Cross-Connections - Revision history

Admin: /* References */

2014-04-09T00:32:47Z

‎References

Admin at 00:22, 9 April 2014

2014-04-09T00:22:12Z

Admin: /* References */

2014-04-08T14:30:38Z

‎References

Admin: /* References */

2014-04-08T12:00:59Z

‎References

Admin: Admin moved page Mtest to Cross-Connections without leaving a redirect

2014-04-05T18:17:36Z

Admin moved page Mtest to Cross-Connections without leaving a redirect

Kssn at 08:25, 4 April 2014

2014-04-04T08:25:01Z

Admin: /* {{anchor|sec-1}}Introduction */

2014-04-04T02:03:38Z

‎{{anchor|sec-1}}Introduction

Admin: /* {{anchor|sec-1}}Introduction */

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‎{{anchor|sec-1}}Introduction

Admin: /* Representations of $\pcon $ by transformations */

2014-04-03T14:37:40Z

‎Representations of $\pcon $ by transformations

Admin: /* Representations of $\pcon $ by transformations */

2014-04-03T14:33:21Z

‎Representations of $\pcon $ by transformations

@@ Line 1,159: / Line 1,159: @@
 | year=1961
 }}
 {{Citation
 | first1=A.R.
@@ Line 1,172: / Line 1,173: @@
 | year=2000
 | postscript=This is a continuing UGC Project
 }}
 {{Citation
 | first1=P.A.
 | last1=Grillet
-| title=Structure of regular semigroups--I: a representation
+| title=Structure of regular semigroups &mdash; I: a representation
 | journal=Semigroup Forum
 | volume=8
-| pages=177--183
+| pages=177-183
 | year=1974a
 }}
 {{Citation
 | first1=P.A.
 | last1=Grillet
-| title=Structure of regular semigroups--II: cross-connections
+| title=Structure of regular semigroups &mdash; II: cross-connections
 | journal=Semigroup Forum
 | volume=8
-| pages=254--259
+| pages=254-259
 | year=1974b
 }}
 {{Citation
 | first1=P.A.
 | last1=Grillet
-| title=Structure of regular semigroups--III: the reduced case
+| title=Structure of regular semigroups &mdash; III: the reduced case
 | journal=Semigroup Forum
 | volume=8
-| pages=260--265
+| pages=260-265
 | year=1974c
 }}
 {{Citation
 | first1=S.
@@ Line 1,208: / Line 1,214: @@
 | year=1971
 }}
 {{Citation
 | first1=K.S.S.
 | last1=Nambooripad
-| title=Structure of regular semigroups--I
+| title=Structure of regular semigroups &mdash; I
 | series=Memoirs of the Amer. Math. Soc.
 | volume=22
@@ Line 1,217: / Line 1,224: @@
 | date=November 1979
 }}
 {{Citation
 | first1=K.S.S.
@@ Line 1,224: / Line 1,232: @@
 | place=Thiruvananthapuram
 | year=1994
-| postscript=This is a revised form of a part of Structure of Regular Semigroups--II, Publications of the Centre for Mathematical Sciences, No. 15, 1989.
+| postscript=This is a revised form of a part of Structure of Regular Semigroups &mdash; II, Publications of the Centre for Mathematical Sciences, No. 15, 1989.
 }}

@@ Line 1,148: / Line 1,148: @@
 ==References==
-<div id="mw-content-text" dir="ltr" class="mw-content-ltr" lang="en"><p><span id="CITEREFCliffordPreston1961" class="citation">Clifford, A.H.; Preston, G.B. (1961), <i>Algebraic Theory of Semigroups</i>, Math. Surveys No.7, Providence, Rhode Island: American Mathmatical Society</span><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fml.sayahna.tk%3AOne&amp;rft.au=Clifford%2C+A.H.&amp;rft.au=Preston%2C+G.B.&amp;rft.aufirst=A.H.&amp;rft.aulast=Clifford&amp;rft.btitle=Algebraic+Theory+of+Semigroups&amp;rft.date=1961&amp;rft.genre=book&amp;rft.place=Providence%2C+Rhode+Island&amp;rft.pub=American+Mathmatical+Society&amp;rft.series=Math.+Surveys+No.7&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&nbsp;</span></span>
+{{Citation
-</p>
+| first1=A.H.
-<p>
+| last1=Clifford
-<span id="CITEREFKrishnan2000" class="citation">Rajan, A.R.; Krishnan, E.; Nambooripad, K.S.S. (2000), <i>Theory of Regular Semigroups</i>, CMS Preprint, Thiruvananthapuram: Center for Mathematical SciencesThis is a continuing UGC Project</span><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fml.sayahna.tk%3AOne&amp;rft.au=Krishnan%2C+E.&amp;rft.au=Nambooripad%2C+K.S.S.&amp;rft.au=Rajan%2C+A.R.&amp;rft.aufirst=A.R.&amp;rft.aulast=Rajan&amp;rft.btitle=Theory+of+Regular+Semigroups&amp;rft.date=2000&amp;rft.genre=book&amp;rft.place=Thiruvananthapuram&amp;rft.pub=Center+for+Mathematical+Sciences&amp;rft.series=CMS+Preprint&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&nbsp;</span></span>
+| first2=G.B.
-</p>
+| last2=Preston
-<p>
+| title=Algebraic Theory of Semigroups
-<span id="CITEREFGrillet1974a" class="citation">Grillet, P.A. (1974a), "Structure of regular semigroups--I: a representation", <i>Semigroup Forum</i> <b>8</b>: 177––183</span><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fml.sayahna.tk%3AOne&amp;rft.atitle=Structure+of+regular+semigroups--I%3A+a+representation&amp;rft.au=Grillet%2C+P.A.&amp;rft.aufirst=P.A.&amp;rft.aulast=Grillet&amp;rft.date=1974a&amp;rft.genre=article&amp;rft.jtitle=Semigroup+Forum&amp;rft.pages=177%E2%80%93%E2%80%93183&amp;rft.volume=8&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;">&nbsp;</span></span>
+| series=Math. Surveys No.7
-</p>
+| publisher=American Mathmatical Society
-<p>
+| place=Providence, Rhode Island
-<span id="CITEREFGrillet1974b" class="citation">Grillet, P.A. (1974b), "Structure of regular semigroups--II: cross-connections", <i>Semigroup Forum</i> <b>8</b>: 254––259</span><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fml.sayahna.tk%3AOne&amp;rft.atitle=Structure+of+regular+semigroups--II%3A+cross-connections&amp;rft.au=Grillet%2C+P.A.&amp;rft.aufirst=P.A.&amp;rft.aulast=Grillet&amp;rft.date=1974b&amp;rft.genre=article&amp;rft.jtitle=Semigroup+Forum&amp;rft.pages=254%E2%80%93%E2%80%93259&amp;rft.volume=8&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;">&nbsp;</span></span>
+| year=1961
-</p>
+}}
-<p>
+{{Citation
-<span id="CITEREFGrillet1974c" class="citation">Grillet, P.A. (1974c), "Structure of regular semigroups--III: the reduced case", <i>Semigroup Forum</i> <b>8</b>: 260––265</span><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fml.sayahna.tk%3AOne&amp;rft.atitle=Structure+of+regular+semigroups--III%3A+the+reduced+case&amp;rft.au=Grillet%2C+P.A.&amp;rft.aufirst=P.A.&amp;rft.aulast=Grillet&amp;rft.date=1974c&amp;rft.genre=article&amp;rft.jtitle=Semigroup+Forum&amp;rft.pages=260%E2%80%93%E2%80%93265&amp;rft.volume=8&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;">&nbsp;</span></span>
+| first1=A.R.
-</p>
+| last1=Rajan
-<p><span id="CITEREFMacLane1971" class="citation">MacLane, S. (1971), <i>Categories for the Working Mathematicians</i>, New York: Springer-Verlag</span><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fml.sayahna.tk%3AOne&amp;rft.au=MacLane%2C+S.&amp;rft.aufirst=S.&amp;rft.aulast=MacLane&amp;rft.btitle=Categories+for+the+Working+Mathematicians&amp;rft.date=1971&amp;rft.genre=book&amp;rft.place=New+York&amp;rft.pub=Springer-Verlag&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&nbsp;</span></span>
+| first2=E.
-</p>
+| last2=Krishnan
-<p>
+| first3=K.S.S.
-<span id="CITEREFNambooripad1979" class="citation">Nambooripad, K.S.S. (November 1979), <i>Structure of regular semigroups--I</i>, Memoirs of the Amer. Math. Soc. <b>22</b> (224 (Second of 3 numbers))</span><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fml.sayahna.tk%3AOne&amp;rft.au=Nambooripad%2C+K.S.S.&amp;rft.aufirst=K.S.S.&amp;rft.aulast=Nambooripad&amp;rft.btitle=Structure+of+regular+semigroups--I&amp;rft.date=November+1979&amp;rft.genre=book&amp;rft.issue=224+%28Second+of+3+numbers%29&amp;rft.series=Memoirs+of+the+Amer.+Math.+Soc.&amp;rft.volume=22&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&nbsp;</span></span>
+| last3=Nambooripad
-</p>
+| title=Theory of Regular Semigroups
-<p><span id="CITEREFNambooripad1994" class="citation">Nambooripad, K.S.S. (1994), <i>Theory of Cross-connections</i>, Thiruvananthapuram: Centre for Mathematical SciencesThis is a revised form of a part of Structure of Regular Semigroups--II, Publications of the Centre for Mathematical Sciences, No. 15, 1989.</span><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fml.sayahna.tk%3AOne&amp;rft.au=Nambooripad%2C+K.S.S.&amp;rft.aufirst=K.S.S.&amp;rft.aulast=Nambooripad&amp;rft.btitle=Theory+of+Cross-connections&amp;rft.date=1994&amp;rft.genre=book&amp;rft.place=Thiruvananthapuram&amp;rft.pub=Centre+for+Mathematical+Sciences&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&nbsp;</span></span>
+| series=CMS Preprint
-</p>
+| publisher=Center for Mathematical Sciences
+| place=Thiruvananthapuram
-</div>
+| year=2000

← Older revision		Revision as of 14:30, 8 April 2014
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@@ Line 1,151: / Line 1,151: @@
 </p>
 <p>
-<span id="CITEREFRajanKrishnanNambooripad2000" class="citation">Rajan, A.R.; Krishnan, E.; Nambooripad, K.S.S. (2000), <i>Theory of Regular Semigroups</i>, CMS Preprint, Thiruvananthapuram: Center for Mathematical SciencesThis is a continuing UGC Project</span><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fml.sayahna.tk%3AOne&amp;rft.au=Krishnan%2C+E.&amp;rft.au=Nambooripad%2C+K.S.S.&amp;rft.au=Rajan%2C+A.R.&amp;rft.aufirst=A.R.&amp;rft.aulast=Rajan&amp;rft.btitle=Theory+of+Regular+Semigroups&amp;rft.date=2000&amp;rft.genre=book&amp;rft.place=Thiruvananthapuram&amp;rft.pub=Center+for+Mathematical+Sciences&amp;rft.series=CMS+Preprint&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&nbsp;</span></span>
+<span id="CITEREFKrishnan2000" class="citation">Rajan, A.R.; Krishnan, E.; Nambooripad, K.S.S. (2000), <i>Theory of Regular Semigroups</i>, CMS Preprint, Thiruvananthapuram: Center for Mathematical SciencesThis is a continuing UGC Project</span><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fml.sayahna.tk%3AOne&amp;rft.au=Krishnan%2C+E.&amp;rft.au=Nambooripad%2C+K.S.S.&amp;rft.au=Rajan%2C+A.R.&amp;rft.aufirst=A.R.&amp;rft.aulast=Rajan&amp;rft.btitle=Theory+of+Regular+Semigroups&amp;rft.date=2000&amp;rft.genre=book&amp;rft.place=Thiruvananthapuram&amp;rft.pub=Center+for+Mathematical+Sciences&amp;rft.series=CMS+Preprint&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&nbsp;</span></span>
 </p>
 <p>
@@ Line 1,171: / Line 1,171: @@
 </div>
-<!--
+UNIQe063670d2486a08e-item-0--QINU

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+$
 \newcommand{\cc}{\mathcal{C}}
 \newcommand{\cd}{\mathcal{D}}

← Older revision	Revision as of 18:17, 5 April 2014
(No difference)

@@ Line 90: / Line 90: @@
 =={{anchor|sec-1}}Introduction==
-The concept of ''cross-connection'' between partially ordered sets was originally introduced by {{harvs|Grillet|year1=1974|year2=1974b|year=31974c|txt=yes}} in order to classify the class of fundamental regular semigroups. Grillet defines a cross-connection between two regular partial ordered sets $I$ and $\Lambda $ as a pair of order-preserving mappings
+The concept of ''cross-connection'' between partially ordered sets was originally introduced by {{harvs|author=Grillet|year1=1974|year2=1974b|year3=1974c|txt=yes}} in order to classify the class of fundamental regular semigroups. Grillet defines a cross-connection between two regular partial ordered sets $I$ and $\Lambda $ as a pair of order-preserving mappings
 \begin{equation}
@@ Line 97: / Line 97: @@
    I^{*}
 \end{equation}
-satisfying certain axioms, where $I^{*}$ denote the dual of $I$. See {{harv|Grillet|1974b}} for definition of regular partially ordered sets, duals and axioms for cross-connections. Every regular semigroup $S$ induces a cross-connection between the partially ordered set $\Lambda (S)$ of all principal left ideals of $S$ under inclusion and the partially ordered set $I(S)$ of all principal right ideals. In
+satisfying certain axioms, where $I^{*}$ denote the dual of $I$. See {{harvtxt|Grillet|1974b}} for definition of regular partially ordered sets, duals and axioms for cross-connections. Every regular semigroup $S$ induces a cross-connection between the partially ordered set $\Lambda (S)$ of all principal left ideals of $S$ under inclusion and the partially ordered set $I(S)$ of all principal right ideals. In
-{{harv|Nambooripad|1994}}, this was generalized to cross-connections between normal categories. Again, given a regular semigroup $S$, category $\lcat $ of all principal left ideals is normal and similarly, the category $\rcat $ of all principal right ideals is also normal (see [[#prop-4.3|Propositions ]][[#prop-4.3|4.3]] and [[#prop-4.4|4.4]] for definitions of these categorioes). We can see that every regular
+{{harvtxt|Nambooripad|1994}}, this was generalized to cross-connections between normal categories. Again, given a regular semigroup $S$, category $\lcat $ of all principal left ideals is normal and similarly, the category $\rcat $ of all principal right ideals is also normal (see [[#prop-4.3|Propositions ]][[#prop-4.3|4.3]] and [[#prop-4.4|4.4]] for definitions of these categorioes). We can see that every regular
 semigroup $S$ induces a cross-connection between categories $\lcat $ and $\rcat $.
-We observe that the configuration that occur in Grillet&#x2019;s definition as well as its generalization in {{harv|Nambooripad|1994}} occures in many areas in mathematics. Our aim here is to describe cross-connections of a more general class of categories. We first identify an appropriate class of categories for which cross-section can be defined. These are categories with subobjects in which objects are sets,
+We observe that the configuration that occur in Grillet&#x2019;s definition as well as its generalization in {{harvtxt|Nambooripad|1994}} occures in many areas in mathematics. Our aim here is to describe cross-connections of a more general class of categories. We first identify an appropriate class of categories for which cross-section can be defined. These are categories with subobjects in which objects are sets,
 morhisms are mappings and inclusions are inclusions of sets. We refer to these categories as ''set-based categories'' ($\relax \mathcal {S}$-category&#x00A0;for short). A cross-connection between to $\relax \mathcal {S}$-categorys $\cc $ and $\cd $ consists of two set-valued bifunctors
 \[ \Ga :\cc \times \cd \to \set \quad \text {and}\quad \De :\cc \times \cd \to \set \]
@@ Line 110: / Line 110: @@
 $.
-In the following discussion, we will follow {{harv|Nambooripad|1994}} for ideas regarding categories, subobject relations, etc. Moreover, to save repetition, we shall assume that categories under consideration are ''small'', unless otherwise provided, so that they may be treated as partial algebras.
+In the following discussion, we will follow {{harvtxt|Nambooripad|1994}} for ideas regarding categories, subobject relations, etc. Moreover, to save repetition, we shall assume that categories under consideration are ''small'', unless otherwise provided, so that they may be treated as partial algebras.
 =={{anchor|sec-2}}Categories with subobjects==

@@ Line 975: / Line 975: @@
 \hfn {\ga }{\inc {c}{d} } = \inc {\hfn{\ga}{c} } {\hfn{\ga}{d} }.
 \]
-In particular, $\hfn {\ga }{1_{c}}=1_{\hfn {\ga }c}$. In view of the remarks following \eqref{eq:17}, this shows that $\hfn {\ga }-:\cc \to \set $ is an inclusion preserving covariant
+In particular, $\hfn {\ga }{1_{c} }=1_{\hfn {\ga }c}$. In view of the remarks following \eqref{eq:17}, this shows that $\hfn {\ga }-:\cc \to \set $ is an inclusion preserving covariant
 functor.

@@ Line 978: / Line 978: @@
 functor.
-To prove the remaining statements, by Yoneda lemma see {{harv|MacLane|1971|loc=Section III.2}} and {{harvtxt|Nambooripad|1994|loc=section I.2}}, it is sufficient to show that $\ga $ is universal element for $\hfn {\ga }-$ in $\hfn {\ga }{c_{\ga }}$. Thus we must show that given any $\ga \ast \epi [f]\in \hfn {\ga }c$, there is a unique $h:c_{\ga }\to c$ such that
+To prove the remaining statements, by Yoneda lemma see {{harv|MacLane|1971|loc=Section III.2}} and {{harvtxt|Nambooripad|1994|loc=section I.2}}, it is sufficient to show that $\ga $ is universal element for $\hfn {\ga}-$ in $\hfn {\ga}{c_{\ga} }$. Thus we must show that given any $\ga \ast \epi [f]\in \hfn {\ga }c$, there is a unique $h:c_{\ga }\to c$ such that
 \[
 (\ga )\hfn {\ga }h=\ga \ast \epi [f]. \]