Splitting p-algebras
A pair (\({\mathcal K}_ 0,{\mathcal K}_ 1)\) of equational subclasses of an equational class \({\mathcal K}\) is splitting in \({\mathcal K}\), if, for every equational subclass \({\mathcal K}_ 2\) of \({\mathcal K}\), either \({\mathcal K}_ 2\subseteq {\mathcal K}_ 0\) or \({\mathcal K}_ 1\subseteq {\mathcal K}_ 2\) in \({\mathcal K}\). If (\({\mathcal K}_ 0,{\mathcal K}_ 1)\) is a splitting pair in \({\mathcal K}\), \({\mathcal K}_ 1\) is generated by a subdirectly irreducible algebra A, called a splitting algebra for \({\mathcal K}\), and \({\mathcal K}_ 0\) is the largest subvariety of \({\mathcal K}\) not containing A, called the conjugate variety. It is generated by a single identity, called the conjugate identity. (This notion is due to \textit{R. McKenzie} [Trans. Am. Math. Soc. 174, 1-43 (1973; Zbl 0265.08006)]; for other structures, see e.g. \textit{A. Day} [Algebra Univers. 5, 153-162 (1975; Zbl 0324.08004)], and \textit{B. Jónsson} [ibid. 15, 273-298 (1982; Zbl 0545.08009)].) In the article under review, splitting algebras in various classes of p- algebras are studied. For \(\omega \geq n\geq 0\), \({\mathcal P}_ n\) is the equational class of p-algebras satisfying Lee's identity \(L_ n\), cf. \textit{K. Lee} [Can. J. Math. 22, 881-891 (1970; Zbl 0244.06009)]. From the numerous interesting results of this article, we state just a few, the choice being due entirely to the reviewer's taste. 1) Every splitting algebra in \({\mathcal P}_ n\) is finite. 2) Every equational subclass of \({\mathcal P}_ n\) generated by a (finite) subdirectly irreducible algebra is (completely) join irreducible in the lattice of subvarieties of \({\mathcal P}_ n\). 3) Let \(n\in\{0,\omega\}\), and L be a finite subdirectly irreducible algebra in \({\mathcal P}_ n\). Then the following are equivalent: i) L is splitting in \({\mathcal P}_ n\), ii) L is a bounded homomorphic image of a free algebra in \({\mathcal P}_ n\) with finitely many generators. iii) Every homomorphism from a finitely generated algebra of \({\mathcal P}_ n\) to L is bounded. In the sequel, a construction due to \textit{A. Day} [Can. Math. Bull. 13, 253-254 (1970; Zbl 0206.297)] is used to exhibit several splitting algebras and their conjugate identities for various \({\mathcal P}_ n\). The article concludes with some examples.
scientific article; zbMATH DE number 3871429