2015/08/21 by Tarek Sayed Ahmed, Ahmed, Tarek Sayed
Computer Science · #Advanced Algebra and Logic #FOS: Mathematics #Logic (math.LO) #Logic, Reasoning, and Knowledge #Logic, programming, and type systems
paper · pdf · doi:10.48550/arxiv.1508.05840
openalex publication_date 2015/08/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We take a long magical tour in algebraic logic, starting from classical\nresults on neat embeddings due to Henkin, Monk and Tarski, all the way to\nrecent results in algebraic logic using so--called rainbow constructions\ninvented by Hirsch and Hodkinson. Highlighting the connections with graph\ntheory, model theory, and finite combinatorics, this article aspires to present\ntopics of broad interest in a way that is hopefully accessible to a large\naudience. The paper has a survey character but it contains new approaches to\nold ones. We aspire to make our survey fairly comprehensive, at least in so far\nas Tarskian algebraic logic, specifically, the theory of cylindric algebras, is\nconcerned. Other topics, such as abstract algebraic logic, modal logic and the\nso--called (central) finitizability problem in algebraic logic will be dealt\nwith; the last in some detail. Rainbow constructions are used to solve problems\nadressing classes of cylindric--like algebras consisting of algebras having a\nneat embedding property. The hitherto obtained results generalize seminal\nresults of Hirsch and Hodkinson on non--atom canonicity, non--first order\ndefinabiity and non--finite axiomatizability, proved for classes of\nrepresentable cylindric algebras of finite dimension>2. We show that such\nresults remain valid for cylindric algebras possesing relativized it clique\nguarded representations that are it only locally well behaved. The paper is\nwritten in a way that makes it accessible to non--specialists curious about the\nstate of the art in Tarskian algebraic logic. Reaching the boundaries of\ncurrent research, the paper also aspires to be informative to the practitioner,\nand even more, stimulates her/him to carry on further research in main stream\nalgebraic logic.\n