2014/01/02 by Tarek Sayed Ahmed, Ahmed, Tarek Sayed · 1 citation
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Topology and Set Theory #FOS: Mathematics #Logic (math.LO) #math.LO #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1401.1103
arXiv admin note: substantial text overlap with arXiv:1308.6165, arXiv:1307.1016, arXiv:1307.4298, arXiv:1309.0681
arxiv created 2014/01/02 · openalex publication_date 2014/01/02 · arxiv updated 2014/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let n be finite >2. We show that any class between S\Nrn\CAn+3 and RCAn is not atom canonical, and any class containing the class of completely representable algebras and contained in Sc\Nrn\CAn+3 is not elementary. We show that there is no finite variable universal axiomatization of many diagonal free reducts of representable cylindric algebras of dimension n, like the varieties of representable diagonal-free cylindric algebras and Halmos' polyadic algebras (without equality). We apply our hitherto obtained algebraic results to show that the omitting types theorem fails for finite variable fragments of first order logic with and without equality, having n variables, even if we count in severely relativized models as candidates for omitting single non-principle types. Finally, we show that for many cylindric-like algebras, like diagonal free cylindric algebras and Halmos' polyadic algebras with and without equality the class of strongly representable atom structures of finite dimension >2 is not elementary.