Algebraically closed algebras in certain small congruence distributive varieties
According to \textit{J. Schmid} [Z. Math. Logik Grundlagen Math. 25, 525--530 (1979; Zbl 0425.06007)], a distributive lattice is algebraically closed iff it is relatively complemented. This characterization is an easy consequence of the much more general main theorem of the present paper: If \(\mathcal V\) is a finitely generated congruence-distributive variety such that any direct product of algebraically closed algebras from \(\mathcal V\) is also algebraically closed, then the algebraically closed algebras in \(\mathcal V\) are precisely the updirected unions of finite direct products of maximal subdirectly irreducibles from \(\mathcal V\); moreover, the class of all algebraically closed algebras in \(\mathcal V\) is elementary, to wit, definable by universal-existential Horn sentences (which is obvious in the example cited above).
scientific article; zbMATH DE number 5646232
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