Karásek, Jiří

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Institute of Mathematics AS CR, v. v. i.

Archivum MathematicumJiří KarásekA remark concerning x-systemsArchivum Mathematicum, Vol. 21 (1985), No. 3, 177--179Persistent URL: http://dml.cz/dmlcz/107230Terms of use:© Masaryk University, 1985Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czARCHIVŮM MATHEMATICUM (BRNO) Vol. 21, No. 3 (1985), 177-180 A REMARK CONCERNING x-SYSTEMS JIŘÍ KARÁSEK, Brno (Received January 2,1984) Abstract. The paper deals with certain closure operators in commutative semigroups introduced by K. E. Aubert. It is shown that they form a complete lattice with respect to a natural ordering. Key words. X-system in a commutative semigroup, x-system of finite character. K. E. Aubert studied so-called x-systems in commutative semigroups in [1]. In Chapter 1, there is a proposition (Proposition 8) claiming that the family 3FS of all x-systems of finite character in a commutative semigroup S forms a complete sublattice of the complete lattice S£s of all x-systems in S with respect to a certain ordering > . In this remark, an example is given showing that the mentioned proposition is incorrect. A weaker theorem is proved below. An x-system in a commutative semigroup S is defined to be a mapping x of the set of all subsets of S into itself satisfying the conditions: 1.1 A £ ^xfor any A £ S, 1.2 A Q Bx=> Ax^ Bx for any A ^ S9 B ^ S9 1.3' ABX £ Bx for any A £ S9 B £ S9 1.3" ABX £ (AB)X for any A £ S, B £ S. An x-system in S is said to be of finite character if, for any A £ S9 Ax == \JNX (N £ A9 N is finite). The family of all x-systems in S (of all x-systems of finite character in S) is denoted by S£S($FS). An ordering > is defined in Se^SF^ as follows: xx >- x2 if AXi £ AX2 for any A £ S. Theorem. The family &rs of all x-systems of finite character in Sforms a sublattice of the complete lattice S£s of all x-systems in S with respect to the ordering > , i.e. when {Xi}iet is a finite family of x-systems of finite character then A *i ar*d iel V xt are both x-systems of finite character. Moreover, 9?s is a meet subsemilattice iel ofses. Proof. In [1], the proof of Proposition 8, it is shown that &s is a meet sub-semilattice ofSPs. Assume that x = V ** where {*<}*6r is a finite family of x-systems iel 177 - - - { . J. KARASEK of finite character. By [1], Proposition 7, Ax = fl .Ax< for all A £ 5. Clearly i e l .4X .2 U .rVx where N denotes a finite set. Conversely, assume that a e Ax = fl AXi. NCA > ' e l Then a e Alx. for all i e I. It follows that for all i e / there is a finite set Nl e A such that a e Nxr Consider N° = U Nl. N° is finite, N° := ,4 and a e N*t for all i s / . Hence a e fl Nx. = Nx £ (J Nx, where N denotes a finite set. Thus x is of finite character. Example. The following example shows that if {Xi}i€j is an infinite family of x-systems of finite character, then x = V xi neec- n o t be an x-system of finite character. Let S be the set of all ordinals less than or equal to the least infinite ordinal (o with their usual ordering. Let M = S — {to}. Define ab = min (a, b) for a e S, b 6 S. S is then clearly a commutative semigroup. Further, define a family {xt}i€ M of mappings of the set of all subsets of S into itself as follows: \S if there is a e A such that i ^ a, [{yeS: 3 ze A: y ^ z} if there is no a e A such that i g a. First, we shall show that {x(}ieM is a family of x-systems in S. Evidently, A £ AXi for all A £ £ and all ieM, so that the condition 1.1 is satisfied. Assume that A £ BXi for some A, B ^ S and some / 6 Af. Let r e AXi. If there is a e .4 s 2?x< such that i ^ a, then we have two possibilities: (a) There is b e B such that i <; b. Then BXi = S. (b) There is z e 2? such that a g z. Then i ^ z and f*X| = S again. Consequently, r e BXi in both cases. If there is z e A ^ BXi such that r ^ z, then we have two possibilities: (a) There is a e B such that i g a. Then fix. = S. (b) There is n e 2? such that z ^ u. Then r ^ « and reBXi. Again, reBXi in both cases. Hence, the condition 1.2 is satisfied. Now, assume that r e ABXi for some A, B ^ S and some i e M. There are s 6 A, t e BXt such that r = jf. If there is a e B such that i g a, then 5X< = 5 and reBxr If there is no a e . 8 such that i g a, then there is ze .5 such that t ^ z. Then, however, r ^ z and reBxr The condition 1.3' is satisfied. Again, assume that reABXi. Then there are s e A, te BXi such that r = st. If there is a e AB such that i g a, then (AB)Xi == 5 and r e (-4.ff)Xl. If there is no a e AB such that i £ a and there is b e B such that i ^ A, then there is no c e A such that i :g c. From this it follows that s < i. Two possibilities can occur: (a) t > y for all yeB. Then t ^ b ;> i > s, so that r = s/ = ^ = sb e Al.fi c £ (AB)xr (b) There is rfe.B such that t <> d. Then r = ^ ^ sdeAB. Thus re(ALB)X|. 178 A REMARK CONCERNING x-SYSTEMS If there is no b e B such that i g b, then there is z e B such that t <J z. From this it follows that r = st S szeAB. Thus r e (AB)Xi. The condition 1.3* is satisfied. Now, we shall show that {Xi}i€M is a family of x-systems of finite character. Assume that A i= S and ie M. Clearly AXi 2 U Nxt where # denotes a finite set. NCA Let r G y4x.. If there is a e A such that i ^ a, then r e {a}Xi = S. If there is no a e -4 such that i :g a, then there is zeA such that r —: z. Thus re {z}X|. Hence re e U NXi where N is a finite set in both cases. NCA x = V *t is, however, not an x-system of finite character. In fact, Mx = ieM = f| MXi = S, while Nx = f| -Vx, = {y e 5: 3 z e N: y £ z} for any N c Af such that 1Vis finite, so that U -Y* = M ^ 5 = Mx. JVCM REFERENCES -i [1] K. E. Aubert: Theory ofx-ideals, Acta Math. 107 (1962), 1-52. J. Kardsek Katedra matematiky FS VUT Gorkeho 13, 602 00 Brno Czechoslovakia 179 

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