J. Morgado.Some results on the closure operators of partially ordered sets. Portugal. Math., 19(2):101--139, 1960.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is a minimal decomposition for a closure ffi (i.e. for all k 2 I and j closure, ae k j) ffi (u i2Infkg ae i ) u j) iff for any k 2 I , ffi (u i2Infkg ae i ) ae k . Among the possible minimal decompositions, we consider those given by the most abstract factors. It is well known (cf. [22]) that if L is a complete lattice, then uco(L) is a dual atomistic complete lattice, viz. for any ae 2 uco(L) ae = u x2ae(L)nf g x , where x = f ; xg. We call the x s atomic closures . We exploit these results, by giving a sufficient condition in order that any closure operator, and ....

J. Morgado. Some results on the closure operators of partially ordered sets. Portug. Math., 19(2):101--139, 1960.

....X is a nonempty subset of Y . Let hP; i be a poset. If x 2 P then x def = fy 2 P j x yg. If x 2 P and Y P then we write x Y when, for any y 2 Y , x y. For any x 2 P , we de ne P x (P ) def = fY P j Y 6= x Y g. Note that if x y then P y (P ) P x (P ) Let us recall from [3] the following de nition of relative quasi in mum in posets, which is the main new notion introduced in Morgado s paper. De nition 2.1 ( 3, De nitions 4 and 5, pp. 118,120] Let P be a poset, x 2 P and Y 2 P x (P ) An element x MY 2 P is the quasi in mum of Y relative to x if the following ....

....If x MY exists for any x 2 P and Y 2 P x (P ) then P is called relatively quasi infcomplete. Thus, x MY is the greatest element in P satisfying conditions (i) and (ii) for x and Y . Note that if a relative quasi in mum exists then this is necessarily unique. Many proofs in Morgado s paper [3] from Section 5 forward, including the proof of the main theorem [3, Theorem 27, p. 138] make crucial use of the following property of relative quasi in ma [3, property (a) p. 118] If x y, Z 2 P y (P ) and hence Z 2 P x (P ) and both x MZ and y MZ exist, then x MZ y MZ. ....

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J. Morgado. Some results on the closure operators of partially ordered sets. Portugal. Math., 19(2):101-139, 1960.

....the complete lattice structure and allows the meet uniform closure to get co additivity. 2 Preliminaries In this section, we briefly introduce some notation used throughout the paper and summarize some definitions and well known properties concerning closure operators (for more details see [2, 11, 12]) Let C and D be sets. The powerset of C is denoted by (C ) and its cardinality by jC j. The set difference between C and D is denoted by C n D . If f is a function defined on C and D C then f (D) ff (x ) j x 2 Dg. Functions will be sometimes denoted by Church s lambda notation. The set C ....

J. Morgado. Some results on the closure operators of partially ordered sets. Portugal. Math., 19(2):101--139, 1960.

....bottom element. It is also known that uco(C ) is dual atomic, where the dual atoms of uco(C ) are all and only those closures OE x for x 2 C n f g, such that OE x (C ) fx ; g, i.e. for any y 2 C , OE x (y) ae x if y x ; otherwise. Further details on closure operators can be found in [48, 64]. 2.3 Logic Programming Notation In the following, Sigma, Pi and Var will respectively denote a set of function symbols, a set of predicate symbols and a denumerable set of variables, defining a fixed first order language L. The set of terms and atoms over L are denoted by Term and Atom, ....

....the disjunctive completion of Sign is the domain Sign 6=0 of Example 4.7. It is then obvious that these two abstractions of (ZZ) are uncomparable. 2 Whenever A is separated in D , the lattice of meet autodependencies Dep (A) enjoys all the well known properties of closure operators (see e.g. [48, 64] for a few of them) In particular, the following lattice theoretic properties are inherited from lco(A) i) Dep (A) is atomic 8 (cf. 64] ii) Dep (A) is distributive iff it is complemented iff it is Boolean iff A is a complete well ordered chain (cf. 49] iii) If A is join continuous ....

J. Morgado. Some results on the closure operators of partially ordered sets. Portug. Math., 19(2):101-- 139, 1960.

....on December 1997. 2 Abstract Interpretation Basics We present in this section a succinct overview of the basic notions on closure operators and abstract interpretation used throughout the article. For more details about closure operators the reader is referred to [Davey and Priestley 1990; Morgado 1960; Ward 1942] while for abstract interpretation to [Cousot and Cousot 1977; 1979b; 1992a] Basic Mathematical Notation. If S and T are sets, then (S ) denotes the powerset of S , jS j the cardinality of S , S n T denotes the set difference between S and T , S ae T denotes strict inclusion, and ....

....checked on f and g . This is important in order to verify full completeness for semantics inductively specified on the syntax of programs, as it is usually done in denotational semantics. Point (vi) is an instance of a more general result later in Theorem 4. 11: In fact it is known (cf. [Morgado 1960]) that if two closures ae and j are such that ae ffi j = j ffi ae then ae ffi j = ae t j. Point (vii) characterizes the set of fully complete abstractions for a function which is itself a closure operator. Finally, point (viii) says that the property of full completeness for an n ary function can ....

Morgado, J. 1960. Some results on the closure operators of partially ordered sets. Portug. Math. 19, 2, 101--139.

....and Sndergaard [13] is shown in [11] Also in this case, the construction does not directly come from the property of groundness, but from a domain more complex than G. 2 Preliminaries Throughout the paper we assume familiarity with lattice theory (e.g. see [3,12] abstract interpretation [5,6,17] and logic programming [1] 2.1 Notation and basic notions Let A, B and C be sets. A n B denotes the set theoretic difference between A and B, A ae B denotes proper inclusion and, if X A, X is the set theoretic complement of X . The powerset of A is denoted by (A) If A is a poset, we usually ....

....equivalence class those objects in A having the same image (meaning) in C. Let L be a complete lattice hL; L ; L ; L ; L ; L i playing the role of the concrete domain. An (upper) closure operator on L is an operator ae : L 7 Gamma L monotonic, idempotent and extensive (viz. 8x 2 L: x L ae(x) [17]. Each closure operator ae is uniquely determined by the set of its fixpoints, which is its image ae(L) ae(L) is a complete lattice with respect to L , but, in general, it is not a complete sublattice of L, since the join in ae(L) might be different from L . ae(L) is a complete sublattice of L ....

J. Morgado. Some results on the closure operators of partially ordered sets. PortugaliaeMathematica, 19(2):101--139, 1960. 15

....does not directly come from the basic property of groundness, but from a domain more complex than G. Moreover, no property of optimality has been taken into account in this construction. 2 Preliminaries Throughout the paper we assume familiarity with lattice theory [3,19] abstract interpretation [6,7,25] and logic programming [1] 6 2.1 Notation and basic notions Let A, B and C be sets. AnB denotes the set theoretic difference, A ae B the proper inclusion and, if X A, X is the set theoretic complement of X. X f A denotes that X is a finite subset of A. The powerset of A is denoted by (A) ....

....class those objects in A having the same image (meaning) in C. Let L be a complete lattice hL; L ; L ; L ; L ; L i playing the role of the concrete domain. An (upper) closure operator on L is a function ff : L 7 Gamma L monotone, idempotent and extensive (i.e. 8x 2 L: x L ff(x) [25]. Each closure operator ff is uniquely determined by the set of its fixpoints, which is its image ff(L) ff(L) is a complete lattice with respect to L , but, in general, it is not a complete sublattice of L, since the join in ff(L) might be different from L . ff(L) is a complete sublattice of L ....

J. Morgado. Some results on the closure operators of partially ordered sets. PortugaliaeMathematica, 19(2):101--139, 1960.

....This shows that Def , which is strictly less expensive than Pos ( 6, 19] always induces the same disjunctive ground dependency analysis, i.e. Omega Gamma Pos) Def . Throughout the paper, we assume familiarity with lattice theory (e.g. see [3, 14] in particular closure operators (see [20, 28]) and abstract interpretation ( 7, 8] 2 Abstract Interpretation and Closure Operators The standard Cousot and Cousot theory of abstract interpretation is based on the notion of Galois connection ( 7, 8] In this section, we briefly introduce some notation and recall some well known notions. ....

J. Morgado. Some results on the closure operators of partially ordered sets. Port. Math., 19(2):101--139, 1960.

....ordered set #L, ## is a monotonic and idempotent operator. An upper closure operator (uco) on L is a retraction # such that #x # L.x # #(x) extensive) a lowerclosure operator (lco) on L is a retraction # such that #x # L. #(x) # x (reductive) More on closure operators can be found in [CC79a, Mor60]. Let #L, #, #, #, #, ## be a nonempty complete lattice, and f : L # L. The upper ordinal powers of f are defined as follows: f # 0(X) X f # #(X) f(f # (# 1) X) for every successor ordinal #; and f # #(X) # # # f # #(X) for every limit ordinal # The first limit ....

J. Morgado. Some results on the closure operators of partially ordered sets.<F4.877e+05> Portugaliæ<F5.303e+05> Mathematica, 19(2):101--139, 1960.

....In x 5 we exploit this result to give sufficient conditions for uco(L) to be pseudo complemented. 2 Preliminaries In this section we summarize some definitions and well known properties concerning (relative) pseudo complementation and closure operators of complete lattices. For more details see [1, 6, 11]. Let L = hL; i be a complete lattice, where L is a set, is the partial ordering relation, is the meet, is the join, is the top element and is the bottom element. In the following, unless stated otherwise, L will stand for the structure hL; i. We will use V and W ....

J. Morgado. Some results on the closure operators of partially ordered sets. Portugal. Math., 19(2):101--139, 1960.

....give sufficient conditions for uco(L) to be pseudo complemented. Technical Report LIX RR 95 04 3 2 Preliminaries In this section we summarize some definitions and well known properties concerning (relative) pseudo complementation and closure operators of complete lattices. For more details see [1, 6, 11]. Let L = hL; i be a complete lattice, where L is a set, is the partial ordering relation, is the meet, is the join, is the top element and is the bottom element. In the following, unless stated otherwise, L will stand for the structure hL; i. We will use V and ....

J. Morgado. Some results on the closure operators of partially ordered sets. Portugal. Math., 19(2):101--139, 1960.

....requires certain hypotheses, ours always holds. The above isomorphisms turn out to be particularly useful for studying complete congruence lattices, because lattices of closure operators have been extensively investigated in the past, and many results on their structure are available, e.g. see [12]. Among the properties of interest that can be inherited from the lattice of closure operators, the property of pseudocomplementedness in complete congruence lattices is particularly important and, to the best of our knowledge, has not been studied yet. We exploit the relation between Con(C) and ....

J. Morgado.Some results on the closure operators of partially ordered sets. Portugal. Math., 19(2):101--139, 1960.

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J. Morgado.Some results on the closure operators of partially ordered sets. Portugal. Math., 19(2):101--139, 1960.

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J. Morgado. Some results on the closure operators of partially ordered sets. Portugal. Math., 19(2):101-139, 1960.

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