SFN:Math-test - Revision history

Cvr: Cvr moved page Math-test to SF:Math-test without leaving a redirect

2014-08-18T09:55:44Z

Cvr moved page Math-test to SF:Math-test without leaving a redirect

Admin at 15:03, 30 March 2014

2014-03-30T15:03:21Z

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Admin: /* {{anchor|sec-2.1.3}}Categories with subobjects */

2014-03-30T11:28:15Z

‎{{anchor|sec-2.1.3}}Categories with subobjects

Admin: /* {{anchor|sec-2.1.3}}Categories with subobjects */

2014-03-30T11:26:55Z

‎{{anchor|sec-2.1.3}}Categories with subobjects

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

2014-03-30T11:26:03Z

‎{{anchor|sec-1}}Introduction

Admin at 11:24, 30 March 2014

2014-03-30T11:24:49Z

Admin at 10:31, 30 March 2014

2014-03-30T10:31:31Z

Admin at 09:25, 30 March 2014

2014-03-30T09:25:32Z

Admin at 09:25, 30 March 2014

2014-03-30T09:25:02Z

Admin at 06:29, 29 March 2014

2014-03-29T06:29:11Z

@@ Line 163: / Line 163: @@
 of partially ordered sets).
-{{defn|name=Remark 2.1:|label=rmk-2.1|statement= Notice that an abstract category $\cc $ may have more than one possible choices of subobjects. For example, if $\cc $ is a small category of
+{{defn|name=Remark 2.1:|label=rmk-2.1|statement=Notice that an abstract category $\cc $ may have more than one possible choices of subobjects. For example, if $\cc $ is a small category of
 sets, the the usual relation of inclusion among vertexes that are morphisms in
 $\cc $, gives a choice of subobjects in $\cc $. Also, identity on $\vrt \cc $
 is also a choice of subobjects which may be different from the one given in the
 last sentence.
 }}

@@ Line 163: / Line 163: @@
 of partially ordered sets).
-{{definition|name=Remark 2.1:|label=rmk-2.1|statement= Notice that an abstract category $\cc $ may have more than
+{{defn|name=Remark 2.1:|label=rmk-2.1|statement= Notice that an abstract category $\cc $ may have more than one possible choices of subobjects. For example, if $\cc $ is a small category of
 sets, the the usual relation of inclusion among vertexes that are morphisms in
 $\cc $, gives a choice of subobjects in $\cc $. Also, identity on $\vrt \cc $

@@ Line 69: / Line 69: @@
 Also for any semigroup $S$ there is a cross-connection between $\lcat $ and
 $\rcat $ and there is a natural representation of $S$ by the cross-connection
-semigrpoup determined by the cross-connection between $\lcat $ and $\rcat
+semigrpoup determined by the cross-connection between $\lcat $ and $\rcat$.
 In the following discussion, we will follow (?) 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 28: / Line 28: @@
-=={{anchor|sec-2}}Introduction==
 The concept of ''cross-connection'' between partially ordered sets was originally
@@ Line 38: / Line 39: @@
 \Gamma:I\to\Lambda^{*}\quad\text{and}\quad\Delta:\Lambda\to
 I^{*}
 \end{equation}
 satisfying certain axioms, where $I^{*}$ denote the dual of $I$. See ? for definition
@@ Line 62: / Line 62: @@
 bifunctors
 \[ \Ga :\cc \times \cd \to \set \quad \text {and}\quad \De
 :\cc \times \cd \to \set \]
 and a natural isomorphism
@@ Line 78: / Line 75: @@
 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.

← Older revision		Revision as of 10:31, 30 March 2014
Line 27:		Line 27:
	$		$

−
−
−	~~=={{anchor\|sec-1}}A==~~
−
−
−	~~==={{anchor\|sec-1.1}}AA===~~
−

	=={{anchor\|sec-2}}Introduction==		=={{anchor\|sec-2}}Introduction==

← Older revision	Revision as of 09:55, 18 August 2014
(No difference)

@@ Line 24: / Line 24: @@
 \newcommand{\dom}{\operatorname{dom}}
 \newcommand{\vmap}[1][{}]{{#1}_{\mathfrak{A}}}
+\def\relax{}
+$

@@ Line 27: / Line 27: @@
+$
-(Sb:b) $U(\im f)=\im U(f)$ for all $f\in\cc$.
+=={{anchor|sec-1}}A==
-(Sb:c) The functor $U$ has the following property: for $c,c'\in\vrt
+==={{anchor|sec-1.1}}AA===
-\cc$ and $x\in U(c)\cap U(c')$ there is $d\in\vrt\cc$ such that
-\[
+=={{anchor|sec-2}}Introduction==
-    d\subseteq c,\quad d\subseteq c'\quad\text{and}\quad x\in U(d).
-\]
+The concept of ''cross-connection'' between partially ordered sets was originally
-}}
+introduced by ??? in order to classify the class of fundamental regular semigroups.
+Grillet defines a cross-connection between two regular partial ordered sets $I$ and
-Notice that the condition (Sb:b) implies that the functor $U$
+$\Lambda $ as a pair of order-preserving mappings
 \begin{equation}
-\label{eq:5}
+\label{eq:1}
- U(f\mid c)=U(f)\mid U(c)\quad\text{and}\quad U\left(f(c)\right)=U(f)\left(U(c)\right)
+\Gamma:I\to\Lambda^{*}\quad\text{and}\quad\Delta:\Lambda\to
 \end{equation}
-for all $f\in\cc$. Moreover, we have
+satisfying certain axioms, where $I^{*}$ denote the dual of $I$. See ? for definition
-\begin{equation}\label{eq:4}
+of regular partially ordered sets, duals and axioms for cross-connections. Every
-  U\left(f^{\circ}\inc{\im f}{\cod f}\right)=U(f)^{\circ}\inc{\im U(f)}{\cod U(f)}
+regular semigroup $S$ induces a cross-connection between the partially ordered set
-\end{equation}
+$\Lambda S$ of all principal left ideals of $S$ under inclusion and the partially
-for any $f\in\cc$ and $c\subseteq\dom f$.
+ordered set $IS$ of all principal right ideals. In ?, this was generalized to
-}}
+cross-connections between normal categories. Again, given a regular semigroup $S$,