2013/04/09 by Tarek Sayed Ahmed, Ahmed, Tarek Sayed
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Logic (math.LO) #math.LO
paper · pdf · doi:10.48550/arxiv.1304.2932
arXiv admin note: text overlap with arXiv:1304.1149
arxiv created 2013/04/09 · openalex publication_date 2013/04/09 · arxiv updated 2013/04/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
While every polyadic algebra (\PA) of dimension 2 is representable, we show that not every atomic polyadic algebra of dimension two is completely representable; though the class is elementary. Using higly involved constructions of Hirsch and Hodkinson we show that it is not elementary for higher dimensions a result that, to the best of our knowledge, though easily destilled from the literature, was never published. We give a uniform flexible way of constructing weak atom structures that are not strong, and we discuss the possibility of extending such result to infinite dimensions. Finally we show that for any finite n>1, there are two n dimensional polyadic atom structures \At1 and \At2 that are L∞,ω equivalent, and there exist atomic \A,\B∈ \PAn, such that \At\A=\At1 and \At\B= \At2, \A∈ \Nrn\PAω and \B∉ \Nrn\PAn+1. This can also be done for infinite dimensions (but we omit the proof)