2021-07-26T05:07:58Z
https://cgasa.sbu.ac.ir/?_action=export&rf=summon&issue=479
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2013
1
1
Front Matter
2013
12
01
https://cgasa.sbu.ac.ir/article_4828_41e2214bc1c00610f598b8812deba79c.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2013
1
1
Countable composition closedness and integer-valued continuous functions in pointfree topology
Bernhard
Banaschewski
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
2013
12
01
1
10
https://cgasa.sbu.ac.ir/article_4262_73b32f9f16cd67536694bb804916b55f.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2013
1
1
Concerning the frame of minimal prime ideals of pointfree function rings
Themba
Dube
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
2013
12
01
11
26
https://cgasa.sbu.ac.ir/article_4263_6f79ee547811c22128d166583042a1da.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2013
1
1
A pointfree version of remainder preservation
Themba
Dube
Inderasan
Naidoo
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
2013
12
01
27
58
https://cgasa.sbu.ac.ir/article_4264_91ce60eb77415d9197885588177906a7.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2013
1
1
Semigroups with inverse skeletons and Zappa-Sz$acute{rm e}$p products
Victoria
Gould
Rida-e-
Zenab
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 $ E\subseteq 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 $E\subseteq U$ 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
2013
12
01
59
89
https://cgasa.sbu.ac.ir/article_4265_12a60e203d8dba10858f7e6a02feadc2.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2013
1
1
A note on semi-regular locales
Wei
He
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
2013
12
01
91
101
https://cgasa.sbu.ac.ir/article_4266_cea1998c803fdf3e9a23488d516a8534.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2013
1
1
A characterization of a pomonoid $S$ all of its cyclic $S$-posets are regular injective
Xia
Zhang
Wenling
Zhang
Ulrich
Knauer
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
2013
12
01
103
117
https://cgasa.sbu.ac.ir/article_4267_6c49d6229329c34167027be1bc728633.pdf
Categories and General Algebraic Structures with Applications
CGASA
2345-5853
2345-5853
2013
1
1
Persian Abstracts
2013
12
01
https://cgasa.sbu.ac.ir/article_4829_4f1aa2c38aa118a25ddddddb53b8e144.pdf