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OF MODULAR ORTHOLATTICES

Lattice of subquasivarieties of variety generated by modular ortholattices MOn, n ∈ ω and MOω is described. In [3] Igošin has proved that any subquasivariety of the variety Mω is a variety. Mω denoted variety generated by modular lattice Mω, where Mω is the lattice of length two with ω 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 MOω form a variety. We give however, a complete description of the lattice of subquasivariety of MOω. 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 WiesÃlaw 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 ∈ K is subdirectly K irreducible iff for every set (Ai : i ∈ I) of algebras from K if A is a subdirect product of (Ai : i ∈ I) then A is isomorphic to Ai0 for some i0 ∈ I. If K is a quasivariety then every algebra A ∈ K is isomorphic to subdirect product of subdirectly K-irreducible algebras. An non trivial algebra A will be called critical iff A is subdirectly Q(A)-irreducible. A finite 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 finitely generated critical algebras. A class K of algebras will be called locally finite if and only if every finitely generated algebra from K is finite.