类圆柱代数的原子正则性和完备表示，以及一阶逻辑的clque保护片断的省略类型
Atom-canonicity and complete representations for cylindric-like algebras, and omitting types for the clque guarded fragment of first order logic
摘要：
固定一个有限序数n>2。我们证明了存在一个原子的、简单的、可数的可表示CA_n，使得它的最小完备化在SNr_ nCA_{n+3}之外.因此，对于任何有限\geq 3，簇SNr_ nCA_{n+k}不是原子正则的，所以具有n+ k-平坦表示的CA_n的簇也不是原子正则的.我们证明了，对于有限\geq 3，S_cNr_ nCA_{n+k}不是初等的，因此具有完全n+3-光滑表示的CA_n类不是初等的.我们得到类似的结果，取代平坦和光滑，分别由（较弱的概念）平方;这给一个更强的结果在这两种情况下，在这里我们可以允许k是无限的。我们的结果证明使用彩虹建设CA的。我们将原子正则性的否定结果提升到超限。本文还证明了对任意序数α \geq ω，对任意有限k \geq 1，对任意r \in ω，存在原子代数A_r \in SNr_ \alphaCA_{α+k} \sim SNr_ nCA_{α+k+1}，使得R_{r/U} A_r \in RCA_α，其中U是ω上的任意非主超滤子.收获我们的代数结果，我们调查了大量的第一逻辑的变体，包括其有限变量片段和包装片段的省略类型定理。
英文摘要：
Fix a finite ordinal n>2. We show that there exists an atomic, simple and countable representable CA_n, such that its minimal completion is outside SNr_nCA_{n+3}. Hence, for any finite k\geq 3, the variety SNr_nCA_{n+k} is not atom-canonical, so that the variety of CA_n's having n+k-flat representations is not atom-canonical, too. We show, for finite k\geq 3, that S_cNr_nCA_{n+k} is not elementary, hence the class of CA_n's having complete n+3-smooth representations is not elementary. We obtain analogous results by replacing flat and smooth, respectively, by (the weaker notion of) square; this give a stronger result in both cases and here we can allow k to be infinite. Our results are proved using rainbow constructions for CA's. We lift the negative result on atom-canonicity to the transfinite. We also show that for any ordinal α\geq ω, for any finite k\geq 1, and for any r\in ω, there exists an atomic algebra A_r\in SNr_\alphaCA_{α+k}\sim SNr_nCA_{α+k+1}, such that Π_{r/U} A_r\in RCA_α where U is any non--principal ultrafilter on ω. Reaping the harvest of our algebraic results we investigate a plethora of omitting types theorems for variants of first logic including its finite variable fragments and its packed fragment.
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