Difference between revisions of "Baker-Beynon duality"
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− | 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 | + | 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 operation]]s. 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 | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110060/b11006010.png" /> is a Riesz ideal, let |
Latest revision as of 22:47, 29 November 2014
The free Riesz space on generators,
, may be described as follows: View the set of all real-valued functions on
as a Riesz space under the pointwise operations. Then
may be identified with the Riesz subspace generated by the
coordinate projections
. This follows from universal algebra and the fact that
generates an equational class of Riesz spaces, see [a4], p. 355. The elements of
are continuous functions that are piecewise-homogeneous linear in polyhedral cones with common vertex
.
If is a Riesz ideal, let
![]() |
If is finitely generated and
, then
is said to be finitely presentable. In this case,
is a polyhedral cone (with vertex
) and
is isomorphic to the Riesz space of all piecewise-homogeneous linear functions on
. Much of this can be found in [a1], but the crucial observation that all piecewise-homogeneous linear functions occur appears first in [a2]. Baker–Beynon duality [a3] states that the category
of finitely presentable Riesz spaces and arbitrary Riesz homomorphisms is dually equivalent to the category
of polyhedral cones (with vertex at
) in some
and with piecewise-homogeneous linear morphisms. The equivalence is via enriched
-functors. For any object
of
,
inherits the structure of a Riesz space from
. In the other direction, if
is in
, then
is in one-to-one correspondence with
.
It can be shown that if also , then
and
are
-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
of polyhedra and piecewise-affine linear mappings, [a3].
Among the first applications of this theory is Baker's proof [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 that every object is an absolute retract (cf. also Collapsibility). Presently (1996), nothing significant is known about projectives that are not finitely generated.
For important recent work related to Abelian -groups (cf.
-group), see [a5].
References
[a1] | K.A. Baker, "Free vector lattices" Canadian J. Math. , 20 (1968) pp. 58–66 |
[a2] | W.M. Beynon, "Combinatorial aspects of piecewise linear functions" J. London Math. Soc. , 7 (1974) pp. 719–727 |
[a3] | W.M. Beynon, "Duality theorems for finitely generated vector lattices" Proc. London Math. Soc. , 31 (1975) pp. 114–128 |
[a4] | G. Birkhoff, "Lattice theory" , Colloq. Publ. , Amer. Math. Soc. (1967) (Edition: Third) |
[a5] | D. Mundici, "Farey stellar subdivisions, ultrasimplicial groups and ![]() ![]() |
Baker-Beynon duality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Baker-Beynon_duality&oldid=35166