Various interplays between relation and cylindric algebras [article]

Tarek Sayed Ahmed
2013 arXiv   pre-print
Using model theoretic techniques that proved that the class of n neat reducts of m dimensional cylindric algebras, _n_m, is not elementary, we prove the same result for _k, k≥ 5, and we show that _k⊂ S_c_k for all k≥ 5. Conversely, using the rainbow construction for cylindric algebra, we show that several classes of algebras, related to the class _n_m, n finite and m arbitrary, are not elementary. Our results apply to many cylindric-like algebras, including Pinter's substitution algebras and
more » ... mos' polyadic algebras with and without equality. The techniques used are essentially those used by Hirsch and Hodkinson, and later by Hirsch in hh and r. In fact, the main result in hh follows from our more general construction. Finally we blow up a little the blow up and blur construction of Andréka nd Németi, showing that various constructions of weakly representable atom structures that are not strongly representable, can be formalized in our blown up, blow up and blur construction, both for relation and cylindric algebras. Two open problems are discussed, in some detail, proposing ideas. One is whether class of subneat reducts are closed under completions, the other is whether there exists a weakly representable ω dimensional atom structure, that is not strongly representable. For the latter we propose a lifting argument, due to Monk, applied to what we call anti-Monk algebras (the algebras, constructed by Hirsch and Hodkinson, are atomic, and their atom structure is stongly representable.)
arXiv:1305.5269v1 fatcat:cxeu6zqwgjfd5evo3i4cybij6u