Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
1
1
2013
12
01
Countable composition closedness and integer-valued continuous functions in pointfree topology
1
10
EN
Bernhard
Banaschewski
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada.
For 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.
Frames,0-dimensional frames,integer-valued continuous
functions on frames,archimedean ${mathbb Z}$-rings,countable
$mathbb {Z}$-composition closedness
https://cgasa.sbu.ac.ir/article_4262.html
https://cgasa.sbu.ac.ir/article_4262_73b32f9f16cd67536694bb804916b55f.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
1
1
2013
12
01
Concerning the frame of minimal prime ideals of pointfree function rings
11
26
EN
Themba
Dube
Department of Mathematical Sciences, University of South Africa, P.O.
Box 392, 0003 Unisa, South Africa.
dubeta@unisa.ac.za
Let $L$ 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.
frame,ring of real-valued continuous functions on a
frame,minimal prime ideal,basically disconnected
https://cgasa.sbu.ac.ir/article_4263.html
https://cgasa.sbu.ac.ir/article_4263_6f79ee547811c22128d166583042a1da.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
1
1
2013
12
01
A pointfree version of remainder preservation
27
58
EN
Themba
Dube
Department of Mathematical Sciences, University of South Africa, P.O. Box 392, 0003 Unisa, South
Africa.
dubeta@unisa.ac.za
Inderasan
Naidoo
Department of Mathematical Sciences, University of South
Africa, P.O. Box 392, 0003 Unisa, South Africa.
naidoi@unisa.ac.za
Recall 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.
frame,remainder preservation,Stone-v{Cech} compactification,regular Lindel"{o}f coreflection,realcompact coreflection,proper map,lax proper map
https://cgasa.sbu.ac.ir/article_4264.html
https://cgasa.sbu.ac.ir/article_4264_91ce60eb77415d9197885588177906a7.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
1
1
2013
12
01
Semigroups with inverse skeletons and Zappa-Sz$acute{rm e}$p products
59
89
EN
Victoria
Gould
Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom.
victoria.gould@york.ac.uk
Rida-e-
Zenab
Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom.
rzz500@york.ac.uk
The 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.
Idempotents,$\mathcal{R}$,$\mathcal{L}$,restriction semigroups,Zappa-Sz$\acute{\rm e}$p products
https://cgasa.sbu.ac.ir/article_4265.html
https://cgasa.sbu.ac.ir/article_4265_12a60e203d8dba10858f7e6a02feadc2.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
1
1
2013
12
01
A note on semi-regular locales
91
101
EN
Wei
He
Institute of Mathematics, Nanjing Normal University, Nanjing, 210097, China.
weihe@njnu.edu.cn
Semi-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.
locale,semi-regular locale,semi-regularization
https://cgasa.sbu.ac.ir/article_4266.html
https://cgasa.sbu.ac.ir/article_4266_cea1998c803fdf3e9a23488d516a8534.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
1
1
2013
12
01
A characterization of a pomonoid $S$ all of its cyclic $S$-posets are regular injective
103
117
EN
Xia
Zhang
School of Mathematical Sciences, South China Normal University, 510631 Guangzhou, China.
xiazhang@scnu.edu.cn
Wenling
Zhang
School of Mathematical Sciences, South China Normal University, 510631 Guangzhou, China.
z$_{-}$wenling@126.com
Ulrich
Knauer
Institut fuer Mathematik, Carl von Ossietzky University, D-26111 Oldenburg, Germany.
ulrich.knauer@uni-oldenburg.de
This work is devoted to give a charcaterization of a pomonoid $S$ such that all cyclic $S$-posets are regular injective.
Promonoid,Regular injectivity,Cyclic $S$-poset
https://cgasa.sbu.ac.ir/article_4267.html
https://cgasa.sbu.ac.ir/article_4267_6c49d6229329c34167027be1bc728633.pdf