Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58531120131201Countable composition closedness and integer-valued continuous functions in pointfree topology1104262ENBernhardBanaschewskiDepartment of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada.Journal Article20140118For any archimedean$f$-ring $A$ with unit in whichbreak$awedge (1-a)leq 0$ for all $ain A$, the following are shown to be equivalent: 1. $A$ is isomorphic to the $l$-ring ${mathfrak Z}L$ of all integer-valued continuous functions on some frame $L$. 2. $A$ is a homomorphic image of the $l$-ring $C_{Bbb Z}(X)$ of all integer-valued continuous functions, in the usual sense, on some topological space $X$. 3. For any family $(a_n)_{nin omega}$ in $A$ there exists an $l$-ring homomorphism break$varphi :C_{Bbb Z}(Bbb Z^omega)rightarrow A$ such that $varphi(p_n)=a_n$ for the product projections break$p_n:{Bbb Z^omega}rightarrow Bbb Z$. This provides an integer-valued counterpart to a familiar result concerning real-valued continuous functions.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58531120131201Concerning the frame of minimal prime ideals of pointfree function rings11264263ENThembaDubeDepartment of Mathematical Sciences, University of South Africa, P.O.
Box 392, 0003 Unisa, South Africa.Journal Article20140118Let $L$ be a completely regular frame and $mathcal{R}L$ be the ring of continuous real-valued functions on $L$. We study the frame $mathfrak{O}(Min(mathcal{R}L))$ of minimal prime ideals of $mathcal{R}L$ in relation to $beta L$. For $Iinbeta L$, denote by $textit{textbf{O}}^I$ the ideal ${alphainmathcal{R}Lmidcozalphain I}$ of $mathcal{R}L$. We show that sending $I$ to the set of minimal prime ideals not containing $textit{textbf{O}}^I$ produces a $*$-dense one-one frame homomorphism $beta Ltomathfrak{O}(Min(mathcal{R}L))$ which is an isomorphism if and only if $L$ is basically disconnected.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58531120131201A pointfree version of remainder preservation27584264ENThembaDubeDepartment of Mathematical Sciences, University of South Africa, P.O. Box 392, 0003 Unisa, South
Africa.InderasanNaidooDepartment of Mathematical Sciences, University of South
Africa, P.O. Box 392, 0003 Unisa, South Africa.Journal Article20140118Recall that a continuous function $fcolon Xto Y$ between Tychonoff spaces is proper if and only if the Stone extension $f^{beta}colon beta Xtobeta Y$ takes remainder to remainder, in the sense that $f^{beta}[beta X-X]subseteq beta Y-Y$. We introduce the notion of ``taking remainder to remainder" to frames, and, using it, we define a frame homomorphism $hcolon Lto M$ to be $beta$-proper, $lambda$-proper or $upsilon$-proper in case the lifted homomorphism $h^{beta}colonbeta Ltobeta M$, $h^{lambda}colonlambda Ltolambda M$ or $h^{upsilon}colonupsilon Ltoupsilon M$ takes remainder to remainder. These turn out to be weaker forms of properness. Indeed, every proper homomorphism is $beta$-proper, every $beta$-proper homomorphism is $lambda$-proper, and $lambda$-properness is equivalent to $upsilon$-properness. A characterization of $beta$-proper maps in terms of pointfree rings of continuous functions is that they are precisely those whose induced ring homomorphisms contract free maximal ideals to free prime ideals.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58531120131201Semigroups with inverse skeletons and Zappa-Sz$acute{rm e}$p products59894265ENVictoriaGouldDepartment of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom.Rida-e-ZenabDepartment of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom.Journal Article20140118The aim of this paper is to study semigroups possessing $E$-regular elements, where an element $a$ of a semigroup $S$ is {em $E$-regular} if $a$ has an inverse $a^circ$ such that $aa^circ,a^circ a$ lie in $ Esubseteq E(S)$. Where $S$ possesses `enough' (in a precisely defined way) $E$-regular elements, analogues of Green's lemmas and even of Green's theorem hold, where Green's relations ${mathcal R},{mathcal L},{mathcal H}$ and $mathcal D$ are replaced by $widetilde{{mathcal R}}_E,widetilde{{mathcal L}}_E, widetilde{{mathcal H}}_E$ and $widetilde{mathcal{D}}_E$. Note that $S$ itself need not be regular. We also obtain results concerning the extension of (one-sided) congruences, which we apply to (one-sided) congruences on maximal subgroups of regular semigroups. If $S$ has an inverse subsemigroup $U$ of $E$-regular elements, such that $Esubseteq U$ and $U$ intersects every $widetilde{{mathcal H}}_E$-class exactly once, then we say that $U$ is an {em inverse skeleton} of $S$. We give some natural examples of semigroups possessing inverse skeletons and examine a situation where we can build an inverse skeleton in a $widetilde{mathcal{D}}_E$-simple monoid. Using these techniques, we show that a reasonably wide class of $widetilde{mathcal{D}}_E$-simple monoids can be decomposed as Zappa-Sz$acute{rm e}$p products. Our approach can be immediately applied to obtain corresponding results for bisimple inverse monoids.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58531120131201A note on semi-regular locales911014266ENWeiHeInstitute of Mathematics, Nanjing Normal University, Nanjing, 210097, China.Journal Article20140118Semi-regular locales are extensions of the classical semiregular spaces. We investigate the conditions such that semi-regularization is a functor. We also investigate the conditions such that semi-regularization is a reflection or coreflection.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58531120131201A characterization of a pomonoid $S$ all of its cyclic $S$-posets are regular injective1031174267ENXiaZhangSchool of Mathematical Sciences, South China Normal University, 510631 Guangzhou, China.WenlingZhangSchool of Mathematical Sciences, South China Normal University, 510631 Guangzhou, China.UlrichKnauerInstitut fuer Mathematik, Carl von Ossietzky University, D-26111 Oldenburg, Germany.Journal Article20140118This work is devoted to give a charcaterization of a pomonoid $S$ such that all cyclic $S$-posets are regular injective.