On neat atom structures for cylindric like algebras

Abstract: (1) Let 1\leq k\leq \omega. Call an atom structure \alpha weakly 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=\alpha, \A\in \Nr_n\CA_{\omega} and for every algebra $\B$ based on this atom structure there exists k\in \omega$, k\geq 1, such that \B\in \Nr_n\CA_{n+k}. (2) Let k\leq \omega. Call an atom structure \alpha k complete, if there exists \A such that \At\A=\alpha and \A\in S_c\Nr_n\CA_{n+k}. (3) Let k\leq \omega. Call an atom structure $\alpha$ k neat if there exists \A such that \At\A=\alpha, and \A\in \Nr_n\CA_{n+k}. (4) Let K\subseteq \CA_n, and \L be an extension of first order logic. We say that \K is well behaved w.r.t to \L, if for any \A\in \K, A atomic, and for any any atom structure \beta such that \At\A is elementary equivalent to \beta, for any \B, \At\B=\beta, 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.