Abstract
A description is given of then-generated free algebras\(F_{\mathcal{M}\mathcal{O}_k } (n),{\mathbf{ }}n > 2\), in the varieties\(\mathcal{M}\mathcal{O}_k ,{\mathbf{ }}k > 2\), of modular ortholattices generated by the ortholatticesMO k of height 2 with 2k atoms. Algebraic methods of the theory of orthomodular lattices are combined with natural duality theory for varieties of algbras. The procedures involved in the analysis of\(F_{\mathcal{M}\mathcal{O}_k } (n)\) generalize the techniques applied in the preceding paper, where the casesk=2,n>2 were solved. The free algebras are decomposed by central elements into products of canonical intervals. Previous methods are refined to accommodate the fact that the decompositions of\(F_{\mathcal{M}\mathcal{O}_k } (n)\) lead to intervals ofk−1 different types. Their structures are obtained from natural dualities for the varieties\(\mathcal{M}\mathcal{O}_k ,k > 2\). Finally, Stirling numbers of the second kind are used to count the number of intervals. The structures of the free algebras\(F_{\mathcal{M}\mathcal{O}_k } (n)\) fork, n≤10 are explicity displayed in a table.
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Haviar, M., Konôpka, P. & Wegener, C.B. Finitely generated free modular ortholattices. II. Int J Theor Phys 36, 2661–2679 (1997). https://doi.org/10.1007/BF02435705
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DOI: https://doi.org/10.1007/BF02435705