Quasivarieties of Modular Ortholattices

Lattice of subquasivarieties of variety generated by modular ortholattices MOn, n 2 ! and MO! is described. In [3] Igo sin has proved that any subquasivariety of the variety M! is a variety. M! denoted variety generated by modular lattice M!, where M! is the lattice of length two with ! atoms. For short proof of this fact see [2]. In this note we present an orthomodular counterpart of this result. In the case of modular ortholattices not every subquasivariety of the variety MO! form a variety. We give however, a complete description of the lattice of subquasivariety of MO!. This description is a consequence of some known general results, summarized here in lemmas 1, 2, 3, not published up to now, however. For basic fact from Universal algebra we refer to [1]. The author is very indebted to Wies law Dziobiak for many helpful discussions and especially for the proof of lemma 1. Let K be a class of algebras, by Q(K) we denote the least quasivariety containing the class K. For every class K of algebras an algebra A 2 K is subdirectly K irreducible i for every set (Ai : i 2 I) of algebras from K if A is a subdirect product of (Ai : i 2 I) then A is isomorphic to Ai0 for some i0 2 I. If K is a quasivariety then every algebra A 2 K is isomorphic to subdirect product of subdirectly K-irreducible algebras. An non trivial algebra A will be called critical i A is subdirectly Q(A)-irreducible. A nite algebra is critical if and only if it does not belong to the quasivariety generated by its proper subalgebras. Every quasivariety of algebras is generated by its nitely generated critical algebras. A class K of algebras will be called locally nite if and only if every nitely generated algebra from K is nite.