## Full subdirect and weak direct products of algebras.(English)Zbl 0797.08005

Let $$P(I)$$ be the Boolean algebra of all subsets of a set $$I$$. Fix an ideal $$L$$ of $$P(I)$$. A subalgebra $$A$$ of the direct product $$\prod_{i\in I} A_ i$$ is called $$L$$-restricted full subdirect product of algebras $$A_ i$$, $$i\in I$$, if (i) for any $$i\in I$$ and any elements $$x,y\in A$$ there exists an element $$z\in A$$ such that $$z(i)= x(i)$$ and $$z(j)= y(j)$$ for all $$j\in T\backslash i$$; (ii) if $$x,y\in A$$ then the set of all $$j\in I$$ such that $$x(j)\neq y(j)$$ belongs to $$L$$. Here $$a(j)$$ denotes the projection of an elements $$a\in A$$ into $$A_ j$$. The first result of the paper characterizes $$L$$-restricted full subdirect products in terms of kernels of projections $$A\to A_ i$$, $$i\in I$$. For any algebra $$A$$ denote by $$\text{DCon}(A)$$ the set of all congruences $$\alpha$$ on $$A$$ such that $$\alpha\cap\beta= 0$$, $$\alpha\circ\beta= 1$$ for some congruence $$\beta$$ on $$A$$. If $$A$$ is congruence-distributive and $$\text{DCon}(A)$$ is closed under unions then $$A$$ is isomorphic to an $$L$$- restricted full subdirect product of directly indecomposable algebras $$A_ i$$, $$i\in I$$, where $$L$$ is an ideal of $$P(I)$$ containing all finite subsets of $$I$$. Some applications of these results are given.

### MSC:

 08B26 Subdirect products and subdirect irreducibility 08B25 Products, amalgamated products, and other kinds of limits and colimits 08B10 Congruence modularity, congruence distributivity

### Keywords:

subdirect product; kernels of projections
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### References:

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