FREE NON-DISTRIBUTIVE MORGAN-STONE ALGEBRAS

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1 NEW ZEALAND JOURNAL OF MATHEMATICS Volume 25 (1996), FREE NON-DISTRIBUTIVE MORGAN-STONE ALGEBRAS D a n i e l S e v c o v i c (Received July 1994) Abstract. In this paper we investigate free non-distributive Morgan-Stone algebras. W e construct the free non-distributive Morgan-Stone algebra as a free lattice generated by a suitable partially ordered set endowed by a unary operation of involution. A positive answer to the word problem is also proven. 1. Introduction In [2] Blyth and Varlet have studied a new variety of so-called Morgan-Stone algebras as a common abstraction of the well known classes of De Morgan and Stone algebras. Such algebras are bounded distributive lattices with a unary operation of involution fulfilling certain identities. The aim of this note is to investigate a larger variety of algebras containing, in particular, Morgan-Stone algebras. In such algebras the distributive identity need not be necessarily satisfied. We are mainly concerned with the construction of free non-distributive Morgan-Stone algebras. The idea of construction is based on the concept of a free lattice generated by a partially ordered set P and preserving bounds prescribed by chosen subsets of P due to Dean [3]. We then analyze the word problem for the varieties under consideration. We show that there is an algorithm for deciding when two words in a free algebra are equal. The approach to the construction of free algebras was significantly influenced by the work of Katrinak. In [5] he has treated a similar task for the class of p-algebras. Based on the characterization of a free p-algebra Katrinak and the author were able to characterize projective p-algebras [6] as well as bounded endomorphisms of free p-algebras [7]. It is hoped that an analogous technique can be also applied in the study of projective non-distributive Morgan-Stone algebras. The outline of the paper is as follows. In Section 2 we recall definitions of De Morgan and Morgan-Stone algebras. New varieties of non-distributive De Morgan and Morgan-Stone algebras are introduced. We also present some of results due to Dean [3] regarding free lattices generated by partially ordered sets. Section 3 is focused on the construction of a free non-distributive De Morgan algebra. In Section 4 we construct a free non-distributive Morgan-Stone algebra and give the positive answer to the word problem in this variety. Finally, some examples showing free non-distributive De Morgan and Morgan-Stone algebras with a simple generator are also presented A M S Mathematics Subject Classification: Primary 06C16, Secondary 06B25. Keywords and Phrases: Nondistributive Morgan-Stone algebras, Free algebras, Free lattice generated by a poset, Word problem.

2 86 DANIEL SEVCOVIC 2. Prelim inaries We start by recalling definitions of De Morgan algebras, Stone algebras and Morgan-Stone algebras. By De Morgan algebra we understand a universal algebra (M, V, A, ~, 0,1) where (M, V, A, 0,1) is a bounded distributive lattice and the unary operation of involution satisfies the identities: x = x, (xa y)~ = x~ Vy_, I- = 0. Stone algebra is a universal algebra (5, V, A, *, 0,1) where (5, V, A, 0,1) is a bounded distributive lattice and the unary operation of complementation satisfies: xax* = 0, (xay)* x*\/y*, 0* = 1. Finally, Morgan-Stone algebra (or MS algebra) is a universal algebra (M, V, A,, 0,1) where (M, V, A, 0,1) is a bounded distributive lattice and the unary operation of involution satisfies: x < x, (x A y) = x V y, 1 = 0. We refer to a book by Balbes and Dwinger [1] for a broader discussion regarding De Morgan and Stone Algebras. Now we introduce two new varieties of so called generalized De Morgan and Morgan-Stone algebras in such a way that all the identities for the unary operation of complementation (involution) are preserved. We will consider a larger equational class of algebras satisfying all the above identities of Morgan-Stone algebras lattice skeletons of which are not assumed to be distributive lattices. Definition 1. A generalized De Morgan algebra (or GM - algebra) is a universal algebra (M, V, A,, 0,1) where (M, V, A, 0,1) is a bounded lattice and the unary operation of involution satisfies the identities: GMi : x = x, GM2 : (x A y)~ = x~ V y_, GM3 : I - = 0. D efinition 2. A generalized Morgan-Stone algebra (or GMS - algebra) is a universal algebra (M, V, A,, 0,1) where (M, V, A,0,1) is a bounded lattice and the unary operation of involution satisfies the identities: GMSi : x < x, GMS2 : (x A y) = x V y, GMS3 : 1 = 0. Let L be a GMS - algebra. We define the skeleton (the set of closed elements) S(L) of L as follows: S(L) := {x E L,x = x }. One can easily verify that the set S(L) endowed with induced operations from L is a GM - algebra. More precisely, Lem m a 1. Let (L, V, A,, 0,1) be a GMS - algebra. (S(i),V, A, o,0,1) is a GM - algebra. Then the skeleton Throughout the paper the following simple rules for computation in GMS - algebras will be frequently used: if x < y then y < x, x 000 =x, (x V y) = x A y, 0 = 1. The construction of free GM - as well as GMS - algebras is based on the well known characterization of free lattices generated by partially ordered sets and preserving bounds due to Dean [3]. Let us therefore summarize his results.

3 FREE NON-DISTRIBUTIVE MORGAN-STONE ALGEBRAS 87 Let P be a partially ordered set (poset) with an order relation families of subsets of P such that if p -< q, p, q P then {p, q} U and {p, q} C if S Eli (S C) then there is supp S (infp S) in the poset P. Let U,C be According to [3, Theorem 6] there exists a free lattice FL(P,U, C) generated by P and preserving bounds prescribed by the sets from U and C. We also recall that, by [3, Theorem 10], the word problem in FL(P, U, C) has an affirmative solution if there is an affirmative solution to the problem of determining whether two ideals of P of the form M(a) = {p P,p > a}, J(b) = {p P,p < b} have a common element. More precisely, in the free lattice FL(P,U, C) a < b if and only if one or more of the following hold: a = ai V a2 and ai < b for i = 1 and i 2, a = ai A a2 and di <b for i 1 or i = 2, (W) b = b\\zb2 and a < bi for i = 1 or 2 = 2, b = 61 A b2 and a < bi for i = 1 and i = 2, there is a p P such that a < p < b, (c.f. [3, Theorem 7]). With regard to [3, Definition 2] the order relation p < b for p P means p < b(k) for some integer A: > 0 where p < 6(0) iff b = q P and p -< q in P; p < b(k) iff either b = bi V b2 and p < bi(k 1) for i = 1 or i = 2, ( j ) or b = b\ A 62 and p < bi(k 1) for i = 1 and i = 2, or there is a S Eli such that p -< sups and p s < b(k 1) for all s S. Analogously, a < p for p P means a < p(k) for some integer k > 0 where a < p(0) iff a = q P and q -< p in P; a < p(k) iff either a = ai V a2 and ai < p(k 1) for i 1 and i 2, (jvf) or a = ai A a2 and a* < p(k 1) for i 1 or i = 2, or there is a 5 such that inf S -< p and p a < s(k 1) for all s S. Suppose that p -< q in P. With regard to the above definition of the ordering in the free lattice FL we observe that if q < b(k) then p < b(k) also. Similarly, if a < p(k) then a < q(k). The proof utilizes an induction argument with respect to k > 0. Let L be a K - algebra in the variety K. By [X]k we denote the K - subalgebra of L generated by a subset X C L. As usual, by [a, b] we denote the interval [a, b] {c, a < c < b}.

4 88 DANIEL SEVCOVIC 3. Free Generalized De Morgan Algebras In this section we study free algebras in the variety of all generalized De Morgan algebras. We begin with a general result due to Katrinak regarding free algebras in a variety of K algebras. Lemma 2. ([5, Lemma 1]) Let K be a class of algebras, X any set, and Fk (X) the free algebra in K freely generated by the set X. Suppose A G K is also generated by X and there exists a K-homomorphism h : A > Fk (X) which is the identity function on X. Then h is an isomorphism. Let L be a GM - algebra and X C L. We denote X~ the set X~ {x~,x G X}. By a straightforward induction on the rank of a GM - term p one can easily prove that for any a\, a2,,an G X there exist &i, b2,...,bm G X U X~ and a lattice term q such that p(ai,a2,..., an) = q(bi,b2,..., bm). In other words, we have - Lemma 3. If a GM - algebra L is generated by the set X, i.e. L = [X]g m, then the set X U X~ generates L in the variety BL of bounded lattices, i.e. L = [X U X~]b l- Now we are in a position to define a poset P = Pq m (X) and two families Mg m -, gm of subsets of Pg m (X) in such a way that the free lattice generated by the poset Pq m (X ) and preserving bounds from Ug m, gm will admit a unary operation of involution with the property that the resulting algebra is free in the category of GM - algebras. Let X be a set. Let I be a disjoint copy of X, i.e. X {x, x G X } and l n l n { 0, l } = 0. Define the set Pg m (X) = X U X U {0,1} and the ordering -< on Pg m {X) as follows: 0 -< x -< 1, 0 -*< x ~< 1 for any x,x G Pg m (X). The families Ug m, gm are defined as Ugm = gm = {{p,q} C Pg m {X), p -< q in Pg m {X)}. Then there is a free lattice FL{Pg m {.X),Ug m, g m) generated by the poset Pg m {X). In what follows, we will show that there is an operation of involution with the property that the free lattice FL(Pg m {X),Ug m, g m) endowed with such a unary operation is a free GM - algebra. To this end, we first introduce the mapping 9 : FL(Pg m {X),Ug m, g m) -* FL(Pg m {X),Ug m, Cg m ) defined on the set of generators as follows: 9(x) x, 6(x) = x, 0(0) = 1, 9(1) = 0. (3.1) This mapping extends to a dual endomorphism of FL(Pg m (X),Ug m, g m ) preserving bounds prescribed by sets from Ug m i^ gm ([3, Theorem 6]). Let p be any lattice term. By p we denote a lattice term which is obtained from p by replacing all the symbols A by V, V by A, 0 by 1 and 1 by 0. Using the properties (W'), (J) and (M ) of FL(Pg m (X), Ug m > g m ) and recalling that S = {p, <?}, p -< q in Pg m (X) for any S G Ugm &GM-, we obtain by a straightforward induction on the rank of lattice terms p, q : p(a\, a2,..., an) < g(ai,a2,...,an) in FL(Pg m (X),Ug m, g m ), where ai, a2,..., an G Pg m (X), implies p(9a\, 9a2,..., 9an) > q(9ai,9a2,..., 9an). Now, it should be obvious that the mapping 9 : FL FL satisfies the identities 9(a A b) 9(a) V 9(b), 9(a V b) = 9(a) A 9(b) and 9(9(a)) = a for any a,b G FL.

5 FREE NON-DISTRIBUTIVE MORGAN-STONE ALGEBRAS 89 Let us denote a~ := 9{a). (3.2) The lattice FL = FL(Pg m (X),Ug m, g m) endowed with such a unary operation is a GM - algebra. Theorem 4. Let X be any set. Then the free lattice F L = FL(Pgm{X),Ugm, Cg m) endowed with the unary operation defined in (3.2) is a free GM - algebra, i-e. FGM(X) = (FL, ). Proof. Note that FL = [X]gm [X U X]bl- Let us define the mapping h : Pgm(X) >Fgm(X) as follows: h(x) := x, h(x) := x~, h(0) - 0, h( 1) = 1. According to [3, Theorem 6] the mapping h can be extended to a homomorphism h : FL >Fgm(X). By an induction on the rank of a lattice term a G FL we will show that h(a~) = h(a)~. If a G Pqm{X) the statement is obvious. If a = a\ A a2 or a = ai V a2 then the statement follows from the induction hypothesis made on terms ai,a2 and the properties of the mapping 9. Hence h : FL * F q m {X ) is a GM - homomorphism which is an identity function on X. According to Lemma 2 h is an isomorphism and the proof of theorem follows. We end this section by proving that the word problem in Fqm{X ) has an affirmative solution. Lemma 5. Let Fg m (X) be a free GM - algebra. Then any element pglu X ~ U {0,1} is join and meet irreducible. Proof. With regard to Theorem 4 it is sufficient to show that any p G Pq m {X ) is join and meet irreducible in the lattice FL. We will proceed by an induction on the rank. According to (J) p < a\ V a2, p G Pgm(X), ai,a2 G FL iff p < a\ V a2(k) for some integer k > 0. This means that either p < ai(k 1) i = 1 or i = 2, or there is S G Ug m, S = {gi, q2}, qi -< q2 such that p -< supp S = q2 and q2 < a\ V a2(k 1). The latter implies p < a\ V a2(k 1) and so one can again decrease the rank (k 1) by one. Since p < ai V 02(0) is impossible we may conclude that either p < a\ or p < a2. Hence p is join irreducible. The proof of meet irreducibility of a P ^ Pgm{ X ) utilizes a dual argument because Ugm = g m - The next lemma is an immediate consequence of the property (W) and Lemma 5. Lemma 6. a\ A a2 < b\ V b2 in Fg m {X) if and only if [ai A a2,b\ V b2] n{ai,a2, 61, 62} Knowing the above characterization of the ordering in Fg m {X) we are in a position to state the following theorem. Theorem 7. The word problem in Fg m {X) has an affirmative solution. Proof. Let a = a(x\, x 2,...,xn) G Fg m {X) be arbitrary GM - term, Xi G X. Taking into account the identities GMi,GM2,GMs one can construct a lattice term a = a(x\, x2,...,xn,x^,x^,...,x~) such that a = a. According to Lemmas 5 and 6 there is an effective algorithm for decision when a = 6 in I

6 90 DANIEL SEVCOVIC FL(Pg m {X),Ugmi g m )- From this we can conclude that the word problem in Fg m (X) has a solution. I 4. Free Generalized M organ-s tone A lgebras In this section we focus on the construction of and analyzing the word problem for the free GMS - algebra Fg m s(x) freely generated by a set X. The next lemma gives us the complete characterization of the skeleton 5(F gm s(x )). Lem m a 8. The skeleton S ( F g m s { X )) = {a G F g m s { X ), c l a00} is isomorphic to the free GM - algebra F g m ( X ) generated by the set X = {x,x G X }. P roof. The proof is essentially the same as that of [5, Lemma 2 and 3] for the variety of p-algebras. We therefore only sketch the main ideas. First, we observe that the skeleton S(Fgms{X)) is a GM - algebra generated by the set X. Indeed, it suffices to show that a G [X )gm whenever a G Fgms(X). This can be readily verified by an induction on the rank of a term a Fgms{X). Furthermore, let us consider the GMS - homomorphism h : Fgms(X)» Fgm{X ) defined as h(x) := x for any x X. The mapping h restricted to the set S(Fg m s{x)) is a GM - homomorphism which is an identity mapping on X. With regard to Lemma 2 we obtain that S(Fgms(X)) is isomorphic to Fgm{X ). Following the same idea of the construction of Fgm (X) as a free lattice FL(P,U, C) generated by some poset P and preserving bounds prescribed by sets from U and C we will define the poset Pg m s{x ) and two families Ugm s, Cgms of finite subsets of Pgms(X) and a unary operation of involution in such a way that the resulting free lattice FL(Pgm s{,x),ugm s-, 'GMs) with the operation is isomorphic to the free GMS - algebra Fgm s{x) freely generated by the set X. Let X be a set. Take a disjoint copy X := {x, x G X } of X with the property X fl Fgm(X) = 0. Let us define the poset Pg m s{x) = X U Fg m {X) and the order relation -< on -Pgms(X) as follows: 0 -< a -< 1 for any a G P g m s ( X ); x -< a, x G X, a G Fg m (X) iff x < a in Fg m {X)-, a -< b, a,b G Fg m {X) iff a < b in Fq m (X). The sets UqmSi gms are defined as follows: a finite subset S C Pgm s(x) belongs to U q m s gms iff either S = {p, q}, p -< q in Pgm s{x ) or S' is a subset of FGm(X). Lem m a 9. Let F L = FL(Pgms{X),Ugms, gms) be the free lattice generated by the poset Pgms{X) and preserving bounds from Uqms and Cgms Then there exists the unique lattice epimorphism ix : F L >Fgm (X) such that n(x) = x for any x G X and 7r(a) = a for any a G Fqm{X). Moreover, a < n(a) for any a G FL. P roof. The mapping r : Pg m s{x) Fgm (X) defined by r(a;) = x for any x G X and r(a) = a for any a G Fgm (X) preserves l.u.b. s and g.l.b. s prescribed by sets from Ugms and Cgm s, respectively, i.e. r(supp 5) = \J t(s) and r(infps') = A T{S) for any S G Ugms = gm s- Hence, by [3, Theorem 6] there is a unique lattice homomorphism -k : FL Fgm (X) extending the mapping r.

7 FREE NON-DISTRIBUTIVE MORGAN-STONE ALGEBRAS 91 Since Fg m {X ) C Pg m s(x) the homomorphism 7r is also onto. Furthermore, the inequality a < tt(a) is satisfied for any a G Pg m s(x). Then one can proceed by an induction on the rank of a lattice term a G FL to verify that a < tt(a) for any a G FL. For any a G FL(Pgm s{x),ugm s, gms) we set a = (7r(a))~ (4.1) where the unary operation " is taken in the free GM - algebra Fg m (X). Notice that a = a~ for any a G Fg m {X). Theorem 10. Let X be any set. Then the free lattice FL = FL(Pgms(X), Mgms, gms) endowed with the unary operation defined in (4.1) is a free GMS - algebra freely generated by the set X, i.e. (F L, ) = Fg m s{x). P roof. First we will verify the identities GMSi, GMS2 and GMS3. By Lemma 9, we have a = (7r(7r(a)- )) = 7r(a) = 7r(a) > a. Further (a A b) = (n(a A b)) = (7r(a) A 7r(6)) = 7r(a) V 7r{b)~ = a V b. Finally, 1 = 7r(l)~ = I- = 0. As x = (tt(tt(x)- )) = x = x w e h a v e? GM s W = X ufgm (X) C [X]Gms and so (FL, ) = [.X]qm s Let us define the mapping h : Pgm s{x) > Fgm s(x) as follows: h(x) := x for x G X, h(x) := x for x G X. Since Fg m {X ) is a free GM - algebra the mapping h : X > Fqm s{x) uniquely extends to a GM - homomorphism h : F q m ( X ) F q m s { X ). Hence the mapping h : P g m s ( X ) > F g m s ( X ) is well defined. Moreover, it preserves all bounds prescribed by sets from U g m s = g m s Again due to [3, Theorem 6] the mapping h can be extended to a lattice homomorphism h : F L ( P g m s { X ), U g m s, g m s ) > F g m s ( X ). In what follows, we will show that h is even a GMS - homomorphism, i.e. h(a ) = h(a). We will argue by an induction on the rank of the lattice term a G F L. For an a G P g m s ( X ) the statement is obvious. If a = a\ V a2 then h(a ) = h{n(a\ V a2)~) = h(ir(ai)~ A 7r(a2)~) = h(n(ai)~) A h(ir(a2)~) = h(al) Ah(a2) = h(a\) Ah(a2) = (h(ai) Vh(a2)) = h(a). The case a = ai Aa2 is similar. This way we have shown that h is a GMS - homomorphism. As h(x) = x for any x G X and Fqm s{x) = [X]gms we infer that h is a surjection. According to Lemma 2 h is a GMS - isomorphism and the proof of theorem follows. In accord to the previous theorem we will henceforth identify Fg m s(x) with the free lattice FL(Pgm s(x),ugmsi gms) endowed with the unary operation 0 defined in (4.1). In the following lemma, we examine the property of join and meet irreducibility of elements of Fqm s(x) belonging to the set X U S(Fg m s{x)).

8 92 DANIEL SEVCOVIC Lemma 11. Let Fqm s{x) be a free GMS - algebra. Then any U S(Fgm s(x )) is both join and meet irreducible. Proof. Notice that, as X = X in FL we obtain S(Fgm s(x)) = Fg m {X). We will distinguish two cases: p = x G X and p G Fg m (X). Let us consider the case p = x G X. From the definition of the ordering in FL we know that x < a\ V a2 means x < a\ V a2(fc) for some integer fc > 0. By (J), either < ai{k 1), i = 1 or i = 2 or there exists an S G Ugms such that x -< supp S and s < a\ V a2(fc 1) for any s G S. The first event immediately implies x < a*, i 1 or i = 2. On the other hand, if S' = {p,q}, p -< q in Pgm s{x), then we obtain x < a\ V a2 (A: 1). If S' is a finite subset of Fg m {X) then with regard to the definition of the order relation we infer that x < x < supp S = \J S. Since any x is join irreducible in Fg m (X) (see Lemma 5) there is an s G S such that x < s. This however means that x < a\ V a2 (k 1). Now, decreasing step by step the rank k by one and taking into account that the case x < a\ V a2(0) is impossible we end up with the claim that either x < a\ or x < a2. Hence any x G X is join irreducible in Fgm s{x). To prove meet irreducibility of an x G X we cannot employ a dual argument because the ordering in P g m s { X ) is not symmetric (x < x). Nevertheless, if ai A a2 < x then there is a k > 0 such that di Aa2 < x(k). By (M ), either ai < x, i = 1 or i 2 or there is a finite subset S C Fg m {X) with the property inf p S -< x and a\ A a2 < s(k 1) for any s G S. Since there are no nonzero elements in Fg m {X) less than x we have infp S = 0 and so ai A a2 = 0. Thus a^0 A a2 = 0 in Fg m (X). We remind ourselves that 0 is meet irreducible in Fq m {X). Hence either ai < ^1 = 0 < x or a2 < a^0 = 0 < x. Hence x G X is also meet irreducible. The proof of join and meet irreducibility of an element p G Fq m {X) again follows from Lemma 5. Indeed, if p < a\ V a2(fc), k > 0 then either p < ai(k 1), i = 1 or i = 2 or there is a finite subset S C Fg m {X) such that p -< supp S and s < ^ V a2(fc - 1) for any s G S. Taking into account join irreducibility of any s G Fg m (X) we infer that either p < ai(k 1), i = 1 or i 2 or p < a\ Va2(fc 1). Now the standard argument enables us to conclude that p < a\ or p < a2. On the other hand, if a\ A a2 < p then a\ A a^0 < p p. As the element p G Fq m {X) is meet irreducible in Fq m {X) we obtain ai < a < p for i = 1 or i 2. This means that p G Fq m {X ) is meet irreducible. The proof of lemma is complete. The next lemma is a direct consequence of Lemma 11 and the property (W). Lemma 12. ai A a2 < b\ V &2 in Fqm s{x ) if and only if [ai A a2,6i V &2] n{ai,a2, 6i,62} 7^ 0. We conclude this paper with the following result proving that the word problem in the free algebra Fqm s{x ) has a solution, i.e. there is an algorithm for decision when two word in Fg m s(x) are equal. Theorem 13. Let X be any set. Then the word problem in the free generalized Morgan - Stone algebra Fgms(X) has an affirmative solution.

9 FREE NON-DISTRIBUTIVE MORGAN-STONE ALGEBRAS 93 P roof. Let Fgms{X) = (FL, ) be a free GMS - algebra and let a, b G FL be two GMS - terms. Since the identities x x000, x A y = (xv y), x V j/ = (x A y) are fulfilled in any GMS - algebra one can construct lattice terms a, b with letters belonging to the set X U X U X and such that a a, b = b. As a = b iff a < b and b < a one can recursively apply Lemma 12 in order to decrease the rank of the lattice terms a, b. After performing finite number of steps we end up with the problem to decide whether p < q in Fgms{X) for letters from the set X UX UX00. With regard to the definition of the ordering in Pgms(X), this is true iff either p = q or p = x and q x for some x X. Hence there is an algorithm for determining whether a = b in Fgms{X). Exam ples. The free GMS and GM - algebras with X = 1 have diagrams as shown by the figures below: Fgm s( 1) Fg m ( 1) x A iv (: x x * 0 0 Acknow ledgem ents. The author wishes to thank to Professor T. Katrinak for his helpful comments.

10 94 DANIEL SEVCOVIC References 1. R. Balbes and R Dwinger, Distributive Lattices, Univ. Missouri Press, Columbia, T.S. Blyth and J.C. Varlet, On a common abstraction of De Morgan algebras and Stone algebras, Proc. Roy. Soc. Edinburgh A 94 (1983), R.A. Dean, Free lattices generated by partially ordered sets and preserving bounds, Canad. J. Math. 16 (1964), G. Gratzer, General Lattice Theory, Birkhauser Verlag, T. Katrinak, Free p-algebras, Algebra Univ. 15 (1982), T. Katrinak and D. Sevcovic, Projective p-algebras, Algebra Univ. 28 (1991), D. Sevcovic, Bounded endomorphisms of free p-alqebras, Glasgow Math. Journal 34 (1992), Daniel Sevcovic Institute of Applied Mathematics Comenius University Mlynska dolina Bratislava SLOVAK REPUBLIC sevcovic@fmph.uniba.sk