isomorphic to a top o logical paragr oup [ X , G, Y ] σ for some topo logical spaces X a nd Y and s ome top ologic al group G . The spa c es X , Y , G , b eing homeomo rphic to clos ed subspa ces o f S , are r egular se q uent ial and countably co mpa ct. (2) ⇒ (1 ) Ass ume that S is top olog ically isomorphic to a to po logical pa ragr oup [ X , G, Y ] σ for some regular sequential countably co mpa ct top olog ical spa ces X , Y and some sequential co unt ably compa c t top o logical g roup G . It is clear that those spaces a re sequentially co mpa ct. It follows from Bo ehme Theo rem [10, 3.10.J(c)] that the pro duct X × G × Y is sequential. Since the pr o duct of sequentially compact spaces is se q uent ially compact (a nd hence countably compact), the space S , b eing homeomorphic to X × G × Y , is re g ular seque ntial and countably compact.  Open Problems Problem 1. L et X b e a doubly c ountably c omp act sp ac e. Is the squar e X × X c oun tably c omp act? Problem 2. L et S b e a (simple) semitop olo gic al s emigr oup with c ountably c omp act p ower S c . Has S an idemp otent? Problem 3. Assume t hat a simple T ychonov c ountably c omp act top olo gic al semi- gr oup S c ontains an idemp oten t. Is S c ompletely simple? Equival ently, is S a top olo gic al p ar agr oup? It is known that each (top olog ical) se mig roup em b eds int o a simple (top ologica l) semigroup, s e e [5], [7, Theor e m 8.45 ] and [1 3]. Problem 4. Is it true that e ach c ountably c omp act top olo gic al semigr oup emb e ds into a simple c oun tably c omp act top olo gic al semigr oup? References [1] O.T. Alas, M. Sanchis, Countably c omp act p ar atop olo gic al gr oups , Semigroup F orum, 74 (2007), 423—438. [2] O. Andersen, Ein Beric ht ¨ ub er die Strukt ur abstr akter Halb grupp en , P hD Thesis, Hamburg, 1952. [3] T. Banakh, S.D. 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Rupp ert, Comp act Semitop olo gic al Semigr oups: An Intrinsic The ory , LNM 1079 , Springer, 1984. [26] A. H. T omita, The Wal lac e pr oblem: A co unter example fr om MA counta ble and p -c omp actness , Canad. Math. Bull 39 :4 (1996), 486—498. [27] A. D. W all ace, A note on mobs, I , Anais. A cad. Brasil. Ci. 24 (1952), 329—334. [28] A. D. W allace, The Suschkewitsch-R ess structur e the or em for c omp act simple semigr oups , Pro c. Nat. Acad. Sci. 42 (1956), 430—432. [29] A.D. W allace, R et r act i ons in semigr oup s Pa cific J. Math. 7 (1957) , 1513—1517. Dep a r tm ent of Ma thema tics, L viv Na tional University, Universytetska 1, L viv, 79000, Ukraine E-mail addr ess : tbanakh@ya hoo.com Na tional Technical University ”Kharkov Pol ytechnical Institute”, Frunze 21, Kharkiv, 61002, Ukraine Dep a r tm ent of Mechanics an d Ma thema tics, Iv a n Franko L viv Na tional University, Universytetska 1, L viv, 7900 0, Ukraine E-mail addr ess : o gutik@fr anko.lviv. ua