### Properties of fixed point set of a multivalued map.

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Some characterizations of random approximations are obtained in a locally convex space through duality theory.

Let $(G,\tau )$ be a commutative Hausdorff locally solid lattice group. In this paper we prove the following: (1) If $(G,\tau )$ has the $A$(iii)-property, then its completion $(\widehat{G},\widehat{\tau})$ is an order-complete locally solid lattice group. (2) If $G$ is order-complete and $\tau $ has the Fatou property, then the order intervals of $G$ are $\tau $-complete. (3) If $(G,\tau )$ has the Fatou property, then $G$ is order-dense in $\widehat{G}$ and $(\widehat{G},\widehat{\tau})$ has the Fatou property. (4) The order-bound topology on any commutative lattice group is the finest locally solid topology on...

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