An equivalence functor between local vector lattices and vector lattices

Document Type: Research Paper


Département de Mathématiques Faculté des Sciences de Tunis Université Tunis-El Manar Campus Universitaire


We call a local vector lattice any vector lattice with a distinguished positive strong unit and having exactly one maximal ideal (its radical). We provide a short study of local vector lattices. In this regards, some characterizations of local vector lattices are given. For instance, we prove that a vector lattice with a distinguished strong unit is local if and only if it is clean with non no-trivial components. Nevertheless, our main purpose is to prove, via what we call the radical functor, that the category of all vector lattices and lattice homomorphisms is equivalent to the category of local vectors lattices and unital (i.e., unit preserving) lattice homomorphisms.


[1] Aliprantis, C.D. and Burkinshaw, O., “Locally Solid Riesz Spaces with Applications to Economics”, Mathematical Surverys and Monographs 105, Amer. Math. Soc., 2003.
[2] Aliprantis, C.D. and Burkinshaw, O., “Positive Operators”, Springer, 2006.
[3] Aliprantis, C.D. and R. Tourkey, R., “Cones and Duality”, Graduate Studies in  athematics 84, Amer. Math. Soc., 2007.
[4] Boulabiar, K. and Smiti, S., Lifting components in clean abelian `-groups, Math. Slovaca, 68 (2018), 299-310.
[5] Hager, A.W., Kimber, C.M., and McGovern, W.W., Clean unital `-groups, Math. Slovaca 63 (2013), 979-992.
[6] Herrlich, H. and Strecker, G.E., “Category Theory: an Introduction”, Allyn and Bacon Inc., 1973.
[7] Luxemburg, W.A.J. and Zaanen, A.C., “Riesz spaces” I, North-Holland Research Monographs, Elsevier, 1971.
[8] Matsumura, H., “Commutative Ring Theory”, Cambridge University Press, 1989.