Gratzer.G, Lattice theory, Freeman, (1971).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(W m;1 ; t) t Gamma 1) t Gamma m Gamma 1) Note that W m;n is ranked in the ordinary sense and Greene computed (W m;n ; t) using the usual rank function. But, by Proposition 6.6, this rank function coincides with ours. For our second example we will use the Tamari lattices [8, 9, 10, 11, 14]. Consider all proper parenthesizations of the word x 1 x 2 : x n 1 . It is well known that the number of these is the Catalan number C n . Partially order this set by saying that oe covers whenever = AB)C) and oe = A(BC) for some subwords A; B;C. The ....

G. Gratzer, "Lattice Theory," Freeman and Co., San Francisco, CA, 1971, pp. 17--18, problems 26-36.

.... atom from each A i andallNBBbasesofagivenx # L have the same cardinality denoted #(x) 2) If we define #(L, t) x#L with # as in (1) then #(L, t) t A 1 ) t A 2 ) t A n ) This theorem can be used on lattices where Stanley s theorem does not apply, e.g. the Tamari lattices [22, 23, 29] and certain shu#e posets [24] Note also that we cannot drop the level condition which replaced semimodularity completely: If one considers the non crossing partition lattice then it has the same modular chain as # n . However, it does not satisfy the level condition and its characteristic ....

G. Gratzer, "Lattice Theory," Freeman and Co., San Francisco, CA, 1971, pp. 17--18, problems 26-36.

....r : V 7 Omega (the ranking function) such that s OE t iff r(s) r(t) The proof is simple and will not be given. A partial order satisfying any of the conditions of Lemma 14 will be called modular (this terminology is proposed in [12] as an extension of the notion of modular lattice of [13]) Definition 14 A ranked model W is a preferential model hV; l; OEi for which the strict partial order OE is modular. 19 Those models are called ranked since the effect of function r of property 4 of Lemma 14 is to rank the states: a state of smaller rank being more normal than a state of ....

George Gratzer. Lattice Theory. W. H. Freeman, San Francisco, 1971.

....(A Gamma B) The properties of the relation are many and delicate to prove. Again no proofs will be given. The main result we need about (all needed properties will easily follow) is that is modular, i.e. if A C, then, for any B, either A B, or B C. The term modular is taken from Gratzer [ Gratzer, 1971 ] Lemma 15 ffl If A B C, then A C. ffl If A B C, then A C. ffl If A B, then A B. ffl Assume B C. If A C, then either A B or B C. ffl Assume B C = If A C, then either A B or B C. ffl If A C, then, for any B, either A B or B C. ffl If A B C D, then A D. ffl ....

George Gratzer. Lattice Theory. W. H. Freeman, San Francisco, 1971.

....influence on the measurement result insofar as the outcome of a particular measurement is concerned. 2 Orthomodular Posets The appropriate algebraic structures to describe the logic of automata are found in the theory of orthomodular posets. Orthomodular structures arose from lattice theory [2, 10, 28] and quantum logic [1, 9] The basic notion of orthomodular posets will be defined first. Then, a new type of logic, termed partition logic, will be introduced. We shall prove a representation theorem, which identifies certain 3 orthomodular posets with partition logics. Some examples of the new ....

Gratzer, G. (1971). Lattice Theory, Freeman, San Francisco.

....is crucial in the proof of the representation result of Section 8.5. Lemma 8.3 If OE is a partial order on a set V , the following conditions are equivalent. A partial order satisfying them is called modular (this terminology is proposed in [9] as an extension of the notion of modular lattice of [10]) 1. for any x; y; z 2 V such that x 6OE y, y 6OE x and z OE x, then z OE y, 2. for any x; y; z 2 V such that x OE z, either x OE y or y OE z, 3. the relation 6OE is transitive, 4. there is a totally ordered set Omega (the strict order on Omega will be denoted by ) and a function r : V ....

George Gratzer. Lattice Theory. W. H. Freeman, San Francisco, 1971.

....We have (W m;1 ; t) t Gamma 1) m (t Gamma m Gamma 1) Note that W m;n is ranked in the ordinary sense and Greene computed (W m;n ; t) using the usual rank function. But, by Proposition 6.6, this rank function coincides with ours. For our second example we will use the Tamari lattices [8, 9, 10, 11, 14]. Consider all proper parenthesizations of the word x 1 x 2 : x n 1 . It is well known that the number of these is the Catalan number C n . Partially order this set by saying that oe covers whenever = AB)C) and oe = A(BC) for some subwords A; B;C. The ....

G. Gratzer, "Lattice Theory," Freeman and Co., San Francisco, CA, 1971, pp. 17--18, problems 26-36.

.... denoted ae(x) 2) If we define (L; t) P x2L (x)t ae( 1) Gammaae(x) with ae as in (1) then (L; t) t Gamma jA 1 j) t Gamma jA 2 j) Delta Delta Delta (t Gamma jA n j) This theorem can be used on lattices where Stanley s theorem does not apply, e.g. the Tamari lattices [22, 23, 29] and certain shuffle posets [24] Note also that we cannot drop the level condition which replaced semimodularity completely: If one considers the non crossing partition lattice then it has the same modular chain as Pi n . However, it does not satisfy the level condition and its characteristic ....

G. Gratzer, "Lattice Theory," Freeman and Co., San Francisco, CA, 1971, pp. 17--18, problems 26-36.

....= Gamma1) jJj if w = w 0 (J) for some J S, 0 else. Bjorner actually derives the Mobius function from any interval [v; w] in PW . But this follows easily from the preceding proposition since there is a poset isomorphism [v; w] j [ 0; v Gamma1 w] Next we consider the Tamari lattices [5, 7, 9]. Consider the set of all proper parenthesizations of the word x 1 x 2 : x n 1 . It is well known that the number of such is the Catalan number C n = 1 n 1 i 2n n j . Partially order this set by saying that is covered by oe if = AB)C) and oe = A(BC) for ....

G. Gratzer, "Lattice Theory," Freeman and Co., San Francisco, CA, 1971, pp. 17--18, problems 26-36.

.... denoted ae(x) 2) If we define (L; t) P x2L (x)t ae( 1) Gammaae(x) with ae as in (1) then (L; t) t Gamma jA 1 j) t Gamma jA 2 j) Delta Delta Delta (t Gamma jA n j) This theorem can be used on lattices where Stanley s theorem does not apply, e.g. the Tamari lattices [16, 17, 20] and certain shuffle posets [18] Note also that we cannot drop the level condition which replaced semimodularity completely: If one considers the non crossing partition lattice then it has the same modular chain as Pi n . However, it does not satisfy the level condition and its characteristic ....

G. Gratzer, "Lattice Theory," Freeman and Co., San Francisco, CA, 1971, pp. 17--18, problems 26-36.

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Gratzer.G, Lattice theory, Freeman, (1971).

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