G. Gratzer, General Lattice Theory (Akademie-Verlag, Berlin, 1978).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....w z, there is an element p A satisfying w z.InG.Gratzer and H. Lakser [4] the following observation was made: Lemma 1. In a strongly atomic complete lattice A, the equality pi # = # holds for any complete congruence # of A. We refer the reader to Crawley and Dilworth [1] and G. Gratzer [2] for the standard notation in lattice theory. The next lemma gives a simple characterization of complete congruences on a complete lattice. Recall that a lattice homomorphism from K to K # is said to be a bounded homomorphism if the set of inverse images of each element of K # has a least and ....

G. Gratzer, General Lattice Theory, Academic Press, New York, N. Y., 1978; Birkhauser Verlag, Basel; Akademie Verlag, Berlin.

....notation and background information for the general theory of algebras and varieties. The books of Gratzer [16] and Burris and Sankappanavar [7] are also valuable references. More information on lattices can be found in the books of Crawley and Dilworth [9] Davey and Priestley [10] Gratzer [15], Freese, Jezek, and Nation [13] and the forthcoming text of Nation [33] The authors are grateful to the referee for pointing to parts of our proofs that required greater explanation. 2 Extending Day s Doubling Construction Let L be a lattice, let F be a convex subset of L,andletGbe a lattice ....

G. Gratzer, General Lattice Theory, Academic Press, New York, 1978.

....a knowledge base forms a bounded distributive lattice. At this point it may be useful to recall some definitions. For the most part, our terminology follows Davey and Priestley [DP90] which is a good introduction to lattice theory. Other good books on lattice theory are Gratzer [G71] and Gratzer [G78]. A lattice is defined to be a partially ordered nonempty set in which every pair elements has both a least upper bound (or join) and a greatest lower bound (or meet) A lattice homomorphism is a function between lattices that preserves binary meets and binary joins. A lattice is bounded if it has ....

....KB where C 1 and C 2 are syntactic concepts, from a correct Birkhoff implementation for B. The key lattice theoretic result we need to solve this problem is an elegant theorem due to G. Gratzer and E. T. Schmidt that was originally presented in [GS58] A proof can also be found on p. 74 of [G78]. Theorem 6.1 (Gratzer and Schmidt) Let L be a distributive lattice, and let a; b 2 L be such that a b. Let j be the least congruence relation on L such that a j b. Then, for all x; y 2 L, the following are equivalent: 1. x j y. 2. x a = y a and x b = y b. Let P and Q be posets. We say ....

Gratzer, G., General Lattice Theory, Academic Press, New York, 1978.

....corollary 3.10. If its converse is true then by lemma 3.5 bisimulations will be closed under composition, thus L ( will preserve weak pullbacks. In this section we shall show that this and related properties hinge on a certain distributivity condition on the lattice L. Definition 4. 1 ([Gra98]) A lattice is called join infinite distributive (in short JID) if it satisfies the law x # # x i i # I = # x # x i i # I . If L is a complete semilattice, we shall say that L is JID i# this is the case for the lattice induced on L. Whenever L is JID, the ....

....m # # i#I l i = e # # i#I (e # l i ) # i#I (m # l i ) finishing the proof. The implication (i) # (iv) is due to S. Pfei#er [Pfe99] A finite lattice is distributive i# it does not contain one of the characteristic five element nondistributive sublattices M 3 or N 5 , see ([Gra98]) By separately excluding these cases, she also obtained the converse (iv) # (i) in case that L is finite and distributive. Observe that nonempty weak pullbacks along injective maps, resp. nonempty pullbacks of an arbitrary collection of injective maps, are always preserved, 11 without ....

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G. Gratzer, General lattice theory, Birkhauser Verlag, 1998.

....to build some algebraic structure on L(P ) From now on, let P = #P, ## be a fixed finite quasiordered set, i.e. # is a reflexive and transitive relation on the finite set P . For x, y # P , x y means that y # x and x ## y. For undefined terminology the reader is referred to Gratzer [4]. Even without explicit mentioning, all sets occurring in this paper are assumed to be finite. The set of all subsets, alias coalitions, of P is denoted by L(P ) For X, Y # L(P ) a map #: X # Y is called an extensive map if # is injective and for every x # X we have x # #(x) Let X ....

....lattice, is distributive, too. Since L n can be obtained from I n , D n 1 # # n (I n 1 ) and D n by using the Hall Dilworth gluing construction (cf. Hall and Dilworth [3] or, e.g. Gratzer [4, page 31, Ex. 20, 21] twice, and this gluing is well known to preserve distributivity, cf. 3] and [4], L n is also distributive. Secondly, let P = c 1 c 2 . c n and consider the chain Q = c 0 c 1 . c n . It is easy to show that S = #x 1 , x n # # Q n : x 1 # x 2 # . # x n 8 G ABOR CZ EDLI AND GY ORGY POLL AK and, for all i 1, x i ....

G. Gratzer, General Lattice Theory, Akademie-Verlag, Berlin, 1978.

....It is the goal of this paper to obtain analogous results for the automorphism groups of the countable universal homogeneous distributive lattice and the atomless generalized Boolean lattice. In lattice theory, the amalgamation property has been well investigated for many classes of lattices. See [13] for a survey. Since the class of finite distributive lattices (with lattice embeddings) has this property, by general model theoretic results ( 9] see also Theorem 7.1.2 of [15] there exists a (unique up to isomorphism) countable distributive lattice D which is homogeneous, i.e. any isomorphism ....

.... 0, again there is a unique countable universal homogeneous generalized Boolean lattice B : There is a sense in which B , as a generalized Boolean lattice, is gen2 erated by the rational interval [0; 1) whereas the atomless countable Boolean algebra is generated by the interval [0; 1] see [13]. These lattices D and B can be viewed as intermediates between the universal homogeneous poset and the atomless Boolean algebra; the two latter structures have simple automorphism groups by [11] and [2] However, for the universal homogeneous poset, the expected small index result has so far ....

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G. Gratzer: `General Lattice Theory', Birkhauser, Basel, 1978.

.... of such varieties have been exhibited by McKenzie in [30] Goralc ik, Koubek, and Sichler [13] represents the final result in a long sequence of papers which began with Gratzer and Sichler [21] For further references see [13] and, for further information on lattices in general, see Gratzer [19]. For any given variety V of (0; 1) lattices containing a finite non distributive simple (0; 1) lattice, the functor F : G Gamma V employed by Goralc ik, Koubek, and Sichler is the composition of some six functors as given in [23] 5] 6] and [13] As an application of 2.4, in this x we ....

G.Gratzer, General Lattice Theory (2nd Edition), Birkhauser Verlag, Basel-Boston, 1998.

....and denoted by a # b. A lattice is complete when each of its subsets has a l.u.b. and g.l.b. in #L, ##. Sometimes one wants to treat lattices as algebras. The triple #L, #, ## is equivalent to #L, ## if both binary operations # and # are idempotent, commutative and associative (see [6] Gratzer, 1998 for proof and further elaboration. Definition 7 (Semilattice) A poset #S, ## is a meet semilattice (or dually join semilattice) i# g.l.b. dually l.u.b. exists in S for any two elements in S. Definition 8 (Bilattice) A bilattice #L, # 1 , # 2 # is a poset with two ....

Gratzer, George (1998). General lattice theory, 2nd edition, Basel, Bosten, Berlin: Birkhauser Verlag.

....of the function x specified by the Pi expression will be the set of farmers, and its codomain will be the set of owned donkeys. If we adopt a lattice theoretic approach to the semantics of plural terms (Link, 1983) then the the pronoun can be resolved by replacing it by either the supremum (Gratzer, 1978) of the domain of x, to refer to the farmers, or by the supremum of the codomain of x to refer to the owned donkeys. There are two objections to this proposal. First, the use of the supremum of just the domain or codomain of a function is not sufficient in all cases. In the representations of some ....

Gratzer, G. (1978). General Lattice Theory. Academic Press, New York.

....we recall some definitions and basic properties concerning partially ordered sets and lattices. In order to make the paper self contained, we tried to mention all the results that we will use without proving them. For a rather complete presentation of this material the reader is refered to one of [Bir73, Gra78] and, in particular concerning semimodular lattices, to [Ste91] Let (P; be a partially ordered set and x; y 2 P . Then #x : fy 2 P j y xg is the principal ideal generated by x and x : fy 2 P j y xg denotes the principal filter generated by x. A set A P is downward closed if #a A for ....

G. Gratzer. General Lattice Theory. Birkhauser, 1978.

....The set of equivalence classes of a set of FDs F over R is a partition of F such that two FDs are in the same equivalence class if and only if the closures of their left hand sides are the same [MAIE80, MANN83] Assume that F satisfies the intersection property. We then show that L(F) is exchange [GRAT78] if and only if F satisfies the split freeness property and the cardinality of all the nonempty equivalence classes of F is maximal. Correspondingly, we show that L(F) is antiexchange [JAMI85] if and only if F satisfies the split freeness property and the cardinality of all the nonempty ....

....G is optimum, since jj(G Gamma H) Jjj jjGjj. The result that H is optimum follows. 2 6 The Lattice of Closed Sets Herein we give the definitions of the lattice theoretic concepts used in the rest of the paper. The reader is referred to [DAVE90] for an introduction to lattice theory and to [GRAT78] for more advanced material. The operator C F (see Definition 4.1) which closes sets of attributes in sch(R) is a closure operator in the lattice theoretic sense [DAVE90] It follows by [DAVE90, Theorem 2.21] that the family of all the closed sets in the power set of sch(R) is a lattice partially ....

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G. Gratzer, General Lattice Theory. Academic Press, New York, 1978.

....MATHEMATICUM (BRNO) Tomus 32 (1996) 163 165 SEMIMODULARITY IN LOWER CONTINUOUS STRONGLY DUALLY ATOMIC LATTICES Andrzej Walendziak Abstract. For lattices of finite length there are many characterizations of semimodularity (see, for instance, Gratzer [3] and Stern [6] 8] The present paper deals with some conditions characterizing semimodularity in lower continuous strongly dually atomic lattices. We give here a generalization of results of paper [7] 1. Preliminaries Let L be a lattice. By [a; b] a b; a; b 2 L) we denote an interval, ....

....called (upper) semimodular, if for all a; b 2 L, ab Gamma Gamma a implies b Gamma Gamma a b. It is immediate that modular lattices and geometric lattices are semimodular. There are many semimodular lattices being neither modular nor geometric (see Birkhoff [1] Crawley Dilworth [2] Gratzer [3] and Stern [8] 2. Results First we put J(L) fu 2 L : u = a b implies u = a or u = bg. The elements of J(L) are called the join irreducibles of L. In a strongly dually atomic lattice L the unique lower cover of a join irreducible (0 6= u 2 J(L) is denoted by u 0 . As a preparation we ....

Gratzer, G., General Lattice Theory, Birhauser Basel, 1978.

....surveys related work and discusses open problems. 2. Preliminaries We first review a few basic definitions and concepts of lattice theory that underlie our main results. This background material may be found in any introductory text on lattice theory; for example, Birkhoff [8] or Gratzer [30]. After that we describe the problem specification language SQ to be used in illustrating transformations for computing and recomputing fixed points. 2.1. Definitions A poset (L, is a reflexive, transitive, antisymmetric, binary relation on a set L. A poset (L, has a minimal element y iff ....

Gratzer, G. General Lattice Theory. Birkhauser, 1978.

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G. Gratzer, General Lattice Theory (Akademie-Verlag, Berlin, 1978).

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G. Gratzer, General Lattice Theory (Akademie-Verlag, Berlin, 1978).

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G. Gratzer. General Lattice Theory. Birkhauser, Basel, 1978.

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Gratzer, G.: General Lattice Theory. Second edn. Birkhauser, Basel (1998)

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G. Gratzer. General Lattice Theory. Academic Press, New York, NY, 1978.

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G. Gratzer, General Lattice Theory, Birkhauser, 1978.

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G. Gratzer. General Lattice Theory. Academic Press, New York, NY, 1978.

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G. Gratzer. General Lattice Theory. Brikhauser Verlag, 2nd edition, 1998.

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G. Gratzer. General Lattice Theory. Birkhauser-Verlag, 1978.

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G. Gratzer, General Lattice Theory. Second Edition, Birkhauser Verlag, Basel. 1998. xix+663 pp.

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G. Gratzer, General Lattice Theory. Second Edition, Birkhauser Verlag, Basel. 1998. xix+663 pp.

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G. Gratzer. General Lattice Theory. Birkhauser, second edition, 1998.

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