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

Walendziak, Andrzej

Math. Slovaca 44, No. 1, 45-54 (1994).

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.

Reviewer: V.A.Artamonov (Moskva)

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

References:

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