Lee classes and sentences for pseudo-complemented semilattices
If B is any Boolean algebra, let \^B stand for the structure obtained from B by adjoining a new greatest element. \^B is a distributive pseudocomplemented lattice (a DPCL, for short). It is well-known [see \textit{K. B. Lee}, Can. J. Math. 22, 881-891 (1970; Zbl 0244.06009)] that the class of all DPCL's is a variety, that its subdirectly irreducible members are just the algebras of type \^B and that the classes Ḇ\({}_ n=ISP(\hat B_ n)\) (with \(B_ n\) the n-atom Boolean algebra) for \(n=0,1,2,..\). are the only proper subvarieties. These subvarieties are characterized within DPCL's by single identities, the so-called Lee identities \(L_ n.\) Turning to pseudocomplemented (meet-)semilattices (PCS, for short), it is also known that the class of all PCS's is a variety with algebras of type \^B exhausting the subdirectly irreducibles. However, there are no proper subvarieties except for that of Boolean algebras [see \textit{H. P. Sankappanavar}, Math. Jap. 25, 519-521 (1980; Zbl 0472.06009)]. The present paper considers the classes Ḇ\({}_ n=ISP(\hat B_ n)\) within \^B\({}_ n\) viewed as a PCS. We show that Ḇ\({}_ n\) is a quasivariety for \(n=0,1,2,..\). and provide explicitly a single quasiidentity \(q_ n\) characterizing Ḇ\({}_ n\) within PCS. Moreover, it is shown that the natural analogues \(l_ n\)- necessarily quasiidentities - of the Lee identities \(L_ n\) in the context of PCS fail to characterize the classes Ḇ\({}_ n\) within PCS.
scientific article; zbMATH DE number 4004236