On regularity of sup-preserving maps: generalizing Zareckiĭ s theorem

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1 Semigroup Forum (2011) 83: DOI /s RESEARCH ARTICLE On regularity o sup-preserving maps: generalizing Zareckiĭ s theorem Ulrich Höhle Tomasz Kubiak Received: 7 March 2011 / Accepted: 13 May 2011 / Published online: 8 June 2011 The Author(s) This article is published with open access at Springerlink.com Abstract A sup-preserving map between complete lattices L and M is regular i there exists a sup-preserving map g rom M to L such that g =. In the class o completely distributive lattices, this paper demonstrates a necessary and suicient condition or to be regular. When L = M is a power set, our theorem reduces to the well known Zareckiĭ s theorem which characterizes regular elements in the semigroup o all binary relations on a set. Another application o our result is a generalization o Zareckiĭ s theorem or quantale-valued relations. Keywords Sup-preserving map Regular map Complete distributivity Regular relation Quantale Quantale-valued relation 1 Introduction A morphism A B in a category C is regular i there is a morphism B g A in C with g = (c. [4]). The purpose o this paper is to examine regular morphisms in the category Sup whose objects are all complete lattices and whose morphisms are all sup-preserving Communicated by Boris M. Schein. The second named author grateully acknowledges the grant MTM C02-02 rom the Ministry o Science and Innovation o Spain. U. Höhle Fachbereich C Mathematik und Naturwissenschaten, Bergische Universität, Gaußstraße 20, Wuppertal, Germany uhoehle@uni-wuppertal.de T. Kubiak ( ) Wydział Matematyki i Inormatyki, Uniwersytet im. Adama Mickiewicza, Umultowska 87, Poznań, Poland tkubiak@amu.edu.pl

2 314 U. Höhle, T. Kubiak maps (c. [3]). We give a suicient condition or a morphism o Sup to be regular as well as a necessary one. Both these conditions involve complete distributivity, so that when gathered together they provide a characterization o regular morphisms o the category o all completely distributive lattices and their sup-preserving maps (see Theorem 3.1). Our result can be viewed as a non-atomic generalization o the amous Zareckiĭ s theorem which characterizes regular elements in the semigroup o all binary relations on a set (c. [14]). As a matter o act, we generalize a recent extension o Zareckiĭ s theorem stated in [12] which characterizes regular morphisms in the category o all relations. We recall that ρ X Y is called regular i there exists σ Y X such that ρσρ = ρ. The result o [12] says: A relation ρ X Y is regular i the lattice {ρ(a) A X} is completely distributive (see Sect. 2 or notation i needed and c. Note 1.2 below). In the case X = Y we have the just mentioned Zareckiĭ s theorem. It may be remarked that with an idempotent ρ X X (i.e. ρρ = ρ), the Zareckiĭ s theorem is already seen in [9]. Another application o Theorem 3.1 is a generalization o Zareckiĭ s theorem or quantale-valued relations (see Theorem 4.3). The extensive literature related to Zareckiĭ s theorem include [1, 5, 8, 11 13] among others. In particular, alternative proos are given in [1, 11, 13] all o which use the act that (2 X, ) is an atomic completely distributive lattice (2 X is the power set o X). Note 1.1 When saying that a morphism o Sup is regular with g =,weo course mean that g is a morphism o Sup too. We recall that in Set each map is regular (c. [4]). Note 1.2 Let L M be a morphism in Sup. Then the range (L) is a complete lattice w.r.t. the partial ordering inherited rom M and sups in (L)are ormed in M, but in general not ins. Obviously, (L)together with the inclusion map is the image o w.r.t. the (extremal epi, mono)-actorization property in Sup. In this context, saying that (L) is completely distributive simply means that the complete lattice (L)is completely distributive. 2 Binary relations as sup-preserving maps Consider the category Rel whose objects are sets and ρ : X Y is a morphism i ρ X Y. The composition o ρ X Y and σ Y Z is the relation σρ X Z where σρ ={(x, z) X Z y Y : (x, y) ρ and (y, z) σ }. Given ρ X Y and A X, letρ(a) be deined by ρ(a) = x A ρ(x) where ρ(x) ={y Y (x, y) ρ}. The assignment A ρ(a) preserves arbitrary unions, hence {ρ(a) A X} is a complete lattice w.r.t the partial ordering inherited by 2 Y. Thus, each ρ X Y determines a union-preserving map 2 X (ρ) 2 Y by (ρ)(a) = ρ(a).

3 On regularity o sup-preserving maps: generalizing Zareckiĭ s theorem 315 Conversely (c. [2, 6]), any map 2 X 2 Y determines () X Y by (x, y) () y ({x}). I is union-preserving, then sends () back to (c. property (2) below). The category Rel is isomorphic to a ull subcategory o Sup. In act, the ollowing hold: Proposition 2.1 Let ρ X Y and σ Y Z be relations, let 2 X be arbitrary maps. Then: (1) (σρ) = (σ) (ρ) and (g) = (g) (), (2) ( (ρ)) = ρ, and ( ( )) = provided is union-preserving. From Proposition 2.1 we immediately obtain: 2 Y g 2 Z Fact 2.2 The ollowing statements hold: (1) A relation ρ X Y is regular i 2 X (ρ) 2 Y is regular. (2) A union-preserving map 2 X 2 Y is regular i () X Y is a regular relation. The Zareckiĭ s theorem (in its more general version o [12]) can now be ormulated as ollows: Theorem 2.3 A union-preserving map 2 X 2 Y is regular i (2 X ) is a completely distributive lattice. This ormulation is an invitation to consider it in a more general lattice-theoretic setting by replacing the power sets by completely distributive lattices (see Theorem 3.1). We recall that the proos o Zareckiĭ s theorem given in [1, 11, 13] make an essential use o the act that 2 X is an atomic completely distributive lattice. The proo o Theorem 3.1 is valid or arbitrary completely distributive lattices and is thus dierent rom all o them. 3 Regular and sup-preserving maps It is well known that a complete lattice L is completely distributive i L op is. In particular, or each amily {A i } i I o subsets o L the ollowing holds: Ai = ( ) ϕ(i). i I ϕ A i i I Given a complete lattice L, we write a b i, whenever C L and b C, there is c C with a c (c. [3, 9]). Then L is completely distributive i has the approximation property: i I a = {b L b a}

4 316 U. Höhle, T. Kubiak or each a L. For all a,b,c,d L the ollowing hold (c. [3]): ( 1 ) a b implies a b, ( 2 ) c a b d implies c d, ( 3 ) a b implies a c b or some c L. The ollowing is our generalization o the Zareckiĭ s theorem. Theorem 3.1 Let L and M be completely distributive lattices, and let L M be sup-preserving. Then is regular i (L)is a completely distributive lattice. Since both the i part and the only i part o Theorem 3.1 holdtrueinamore general setting, we wish to split the theorem into two propositions rom which Theorems 3.1 ollows immediately. In act, the only i part is also valid or L a continuous lattice (c. Remark 3.4). Proposition 3.2 Let L and M be complete lattices, and let L M be suppreserving. I both M and (L)are completely distributive, then is regular. Proo For every y M we deine A y ={c L y (c)}. Then we put { G y = b L (b)= } (A y ) where is perormed in (L). Deine a map M g(x) = y x Gy g L by or all x M. LetA M. Byusing( 1 ) ( 3 ),wehavey A i y x or some x A so that g( A) = ( x A y x G y) = x A g(x), hence g is suppreserving. In order to veriy g = we proceed as ollows. As is sup-preserving we have (A y ) = ( G y ) and subsequently ( ) (Ay ) = Gy = g(a), a L. y (a) y (a) Now we make use o the complete distributivity o (L)and obtain: ( ) ϕ(y) = g(a). ϕ (A y ) y (a) y (a) Because o ϕ(y) (A y ) we have y ϕ(y), and the relation (a)= y (a) y g(a)

5 On regularity o sup-preserving maps: generalizing Zareckiĭ s theorem 317 ollows. On the other hand y (a) implies y (a), and consequently (a) (A y ). Because o (A y ) (a)the relation g(a) (a)holds. Proposition 3.3 Let L and M be complete lattices, let L M be sup-preserving. I is regular and L is completely distributive, then (L)is completely distributive too. Proo Let g = where M g L is sup-preserving, and let a L. Then g (a) L and by complete distributivity o L we have g (a) = {b L b g (a)}. We now check that b g (a) in L implies (b) (a)in (L). Indeed, i (a) (C) or some C L, then g (a) g (C) and there is a c C such that b g (c). This yields (b) (c), so that (b) (a). Consequently, (a)= g(a) {(b) (b) (a)} {y (L) y (a)} (a). We have proved that has the approximation property in (L). Remark 3.4 We recall that a complete lattice L is continuous i a = {b L b a} or all a L, where b a i, whenever a A or some A L, there exists a inite subset D A such that b D. Every completely distributive lattice is continuous (see [3]). It is not diicult to observe that Proposition 3.3 maintains its validity or the relation and continuous lattices. 4 A quantalic version o Zareckiĭ s theorem In what ollows we extend our previous considerations in Sect. 2 rom the two-valued to the many-valued setting. More speciically, we will replace maps 2 X 2 Y by maps L X L Y where the complete lattice L replaces the two point lattice 2. Here and elsewhere L X is the set o all maps rom X into L ordered pointwisely. For this purpose we irst recall that a triple Q = (Q,, &) is called a quantale i (Q, ) is a complete lattice, (Q, &) is a semigroup, and & distributes over arbitrary sups in both variables (c. [10]). We begin with the ollowing example (c. [6, 7]). Example 4.1 (Quantale Q(L)) LetL be a complete lattice. On the set Q(L) o all sup-preserving sel-maps on L we deine a partial ordering in a pointwise way. Then (Q(L), ) is again a complete lattice. In particular, the sup o S Q(L) is given by ( S)(a) = σ S σ(a) or all a L. The composition (σ 1 & σ 2 )(a) = σ 2 (σ 1 (a)), a L,

6 318 U. Höhle, T. Kubiak induces a semigroup operation & on Q(L) so that the resulting triple Q(L) = (Q(L),, &) becomes a quantale (in act, a unital quantale a detail not needed or our purposes). Let Q be an arbitrary quantale, and X and Y be sets. A Q-valued relation R : X Y is a map X Y R Q. Thecomposition o Q-valued relations R : X Y and S : Y Zis deined or all x X and z Z by (SR)(x, z) = y Y R(x,y)& S(y,z). When L = 2, the category o all Q(L)-valued relations can be identiied with Rel because Q(2) is just the two-point lattice. Now we observe that each Q(L)-valued relation R : X Y determines a suppreserving map L X (R) L Y deined or all L X and y Y by (R)( )(y) = x X R(x, y)( (x)). On the other hand, each sup-preserving map L X ϕ L Y induces a Q(L)-valued relation X Y (ϕ) Q(L) deined or all a L, x X, and y Y by (ϕ)(x, y)(a) = ϕ(a1 x )(y), where { a, z = x, a1 x (z) =, z x and is the bottom element o L. The ollowing provides a Q(L)-valued counterpart (in act, a generalization rom 2 to L) o Proposition 2.1. Proposition 4.2 Let R : X Y and S : Y Zbe Q(L)-valued relations, and let ϕ L Y ψ L Z be arbitrary maps. Then: L X (1) (SR) = (S) (R) and (ψϕ) = (ψ) (ϕ), (2) ( (R)) = R, and ( (ϕ)) = ϕ provided ϕ is sup-preserving. Proo The proos o (1) and the irst part o (2) ollows immediately rom the deinitions o all the involved compositions. Proving the second part o (2) requires a simple observation that each L X has the ollowing decomposition: = x X (x)1 x.

7 On regularity o sup-preserving maps: generalizing Zareckiĭ s theorem 319 Indeed, ( (ϕ))( )(y) = x X ϕ((x)1 x )(y) ( = ϕ x X = ϕ( )(y) (x)1 x )(y) or all L X and y Y. When L is completely distributive, then so is L X. Using Theorem 3.1 and Proposition 4.2 (also c. Note 1.2), we obtain our quantalic analogue o Zareckiĭ s theorem which when L = 2 yields the characterization o regularity o usual relations as given in Theorem 2.3. Theorem 4.3 Let L be a completely distributive lattice. Then a Q(L)-valued relation R : X Y is regular i (R)(L X ) is a completely distributive lattice. Open Access This article is distributed under the terms o the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited. Reerences 1. Bandelt, H.-J.: Regularity and complete distributivity. Semigroup Forum 19, (1980) 2. Erceg, M.A.: Functions, equivalence relations, quotient spaces and subsets in uzzy set theory. Fuzzy Sets Syst. 3, (1980) 3. Gierz, G., Homann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: A Compendium o Continuous Lattices. Springer, Berlin, Heidelberg, New York (1980) 4. Mac Lane, S.: Categories or the Working Mathematician. Springer, New York, Heidelberg, Berlin (1971) 5. Markowsky, G.: Idempotents and product representations with applications to semigroup o binary relations. Semigroup Forum 5, (1972) 6. Mulvey, C.J., Pelletier, J.W.: A quantisation o the calculus o relations. In: CMS Proceedings, vol. 13, pp Am. Math. Soc., Providence (1992) 7. Pelletier, J.W., Rosický, J.: Simple involutive quantales. J. Algebra 195, (1997) 8. Plemmons, R.J., West, M.T.: On the semigroup o binary relations. Pac. J. Math. 35, (1970) 9. Raney, G.N.: A subdirect-union representation o completely distributive lattices. Proc. Am. Math. Soc. 4, (1953) 10. Rosenthal, K.I.: Quantales and Their Applications. Pitman Research Notes in Mathematics, vol Longman Scientiic & Technical, Longman House, Burnt Mill, Harlow (1996) 11. Schein, B.M.: Regular elements o the semigroup o all binary relations. Semigroup Forum 13, (1976) 12. Xu, X.-Q., Luo, M.-K.: Regular relations and normality o topologies. Semigroup Forum 72, (2006) 13. Yang, J.C.: A theorem on the semigroup o binary relations. Proc. Am. Math. Soc. 22, (1969) 14. Zareckiĭ, K.A.: The semigroup o binary relations. Mat. Sb. 61, (1963) (in Russian)