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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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