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The direct decomposition of \(l\)-algebras into products of subdirectly irreducible factors. (English) Zbl 1045.08002
The paper under review is a continuation of the reviewer and S. El-Assar’s article [Acta Math. Hung. 48, 301–316 (1986; Zbl 0618.08001)]. Recall that \((L; \wedge, 0,1,F)\) is an \(l\)-algebra, if for every \(n\)-ary operation \(f\in F\) and any center element \(c\in\text{Cen}(L)\) there is \(f(x_1,\dots, x_n)\equiv f(y_1,\dots, y_n)(\varphi_c)\) whenever \(x_i\equiv y_i(\varphi_c)\), \(i= 1,\dots, n\), and \(\varphi_c\) is a congruence relation generated by \(c\).
Main results: Let \(L\) be an \(l\)-algebra. (1) \(L\) is a direct product of finitely subdirectly irreducible \(l\)-algebras if and only if \(L\) enjoys property (PCC), \(\text{Con}(L)\) is a Stone lattice and the underlying lattice \(L\) is weakly central-complete with an atomic center. (2) \(L\) is a direct product of subdirectly irreducible \(l\)-algebras if and only if \(L\) enjoys property (PCC), \(\text{Con}(L)\) is an atomic Stone lattice and the underlying lattice \(L\) is weakly central-complete.

MSC:
08A05 Structure theory of algebraic structures
06F99 Ordered structures
06B05 Structure theory of lattices
06B10 Lattice ideals, congruence relations
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