View source for Baker-Beynon duality

The free [[Riesz space|Riesz space]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b1100601.png" /> generators, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b1100602.png" />, may be described as follows: View the set of all real-valued functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b1100603.png" /> as a Riesz space under the pointwise operations. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b1100604.png" /> may be identified with the Riesz subspace generated by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b1100605.png" /> coordinate projections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b1100606.png" />. This follows from [[Universal algebra|universal algebra]] and the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b1100607.png" /> generates an equational class of Riesz spaces, see [[#References|[a4]]], p. 355. The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b1100608.png" /> are continuous functions that are piecewise-homogeneous linear in polyhedral cones with common vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b1100609.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006010.png" /> is a Riesz ideal, let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006011.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006012.png" /> is finitely generated and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006013.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006014.png" /> is said to be finitely presentable. In this case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006015.png" /> is a polyhedral cone (with vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006016.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006017.png" /> is isomorphic to the Riesz space of all piecewise-homogeneous linear functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006018.png" />. Much of this can be found in [[#References|[a1]]], but the crucial observation that all piecewise-homogeneous linear functions occur appears first in [[#References|[a2]]]. Baker–Beynon duality [[#References|[a3]]] states that the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006019.png" /> of finitely presentable Riesz spaces and arbitrary Riesz homomorphisms is dually equivalent to the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006020.png" /> of polyhedral cones (with vertex at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006021.png" />) in some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006022.png" /> and with piecewise-homogeneous linear morphisms. The equivalence is via enriched <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006023.png" />-functors. For any object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006026.png" /> inherits the structure of a Riesz space from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006027.png" />. In the other direction, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006028.png" /> is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006029.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006030.png" /> is in one-to-one correspondence with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006031.png" />.

It can be shown that if also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006032.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006034.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006035.png" />-equivalent. There is an induced duality between the category of finitely presentable Riesz spaces with a distinguished strong unit and unit-preserving morphisms and the familiar category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006036.png" /> of polyhedra and piecewise-affine linear mappings, [[#References|[a3]]].

Among the first applications of this theory is Baker's proof [[#References|[a1]]] that the finitely generated projectives in the category of Riesz spaces are precisely the finitely presented Riesz spaces. This corresponds to the peculiar feature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006037.png" /> that every object is an absolute retract (cf. also [[Collapsibility|Collapsibility]]). Presently (1996), nothing significant is known about projectives that are not finitely generated.

For important recent work related to Abelian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006038.png" />-groups (cf. [[L-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006039.png" />-group]]), see [[#References|[a5]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K.A. Baker,   "Free vector lattices"  ''Canadian J. Math.'' , '''20'''  (1968)  pp. 58–66</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W.M. Beynon,   "Combinatorial aspects of piecewise linear functions"  ''J. London Math. Soc.'' , '''7'''  (1974)  pp. 719–727</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W.M. Beynon,   "Duality theorems for finitely generated vector lattices"  ''Proc. London Math. Soc.'' , '''31'''  (1975)  pp. 114–128</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Birkhoff,   "Lattice theory" , ''Colloq. Publ.'' , Amer. Math. Soc.  (1967)  (Edition: Third)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D. Mundici,   "Farey stellar subdivisions, ultrasimplicial groups and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006040.png" /> of AF <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006041.png" />-algebras"  ''Adv. Math.'' , '''68'''  (1988)  pp. 23–39</TD></TR></table>