The divisible hull and orthocompletion of~lattice ordered modules

V. K. Zakharov

Sbornik. Mathematics, Tome 32 (1977) no. 3, pp. 293-303

Résumé

This work is devoted to investigation of order properties of the regular completion $R(X)$, the orthocompletion $M(X)$, and the divisible (=injective) hull $D_{\mathfrak F_R}(X)$ with respect to dense ideals of an $f$-module $X$ torsion free with respect to the filter $\mathfrak F_R$ of dense ideals over the commutative $f$-ring $R$ without nilpotent elements. The possibility of extending the order from $X$ to $R(X)$, $M(X)$ and $D(X)$ is established. The equivalence is demonstrated of the notions of orthocompleteness and lattice orthocompleteness, divisibility and order divisibility, and injectivitity and lattice injectivity. Also, the equivalence is proved of order divisibility, lattice injectivity and lattice completeness and regularity. Appropriate characterizations of $M(X)$ and $D(X)$ are given. Bibliography: 14 titles.