Cederquist, J.; Coquand, Th. Entailment relations and distributive lattices. Logic Colloquium '98 (Prague), 127-139, Lect. Notes Log., 13,  Home/Search   Document Details and Download   Summary   Related Articles   Check

....(a) Lemma 3.3. If A; B are two nite subsets of S such that a2A (a) b2B (b) then there exists i 2 I and two nite subsets X; Y of D i such that x2X x y2Y y in D i and (i; x) 2 A for x 2 X and (i; y) 2 B for y 2 Y . Proof. This follows for instance from the method presented in [5]. Lemma 3.4. If a 2 D i and i (a) x = 1 in D then there exists y 2 D i such that a y = 1 in D i and i (y) x. Proof. We can assume x to be a disjunction of elements of the form j (b) in which case this follows from lemma 3.3 (with A = We can now prove the theorem. Proof. Assume a ....

....Sp(D j ; j ) Sp(D i ; i ) where each i is a compact covering on the lattice D i such that a b i a b in D i . For i j we have a lattice map f ij : D i Idl(D j ; j ) such that a i U f ij (a) j f ij (U) We then let S be ( i 2 I)D i and be the entailment relation [5] on S de ned by (i 1 ; a 1 ) i n ; a n ) j 1 ; b 1 ) j m ; b m ) i there exists k i 1 ; i n ; j 1 ; j m such that i 1 k (a 1 ) f i nk (a n ) f j 1 k (b 1 ) f jmk (b m ) in the frame Idl(D k ; k ) We let D; m : S D ....

[Article contains additional citation context not shown here]

Cederquist, J.; Coquand, Th. Entailment relations and distributive lattices. Logic Colloquium '98 (Prague), 127-139, Lect. Notes Log., 13,

....2 D i such that oe i (y) x and a y = 1 in D i . COMPACT SPACES AND DISTRIBUTIVE LATTICES 3 Proof. If x 2 D then x can be written as a conjunction m k of disjunctions of elements of the form oe l u l . If we have 1 = oe i (a) x then for each k we have 1 = oe i (a) m k : Using the results of [3], recalled below in the appendix, the disjunction m k should contain oe i v; v 2 L k for some L k fin D i such that u L k = 1 2 D i . We can then take y = L k : Corollary 3.3. Let D i be a family of distributive lattices and D; oe i : D i D its coproduct. If a 2 D i and U 2 Idl(D i ) a OE ....

....same pattern as the previous proof of Tychonoff s theorem. We represent each spaces X i as Sp(D i ; i ) with i OE : For each i j we have a map f ij : D i Idl(D j ; j ) such that a i U f ij (a) f ij (U) Let then S be ( Sigmai 2 I)D i and be the entailment relation on S [3] defined by (i 1 ; a 1 ) i n ; a n ) j 1 ; b 1 ) j m ; b m ) iff there exists k i 1 ; i n ; j 1 ; j m such that f i 1 k (a 1 ) Delta Delta Delta f i nk (a n ) f j 1 k (b 1 ) Delta Delta Delta f jmk (b m ) Let D; m : S D be the ....

[Article contains additional citation context not shown here]

Cederquist, J. and Coquand, Th. Entailment relations and distributive lattices. Logic Colloquium '98 (Prague), 127--139, Lect. Notes Log., 13,

....rules (all but the last one) We have the following characterisation. Theorem 6.2. a 1 ; a n fin b 1 ; b m iff 1 a j ) for some N . In the special case where n = 0 we get that 1 N(b j ) for some N . If m = 0 we get that a i = 0: Proof. We follow the method of [5], and prove that the relation 1 a j ) for some N is an entailment relation, and that it validates all the axioms of fin . We have to prove that, if we have for some positive a 1; x; y and some N a x Nb a N(b x) then a Mb for some M . But we have a Na and Na N(b x) N(b (a ....

J. Cederquist and Th. Coquand. Entailment Relations and Distributive Lattices. Logic Colloquium '98 (Prague), 127--139, Lect. Notes Log., 13.

....but the last one) We have the following characterisation. Theorem 3.2. a 1 ; a n fin b 1 ; b m iff 1 a i N(b j ) for some N . In the special case where n = 0 we get that 1 N(b j ) for some N . If m = 0 we get that a i = 0: Proof. We follow the method of [4], and prove that the relation 1 a i N(b j ) for some N is an entailment relation, and that it validates all the axioms of fin . We have to prove that, if we have for some positive a 1; x; y and some N a x Nb a N(b x) then a Mb for some M . But we have a Na and Na N(b x) ....

J. Cederquist and Th. Coquand. Entailment Relations and Distributive Lattices. In Logic Colloqium, 1998. 10 THIERRY COQUAND

....model over a boolean algebra associated to the ring of parameters. This may be useful to structure some proofs and algorithms, like the one of computing a comprehensive Grobner bases [BW91] 2 Spectrum of a Ring and Constructible Topology We recall the construction of the spectrum of a ring [CC98]. Let A be a commutative ring. The relation A B is defined to mean that the product of the elements of B belongs to the radical of the ideal generated by A. Theorem 2.1 is an entailment relation on A: It is the least entailment relation on A such that: ffl 0; ffl 1 ; ffl x xy; ffl ....

....is an entailment relation on A: It is the least entailment relation on A such that: ffl 0; ffl 1 ; ffl x xy; ffl xy x; y; ffl x; y x y: A point for this entailment relation is exactly a prime ideal of A. We recall next the embedding of a distributive lattice in a boolean algebra [CC98]. Let D be a distributive lattice. We are interested in the following problem: to find a lattice map i : D B from D in a boolean algebra B such that, if B 0 is any boolean algebra and f : D B 0 any lattice map there exists a unique lattice map f 0 : B B 0 such that f 0 i = f: We ....

J. Cederquist and Th. Coquand. Entailment relations and distributive lattices. submitted, 1998.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC