following the categorical approach to universal algebra through algebraic theories, proposed by f.~w.~lawvere in his phd thesis, this paper aims at introducing a similar setting for general topology. the cornerstone of the new framework is the notion of emph{categorically-algebraic} (emph{catalg}) emph{topological theory}, whose models induce a category of topological structures. we introduce t...

Let (K, |·|) be a complete discretely valued field and f : B1(K, 1) → B(K, 1) a nonconstant analytic map from the unit back to itself. We assume that 0 is an attracting fixed point of f . Let a ∈ K with limn→∞ f(a) = 0 and consider the orbit Of (a) := {f (a) : n ∈ N}. We show that if 0 is a superattracting fixed point, then every irreducible analytic subvariety of Bn(K, 1) meeting Of (a) n in a...

We prove that the Büchi topology, the automatic topology, the alphabetic topology and the strong alphabetic topology are Polish, and provide consequences of this. 1998 ACM Subject Classification F.1.1 Models of Computation, F.1.3 Complexity Measures and Classes, F.4.1 Mathematical Logic, F.4.3 Formal Languages

Given a surjective endomorphism $$f: X \rightarrow X$$ on projective variety over number field, one can define the arithmetic degree $$\alpha _f(x)$$ of f at point x in X. The Kawaguchi–Silverman Conjecture (KSC) predicts that any forward f-orbit which is strictly smaller than first dynamical $$\delta _f$$ not Zariski dense. We extend KSC to sAND (= small Arithmetic Non-Density) locus $$Z_f(d)$...

The objective of these notes is to present a few important results about complete ideals in 2–dimensional regular local rings. The fundamental theorems about such ideals are due to Zariski found in appendix 5 of [26]. These results were proved by Zariski in [27] for 2dimensional polynomial rings over an algebraically closed field of characteristic zero and rings of holomorphic functions. Zarisk...

In this paper, we introduce the notions of Belluce lattice associated with a bounded $BCK$-algebra and reticulation $BCK$-algebra. To do this, first, define operations $curlywedge ,$ $curlyvee $ $sqcup on $BCK$-algebras study some algebraic properties them. Also, for $A$ Zariski topology Spec(A)$ induced $tau _{A,Max(A)}$ $Max(A)$. We prove $(Max(A),tau_{A,Max(A)})$ is compact topological space...