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《Journal of Sichuan University (Natural Science Edition)》 1980-04
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INTERSECTION OPERATION ON UNION-PRESERVING MAPPINGS IN COMPLETELY DISTRIBUTIVE LATTIES

Liu Ying-ming  
The completely distributive lattices[3] are a suitable framework to expound theory of L-fuzzy set [2] . Meanwhile, in some recent papers on fuzzy topology[9][10], the importance of investigation about the intersection operation on union-preserving mappings in completely distributive lattices became apparent. Especially, Button' s formula on intersection operation[9] is useful. In this note, we shall show that this formula does not hold for almost all completely distributive lattices by an example, Moreover basing on an added assumption, a proof of this formula is given. We shall also show some properties about the intersection operation which will be needed in our latter works on fuzzy uniformities and fuzzy metrization. We will give following results, where L is a completely distributive lattice, and I is an index set (may be infinite) .Definition. Mapping f L-L will be called order-preserving iff implies for L. Mapping f: L-L will be called union-preserving ifff(a,)forAs given in[9], for union-preserving mappins f, g: L-L we may define the intersection of f and g, denoted by f g.Counterexample. Let L be a completly distributive lattice and contain an clement (denoted by b) different to 0 and 1. Define f1 f L-L byClearly, f1 and f2 satisfy all conditios of [9; Lemma3]. But for a=b, Hutton's formula doea not hold.Theorem. Let f1f2: L-L be union-presesviny mappings satifying f1(0) =f2 (0): Then for every aL, Especially, the associative law for theintersection operation holds.Proposition 1 Let f, g, f1 g1 : L-L be union-preserving mappings; Then about the composition of mappings, we haveProposition 2 Let L1 and L2 be completely distibutive lattices. Let the correspondences G: L1-L2 and H:L1-L2 be union-preserving. Then for every union-preserviny mapping g in L2, defne : L1-L2 byWe have (1) (g) is union-preserving in L1; and(2) uuion-preserving mappings (in L2) g1 and g2:
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