We give conclusive answers to some questions about definability in analytic languages that arose shortly after the work by Denef and van den Dries, [DD], on p-adic subanalytic sets, and we continue the study of nonarchimedean fields with analytic structure of [LR3], [CLR1] and [CL1]. We show that the language LK consisting of the language of valued fields together with all strictly convergent p...

The notion of strict closedness rings was given by J. Lipman [Amer. Math. 93 (1971), pp. 649–685] in connection with a conjecture O. Zariski. present purpose is to give practical method construction strictly closed rings. It also shown that the Stanley-Reisner simplicial complexes (resp. $F$-pure satisfying condition $(\operatorname {S}_2)$ Serre) are weakly Arf)