I looked in all textbooks on vector lattices (Riesz spaces) as well as ordered vector spaces, but couldn't find any mentions of neither inductive nor projective limit for these structures. Googling also didn't help. Since there is some ambiguity in terminology let me state the definition.


Let $\Gamma$ be a directed set and let $\left(E\_{\gamma}\right)\_{\gamma\in\Gamma}$ be a family of partially ordered vector spaces, such that for every $\alpha\le\beta$ there is a designated positive linear operator $T\_{\alpha\beta}:E\_{\alpha}\to E\_{\beta}$, such that $T\_{\gamma\gamma}=Id\_{E\_{\gamma}}$ and $T\_{\beta\gamma}T\_{\alpha\beta}=T\_{\alpha\gamma}$, for every $\alpha,\beta,\gamma$. The inductive limit then is an ordered space $E$ and a collection of positive linear operators $T\_{\gamma}:E\_{\gamma}\to E$ such that $T\_{\beta}T\_{\alpha\beta}=T\_{\alpha}$, and whenever $F,\left(S\_{\gamma}\right)\_{\gamma\in\Gamma}$ has this property, there is a unique $S:E\to F$ such that $S\_{\gamma}=ST\_{\gamma}$.


If we drop "ordered" and "positive", we will get the definition of the inductive limit for vector spaces. In fact, the inductive limit of ordered vector spaces can be constructed as the inductive limit of the underlying vector spaces endowed with the weakest linear order the makes all $T\_{\gamma}$'s positive. It is also not hard to show that if all $E\_{\gamma}$'s are vector lattices, and $T\_{\alpha\beta}$'s are lattice homomorphisms, then $E$ is itself a vector lattice.



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> I need some permanence properties of this construction, which I suspect are known, so I am asking for a reference where it is studied systematically.
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Reference on inductive (direct) limit of ordered vector spaces and vector lattices
There is a paper by Wolfgang Filter form 1980s.


W. Filter, Inductive limits of Riesz spaces, Proceedings of the International Conference held in Dubrovnik, June 23-27, 1987 (Bogoljub Stankovic, Endre Pap, Stevan Pilipovic, and Vasilij S. Vladimirov, eds.), Plenum Press, New York, 1988, pp. 383-392.


Unfortunately I cannot supply a link.


