(1)令1 \leq k \leq ω。称原子结构α为弱k纯可表示的，术语代数在\RCA_n \cap \Nr_n \CA_{n+k}中是可表示的，但复代数是不可表示的。如果存在原子代数\A，使得\At \A =α，\A \in \Nr_n \CA_ω，并且对于基于该原子结构的每个代数$\B$，存在k个\in ω $, k\geq 1, such that \B\in \Nr_n\CA_{n+k}.
(2) Let k\leq ω. Call an atom structure αk complete, if there exists \A such that \At\A=αand \A\in S_c\Nr_n\CA_{n+k}.
(3) Let k\leq ω. Call an atom structure $ α$ k neat，如果存在\A，使得\At \A =α，并且\A \in \Nr_n \CA_{n+k}，则称该原子结构为neat。(4)设K \subseteq \CA_n，K是一阶逻辑的扩充.我们说\K对于β是良好行为的，如果对于任何PDMATHABMPDMATHABMT \K，A原子，对于任何原子结构β，使得\At \A基本等价于β，对于任何\B，\At \B =β，则B \in K。我们调查这种结构的存在，以及相互联系。我们也提出了几个K和L的第二个定义。我们的结果推广到Pinter代数和多进代数的等式和不等式。
英文摘要：
(1) Let 1\leq k\leq ω. Call an atom structure αweakly k neat representable, the term algebra is in \RCA_n\cap \Nr_n\CA_{n+k}, but the complex algebra is not representable. Call an atom structure neat if there is an atomic algebra \A, such that \At\A=α, \A\in \Nr_n\CA_ω and for every algebra $\B$ based on this atom structure there exists k\in ω$, k\geq 1, such that \B\in \Nr_n\CA_{n+k}.
(2) Let k\leq ω. Call an atom structure αk complete, if there exists \A such that \At\A=αand \A\in S_c\Nr_n\CA_{n+k}.
(3) Let k\leq ω. Call an atom structure $α$ k neat if there exists \A such that \At\A=α, and \A\in \Nr_n\CA_{n+k}.
(4) Let K\subseteq \CA_n, and Łbe an extension of first order logic. We say that \K is well behaved w.r.t to Ł, if for any \A\in \K, A atomic, and for any any atom structure βsuch that \At\A is elementary equivalent to β, for any \B, \At\B=β, then B\in K.
We investigate the existence of such structures, and the interconnections. We also present several K's and L's as in the second definition. All our results extend to Pinter's algebras and polyadic algebras with and without equality.
Completeness and interpolation for intuitionistic infinitary predicate logic, in connection to finitizing the class of representable Heyting polyadic algebras
Splitting methods in algebraic logic: Proving results on non-atom-canonicity, non-finite axiomatizability and non-first oder definability for cylindric and relation algebras
Atoms in infinite dimensional free sequence-set algebras
A. Tarski证明了当$m\ge 1$是有限基数，$α$是任意序数时，维数为$α$的柱形代数类$\mathrm{CA}_α$的m-生成自由代数恰好包含$2^m$零维原子.他解释说，当$α$无限大时，就不再有原子了。这一猜想尚未得到证实或否认。本文证明了若$\mathrm{CA}_α$被$\mathrm{D}_α$，$\mathrm{G}_α$取代，Tarski猜想成立，但$m$生成的自由$\mathrm{Crs}_α$代数是无原子的.
