#LyX 2.0.0svn created this file. For more info see http://www.lyx.org/
\lyxformat 381
\begin_document
\begin_header
\textclass article
\begin_preamble
\usepackage[all, knot]{xy}
\xyoption{arc} 
\usepackage{qtree}
\usepackage{tipa}
\end_preamble
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\language english
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\cite_engine basic
\use_bibtopic false
\use_indices false
\paperorientation portrait
\suppress_date false
\index Index
\shortcut idx
\color #008000
\end_index
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\paragraph_indentation default
\quotes_language english
\papercolumns 1
\papersides 1
\paperpagestyle default
\tracking_changes false
\output_changes false
\html_math_output 0
\html_be_strict true
\end_header

\begin_body

\begin_layout Section
Introduction
\end_layout

\begin_layout Standard
The following packages are needed to be installed at your computer in order
 to typeset the document properly and have instant preview working:
\end_layout

\begin_layout LyX-Code
xy.sty, qtree.sty, tipa.sty
\end_layout

\begin_layout Section
Diagrams from by XY(-pic)
\end_layout

\begin_layout Standard
Few examples of diagrams were taken from Aaron Landa's archive of xy-pic
 examples, see 
\begin_inset Flex URL
status collapsed

\begin_layout Plain Layout

http://www.math.columbia.edu/~lauda/xy/
\end_layout

\end_inset

.
\end_layout

\begin_layout Description
Braid
\end_layout

\begin_layout Standard
\begin_inset Preview

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
xy
\end_layout

\begin_layout Plain Layout

(5,10)*{}; (-10,-10)*{} **
\backslash
crv{(6,-2)&(-12,4)}
\end_layout

\begin_layout Plain Layout


\backslash
POS?(.25)*{
\backslash
hole}="x" 
\backslash
POS?(.45)*{
\backslash
hole}="y" 
\backslash
POS?(.6)*{
\backslash
hole}="z";
\end_layout

\begin_layout Plain Layout

"y"+(0,-1); (2,-10)*{} **
\backslash
crv{}
\backslash
POS?(.2)*{
\backslash
hole}="M";
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\begin_layout Plain Layout

(-10,10)*{}; "z" **
\backslash
crv{(-9,0)};
\end_layout

\begin_layout Plain Layout

"z"; "M" **
\backslash
crv{};
\end_layout

\begin_layout Plain Layout

"M"; "x" **
\backslash
crv{(5,0)};
\end_layout

\begin_layout Plain Layout

"x"; "y" **
\backslash
crv{(0,7) & (-5,6)};
\end_layout

\begin_layout Plain Layout


\backslash
endxy 
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Description
Knots
\end_layout

\begin_layout Standard
\begin_inset Preview

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
xy
\end_layout

\begin_layout Plain Layout

(6,9)*{}="1";
\end_layout

\begin_layout Plain Layout

(-8.5,-1)*{}="2";
\end_layout

\begin_layout Plain Layout

"1";"2" **
\backslash
crv{~*=<.5pt>{.} (0,30)}?(.75)*
\backslash
dir{>}+(-2,2)*{z};
\end_layout

\begin_layout Plain Layout

(-6.5,8)*{}="1";
\end_layout

\begin_layout Plain Layout

(-.5,-9)*{}="2";
\end_layout

\begin_layout Plain Layout

"1";"2" **
\backslash
crv{~*=<.5pt>{.} (-28.5,9.3)}?(.7)*
\backslash
dir{>}+(-2,-2)*{u};
\end_layout

\begin_layout Plain Layout

(-9.5,-3.35)*{}="1";
\end_layout

\begin_layout Plain Layout

(8.5,-3)*{}="2";
\end_layout

\begin_layout Plain Layout

"1";"2" **
\backslash
crv{~*=<.5pt>{.} (-17.67,-24.19)}?(.7)*
\backslash
dir{>}+(-1,-3)*{x};
\end_layout

\begin_layout Plain Layout

(1,-10)*{}="1";
\end_layout

\begin_layout Plain Layout

(6.5,7.13)*{}="2";
\end_layout

\begin_layout Plain Layout

"1";"2" **
\backslash
crv{~*=<.5pt>{.} (17.67,-24.19)}?(.7)*
\backslash
dir{>}+(3,-1)*{y};
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\begin_layout Plain Layout

(11,-1)*{}="1";
\end_layout

\begin_layout Plain Layout

(-4,8)*{}="2";
\end_layout

\begin_layout Plain Layout

"1";"2" **
\backslash
crv{~*=<.5pt>{.} (28.5,9.3)}?(.93)*
\backslash
dir{>}+(1,2)*{w};
\end_layout

\begin_layout Plain Layout


\backslash
endxy
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Description
Circles
\end_layout

\begin_layout Standard
\begin_inset Preview

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
xy
\end_layout

\begin_layout Plain Layout

(0,0)*
\backslash
xycircle(14,8){++
\backslash
dir{<}};
\end_layout

\begin_layout Plain Layout

(0,0)*
\backslash
xycircle<10pt>{{.}};
\end_layout

\begin_layout Plain Layout

(0,0)*
\backslash
xycircle(5,2){:};
\end_layout

\begin_layout Plain Layout

(0,15)*
\backslash
xycircle(10,4){--};
\end_layout

\begin_layout Plain Layout


\backslash
endxy 
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Description
Cobordism
\end_layout

\begin_layout Standard
\begin_inset Preview

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
xy
\end_layout

\begin_layout Plain Layout

(0,0)*
\backslash
ellipse(3,1){.};
\end_layout

\begin_layout Plain Layout

(0,0)*
\backslash
ellipse(3,1)__,=:a(-180){-};
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\begin_layout Plain Layout

(-6,-8)*
\backslash
ellipse(3,1){.};
\end_layout

\begin_layout Plain Layout

(6,-8)*
\backslash
ellipse(3,1){.};
\end_layout

\begin_layout Plain Layout

(0,-8)*
\backslash
ellipse(3,1){.};
\end_layout

\begin_layout Plain Layout

(-6,-8)*
\backslash
ellipse(3,1)__,=:a(-180){-};
\end_layout

\begin_layout Plain Layout

(6,-8)*
\backslash
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\begin_layout Plain Layout

(0,-8)*
\backslash
ellipse(3,1)__,=:a(180){-};
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\begin_layout Plain Layout

(-3,6)*
\backslash
ellipse(3,1){-};
\end_layout

\begin_layout Plain Layout

(3,6)*
\backslash
ellipse(3,1){-};
\end_layout

\begin_layout Plain Layout

(-3,12)*{}="1";
\end_layout

\begin_layout Plain Layout

(3,12)*{}="2";
\end_layout

\begin_layout Plain Layout

(-9,12)*{}="A2";
\end_layout

\begin_layout Plain Layout

(9,12)*{}="B2";
\end_layout

\begin_layout Plain Layout

"1";"2" **
\backslash
crv{(-3,7) & (3,7)};
\end_layout

\begin_layout Plain Layout

(-3,0)*{}="A";
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\begin_layout Plain Layout

(3,0)*{}="B";
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\begin_layout Plain Layout

(-3,1)*{}="A1";
\end_layout

\begin_layout Plain Layout

(3,1)*{}="B1";
\end_layout

\begin_layout Plain Layout

"A";"A1" **
\backslash
dir{-};
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\begin_layout Plain Layout

"B";"B1" **
\backslash
dir{-};
\end_layout

\begin_layout Plain Layout

"B2";"B1" **
\backslash
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\begin_layout Plain Layout

"A2";"A1" **
\backslash
crv{(-8,7) & (-3,5)};
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\begin_layout Plain Layout

%REFLECT
\end_layout

\begin_layout Plain Layout

(3,-16)*{}="1";
\end_layout

\begin_layout Plain Layout

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\end_layout

\begin_layout Plain Layout

"1";"2" **
\backslash
crv{(3,-10) & (9,-10)};
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\begin_layout Plain Layout

(-3,-16)*{}="1";
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\begin_layout Plain Layout

(-9,-16)*{}="2";
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\begin_layout Plain Layout

"1";"2" **
\backslash
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\begin_layout Plain Layout

(-15,-16)*{}="A2";
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(-3,-1)*{}="A1";
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\begin_layout Plain Layout

(3,-1)*{}="B1";
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\begin_layout Plain Layout

"A";"A1" **
\backslash
dir{-};
\end_layout

\begin_layout Plain Layout

"B";"B1" **
\backslash
dir{-};
\end_layout

\begin_layout Plain Layout

"B2";"B1" **
\backslash
crv{(13,-6) & (2,-8)};
\end_layout

\begin_layout Plain Layout

"A2";"A1" **
\backslash
crv{(-13,-6) & (-2,-8)};
\end_layout

\begin_layout Plain Layout


\backslash
endxy
\end_layout

\end_inset


\end_layout

\end_inset

 
\begin_inset Preview

\begin_layout Standard
\begin_inset ERT
status collapsed

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\backslash
xy 0;/r.18pc/:
\end_layout

\begin_layout Plain Layout

(20,2)*{
\backslash
bullet}="RU"+(1,3)*{x^{
\backslash
ast}};
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\begin_layout Plain Layout

(16,-3)*{
\backslash
bullet}="RD"+(2.5,2)*{x^{
\backslash
ast}};
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\begin_layout Plain Layout

(-16,2)*{
\backslash
bullet}="LU"+(-1,2)*{x};
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\begin_layout Plain Layout

(-20,-3)*{
\backslash
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\begin_layout Plain Layout

"RU";"RD" **
\backslash
crv{(4,2) & (4,-1)}; ?(.1)*
\backslash
dir{<}; ?(.85)*
\backslash
dir{<};
\end_layout

\begin_layout Plain Layout

"LD";"LU" **
\backslash
crv{(-4,-2) & (-4,1)};
\end_layout

\begin_layout Plain Layout

?(.08)*
\backslash
dir{<}; ?(.85)*
\backslash
dir{<};
\end_layout

\begin_layout Plain Layout

(7.5,0)*{}="x1"; (-7.5,0)*{}="x2";
\end_layout

\begin_layout Plain Layout

"x1"; "x2" **
\backslash
crv{(7,-10) & (-7,-10)};
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\begin_layout Plain Layout

(16,-20)*{
\backslash
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\backslash
ast}};
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\begin_layout Plain Layout

(-20,-20)*{
\backslash
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\begin_layout Plain Layout

(20,-12.5)*{
\backslash
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\begin_layout Plain Layout

(-16,-15)*{
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\end_layout

\begin_layout Plain Layout

(-16,-2.5)*{}="A"; (16.1,-14.9)*{}="B";
\end_layout

\begin_layout Plain Layout

"RD"; "RDD" **
\backslash
dir{-};
\end_layout

\begin_layout Plain Layout

"LD"; "LDD" **
\backslash
dir{-};
\end_layout

\begin_layout Plain Layout

"A"; "LUD" **
\backslash
dir{.};
\end_layout

\begin_layout Plain Layout

"RDD"; "LDD" **
\backslash
crv{(0,-17)}; ?(.57)*
\backslash
dir{>};
\end_layout

\begin_layout Plain Layout

"RU"; "RUD" **
\backslash
dir{-};
\end_layout

\begin_layout Plain Layout

"LU"; "A" **
\backslash
dir{-};
\end_layout

\begin_layout Plain Layout

"B"; "RUD" **
\backslash
crv{(18,-14.15)};
\end_layout

\begin_layout Plain Layout

"B"; "LUD" **
\backslash
crv{~*=<4pt>{.}(0,-18)}; ?(.4)*
\backslash
dir{<};
\end_layout

\begin_layout Plain Layout


\backslash
endxy
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Description
Strings
\end_layout

\begin_layout Standard
\begin_inset Preview

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
xy (-15,0)*{};(15,0)*{};
\end_layout

\begin_layout Plain Layout


\backslash
endxy
\end_layout

\begin_layout Plain Layout


\backslash
qquad = 
\backslash
qquad 
\backslash
xy
\end_layout

\begin_layout Plain Layout

(-10,12)*{}; (-10,-12)*{}; **
\backslash
dir{-}?(.5)*
\backslash
dir{<}+(-4,0)*{F(x)};
\end_layout

\begin_layout Plain Layout

(-2,12)*{}="1";
\end_layout

\begin_layout Plain Layout

(6,-7) *{}="2";
\end_layout

\begin_layout Plain Layout

(14,-7) *{}="3";
\end_layout

\begin_layout Plain Layout

(14,-13)*{}="3'";
\end_layout

\begin_layout Plain Layout

(18,0) *={}="";
\end_layout

\begin_layout Plain Layout

"1";"2" **
\backslash
crv{(-4,-21)&(5,-15)};
\end_layout

\begin_layout Plain Layout

?(.15)*
\backslash
dir{<}+(5,10)*{
\backslash
overline{F(x)}};
\end_layout

\begin_layout Plain Layout

"2";"3" **
\backslash
crv{(6,1)&(14,1)};
\end_layout

\begin_layout Plain Layout

?(0)*
\backslash
dir{<}+(4,8)*{
\backslash
widetilde{F(i_x)}} ;
\end_layout

\begin_layout Plain Layout

?(.95)*
\backslash
dir{<}+(5,-2)*{F(
\backslash
overline x)};
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\begin_layout Plain Layout

"3";"3'" **
\backslash
dir{-};
\end_layout

\begin_layout Plain Layout

(-2,6)*{}="X1";
\end_layout

\begin_layout Plain Layout

(-10,6)*{}="X2";
\end_layout

\begin_layout Plain Layout

"X1";"X2" **
\backslash
crv{~*=<2pt>{.}(-2,-1)&(-10,-1)};
\end_layout

\begin_layout Plain Layout

(-2,-6)*{}="XA";
\end_layout

\begin_layout Plain Layout

(-10,-6)*{}="XB";
\end_layout

\begin_layout Plain Layout

"XA";"XB" **
\backslash
crv{~*=<2pt>{.}(-2,1)&(-10,1)};
\end_layout

\begin_layout Plain Layout


\backslash
endxy
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Description
Categories
\end_layout

\begin_layout Standard
\begin_inset Preview

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
xymatrix{
\end_layout

\begin_layout Plain Layout

(x 
\backslash
otimes 1) 
\backslash
otimes y
\end_layout

\begin_layout Plain Layout


\backslash
ar[rr]^{a_{x,1,y}}
\end_layout

\begin_layout Plain Layout


\backslash
ar[dr]_{r_x 
\backslash
otimes 1_y}
\end_layout

\begin_layout Plain Layout

&& x 
\backslash
otimes(1 
\backslash
otimes y)
\end_layout

\begin_layout Plain Layout


\backslash
ar[dl]^{1_x 
\backslash
otimes
\backslash
ell_y } 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

& x 
\backslash
otimes y }
\end_layout

\end_inset


\end_layout

\end_inset

 
\begin_inset Preview

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
xy
\end_layout

\begin_layout Plain Layout

(-9,0)*+{
\backslash
bullet}="1";
\end_layout

\begin_layout Plain Layout

(0,0)*+{
\backslash
bullet}="2";
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\begin_layout Plain Layout

{
\backslash
ar@/^.9pc/ "1";"2"};
\end_layout

\begin_layout Plain Layout

{
\backslash
ar@/_.9pc/ "1";"2"};
\end_layout

\begin_layout Plain Layout

{
\backslash
ar@/^2pc/ "1";"2"};
\end_layout

\begin_layout Plain Layout

{
\backslash
ar@/_2pc/ "1";"2"};
\end_layout

\begin_layout Plain Layout

{
\backslash
ar@{=>} (-4.5,1.5)*{};(-4.5,-1.5)*{}} ;
\end_layout

\begin_layout Plain Layout

{
\backslash
ar@{=>} (-4.5,7.25)*{};(-4.5,4.75)*{}} ;
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\begin_layout Plain Layout

{
\backslash
ar@{=>} (-4.5,-4.75)*{};(-4.5,-7.25)*{}} ;
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\begin_layout Plain Layout

(0,0)*+{
\backslash
bullet}="1";
\end_layout

\begin_layout Plain Layout

(9,0)*+{
\backslash
bullet}="2";
\end_layout

\begin_layout Plain Layout

(18,0)*+{
\backslash
bullet}="3";
\end_layout

\begin_layout Plain Layout

{
\backslash
ar "1";"2"};
\end_layout

\begin_layout Plain Layout

{
\backslash
ar "2";"3"};
\end_layout

\begin_layout Plain Layout

{
\backslash
ar@/^1pc/ "1";"2"};
\end_layout

\begin_layout Plain Layout

{
\backslash
ar@/_1pc/ "1";"2"};
\end_layout

\begin_layout Plain Layout

{
\backslash
ar@{=>} (4.5,3)*{};(4.5,.75)*{}} ;
\end_layout

\begin_layout Plain Layout

{
\backslash
ar@{=>} (4.5,-.75)*{};(4.5,-3)*{}} ;
\end_layout

\begin_layout Plain Layout


\backslash
endxy 
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Description
Transformations
\end_layout

\begin_layout Standard
\begin_inset Preview

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
xy %FIG.19.
 Natural transformation between functors.
\end_layout

\begin_layout Plain Layout

(-26,-2)*+{
\backslash
bullet}="a";
\end_layout

\begin_layout Plain Layout

(-12,3)*+{
\backslash
bullet}="b";
\end_layout

\begin_layout Plain Layout

(-16,-5)*+{
\backslash
bullet}="c";
\end_layout

\begin_layout Plain Layout

{
\backslash
ar@/^.25pc/ "a";"b"};
\end_layout

\begin_layout Plain Layout

{
\backslash
ar@/_.25pc/"a";"c"};
\end_layout

\begin_layout Plain Layout

{
\backslash
ar@/_.15pc/ "b";"c"};
\end_layout

\begin_layout Plain Layout

%START TOP
\end_layout

\begin_layout Plain Layout

(12,10)*+{
\backslash
bullet}="ta";
\end_layout

\begin_layout Plain Layout

(26,15)*+{
\backslash
bullet}="tb";
\end_layout

\begin_layout Plain Layout

(22,8)*+{
\backslash
bullet}="tc";
\end_layout

\begin_layout Plain Layout

{
\backslash
ar@/^.25pc/ "ta";"tb"};
\end_layout

\begin_layout Plain Layout

{
\backslash
ar@/_.25pc/ "ta";"tc"};
\end_layout

\begin_layout Plain Layout

{
\backslash
ar@/_.15pc/ "tb";"tc"};
\end_layout

\begin_layout Plain Layout

%START bottom
\end_layout

\begin_layout Plain Layout

(12,-10)*+{
\backslash
bullet}="ba";
\end_layout

\begin_layout Plain Layout

(26,-5)*+{
\backslash
bullet}="bb";
\end_layout

\begin_layout Plain Layout

(22,-12)*+{
\backslash
bullet}="bc";
\end_layout

\begin_layout Plain Layout

{
\backslash
ar@/^.25pc/@{.>} "ba";"bb"};
\end_layout

\begin_layout Plain Layout

{
\backslash
ar@/_.25pc/ "ba";"bc"};
\end_layout

\begin_layout Plain Layout

{
\backslash
ar@/_.15pc/ "bb";"bc"};
\end_layout

\begin_layout Plain Layout

%START CONNECT
\end_layout

\begin_layout Plain Layout

{
\backslash
ar@/_.15pc/ "ta";"ba"};
\end_layout

\begin_layout Plain Layout

{
\backslash
ar@/_.15pc/ "tb";"bb"};
\end_layout

\begin_layout Plain Layout

{
\backslash
ar@/_.15pc/ "tc";"bc"};
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\begin_layout Plain Layout

%START ARROWS AND LABELS
\end_layout

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(-12,6)*{}="DF";
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\begin_layout Plain Layout

(6,10)*{}="CF";
\end_layout

\begin_layout Plain Layout

{
\backslash
ar@/^.35pc/^F "DF";"CF"};
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\begin_layout Plain Layout

(-12,-6)*{}="DF";
\end_layout

\begin_layout Plain Layout

(6,-10)*{}="CF";
\end_layout

\begin_layout Plain Layout

{
\backslash
ar@/_.35pc/_G "DF";"CF"};
\end_layout

\begin_layout Plain Layout

(-2,5)*{}="DF";
\end_layout

\begin_layout Plain Layout

(-2,-5)*{}="CF";
\end_layout

\begin_layout Plain Layout

{
\backslash
ar@{=>}^{
\backslash
alpha} "DF";"CF"};
\end_layout

\begin_layout Plain Layout

(29,3)*{D};
\end_layout

\begin_layout Plain Layout

(-23,3)*{C};
\end_layout

\begin_layout Plain Layout


\backslash
endxy 
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Section
IPA
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\begin_layout Standard
\begin_inset Preview

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
textipa{Ekspl@"neIS@n}
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Section
QTree
\end_layout

\begin_layout Standard
\begin_inset Preview

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Tree[.S [.N This ] [.V is ] [.NP [.Det a ] [.N test ] ] ] 
\end_layout

\end_inset


\end_layout

\end_inset


\begin_inset Preview

\begin_layout Standard
\begin_inset ERT
status collapsed

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\backslash
Tree [.$
\backslash
sigma$ $
\backslash
mu$ $
\backslash
mu$ ]
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\end_body
\end_document