Let $K$ be a field. The \'etale open topology on the $K$-points $V(K)$ of $K$-variety $V$ was introduced in our previous work. is non-discrete if and only large. If separably, real, $p$-adically closed then agrees with Zariski, order, valuation topology, respectively. We show that existentially definable sets perfect large fields behave well respect to this topology: such are finite unions subs...

<p>We continue the study of freezing sets in digital topology, introduced [4]. We show how to find a minimal set for "thick" convex disk X plane Z^2. give examples showing significance assumption that is convex.</p>

‎in this article we introduce $mu$-filtered fuzzy module with a family of fuzzy submodules.  it shows the relation between $mu$-filtered fuzzy modules and crisp filtered modules by level sets. we investigate fuzzy topology on the $mu$-filtered fuzzy module and apply that to introduce fuzzy completion. finally we extend krull's intersection theorem of fuzzy ideals by using concept $mu$-adic...

The backing of the fuzzy ideal is normal ideal in some ring and in same time there fuzzy set whose is not fuzzy ideal and it backing set is ideal, i.e., it crisp is normal ideal. Consequently, in this paper we constructing a fuzziness function which defined on fuzzy sets and assigns membership grade for every fuzzy set whose it backing set are crisp ideal. Now, Let  be collection of all fuzzy s...

The paper sets forth in detail categorically-algebraic or catalg foundations for the operations of taking the image and preimage of (fuzzy) sets called forward and backward powerset operators. Motivated by an open question of S. E. Rodabaugh, we construct a monad on the category of sets, the algebras of which generate the fixed-basis forward powerset operator of L. A. Zadeh. On the next step, w...

The pointfree transform offers to the predicate calculus what the Laplace transform offers to the differential/integral calculus: the possibility of changing the underlying mathematical space so as to enable agile algebraic calculation. This paper addresses the foundations of the transform and its application to a calculational approach to extended static checking (ESC) in the context of abstra...