Baker-Beynon duality

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 of AF -algebras" Adv. Math. , 68 (1988) pp. 23–39