# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Grätzer, General lattice theory (1978) [2] DOI: 10.1215/S0012-7094-62-02951-4 · Zbl 0114.01602 [3] DOI: 10.2307/1969328 · Zbl 0036.01802 [4] Burris, A course in universal algebra (1981) · Zbl 0478.08001 [5] DOI: 10.1007/s100120100014 · Zbl 1010.06007 [6] DOI: 10.1007/BF01110457 · Zbl 0167.01902 [7] Maeda, Theory of symmetric lattices (1970) · Zbl 0219.06002 [8] DOI: 10.1007/BF01951357 · Zbl 0618.08001 [9] Katriňák, Mat. Fyz. Časop. SAV 17 pp 20– (1967) [10] Janowitz, Pacific J. Math. 73 pp 87– (1977) · Zbl 0371.06005 [11] DOI: 10.1007/BF01190769 · Zbl 0818.06004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.