With each orthogeometry (P, ⊥) we associate \documentclass[12pt]{minimal}
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\begin{document}$${{\mathbb {L}}(P, \bot)}$$\end{document}, a complemented modular lattice with involution (CMIL), consisting of all subspaces X and X⊥ such that dim X < ℵ0, and we study its rôle in decompositions of (P, ⊥) as directed (resp., disjoint) union. We also establish a 1–1 correspondence between ∃-varieties \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {V}}$$\end{document} of CMILs with \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {V}}$$\end{document} generated by its finite dimensional members and ‘quasivarieties’ \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {G}}$$\end{document} of orthogeometries: \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {V}}$$\end{document} consists of the CMILs representable within some geometry from \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {G}}$$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {G}}$$\end{document} of the (P, ⊥) with \documentclass[12pt]{minimal}
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\begin{document}$${{\mathbb {L}}(P, \bot) \in {\mathcal {V}}}$$\end{document}. Here,\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {V}}$$\end{document} is recursively axiomatizable if and only if so is \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {G}}$$\end{document}. It follows that the equational theory of \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {V}}$$\end{document} is decidable provided that the equational theories of the \documentclass[12pt]{minimal}
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\begin{document}$${\{{\mathbb {L}}(P, \bot)\, |\, (P, \bot) \in \mathcal {G}, {\rm{dim}} P = n\}}$$\end{document} are uniformly decidable.
