Let L be a finite distributive lattice, and let J(L) denote the set of all join-irreducible elements of L. Set j (L) = IJ(Z)l. For each a C J(L), let u(a) denote the number of elements in the prime filter {x C L: x >~a}. Our main theorem is Theorem 1. For any finite distributive lattice L, 4 "(a) ~>j(L)41q ,'2. aEJ(L) The base 4 here can most likely be replaced by a smaller number, but it canno...

This paper studies a fascinating type of filter in residuated lattices, the so-called pure filters. A combination algebraic and topological methods on filters lattice is applied to obtain some new structural results. The notion purely-prime has been investigated, Cohen-type theorem obtained. It shown that spectrum compact sober space, Grothendieck-type demonstrated. proved Gelfand Hausdorff ded...

The use of an arterial line filter (ALF) within the pediatric cardiopulmonary bypass (CPB) circuit is not a new concept. It has always presented the perfusionist with a circuit component that while valuable, increased prime volume. The purpose of this study was to evaluate the change in prime volume and emboli between a conventional oxygenator with separate ALF and a new generation oxygenator w...

The usual definition facilities in theorem provers cannot handle all recursive functions on lazy lists; the filter function is a prime counterexample. We present two new ways of directly defining functions like filter by exploiting their dual nature as producers and consumers. Borrowing from domain theory and topology, we define them as a least fixpoint (producer view) and as a continuous exten...

The usual definition facilities in theorem provers cannot handle all recursive functions on lazy lists; the filter function is a prime counterexample. We present two new ways of directly defining functions like filter by exploiting their dual nature as producers and consumers. Borrowing from domain theory and topology, we define them as a least fixpoint (producer view) and as a continuous exten...

1. Results. Scattered results about countably2 generated ideals in C(X) are established in [2] and [4] : Op is countably generated if and only if pEX and p has a countable base of neighborhoods; Op is both prime and countably generated if and only if Mp is principal, and if and only if pEX and p is isolated; no lower prime ideal is countably generated. We generalize these as follows: if Mp is c...

‎let $r$ be a domain with quotiont field $k$‎, ‎and‎ ‎let $n$ be a submodule of an $r$-module $m$‎. ‎we say that $n$ is‎ ‎powerful (strongly primary) if $x,yin k$ and‎ ‎$xymsubseteq n$‎, ‎then $xin r$ or $yin r$ ($xmsubseteq n$‎ ‎or $y^nmsubseteq n$ for some $ngeq1$)‎. ‎we show that a submodule‎ ‎with either of these properties is comparable to every prime‎ ‎submodule of $m$‎, ‎also we show tha...

let $n$ be a submodule of a module $m$ and a minimal primary decomposition of $n$ is known‎. ‎a formula to compute baer's lower nilradical of $n$ is given‎. ‎the relations between classical prime submodules and their nilradicals are investigated‎. ‎some situations in which semiprime submodules can be written as finite intersection of classical prime submodule are stated‎.

in this work, by employing the krasnosel'skii fixed point theorem, we study the existence of positive solutions of a three-point boundary value problem for the following fourth-order differential equation begin{eqnarray*} left { begin{array}{ll} u^{(4)}(t) -f(t,u(t),u^{prime prime }(t))=0 hspace{1cm} 0 leq t leq 1, & u(0) = u(1)=0, hspace{1cm} alpha u^{prime prime }(0) - beta u^{prime prime...