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.


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


[1] DRAŠKOVIČOVÁ H.: Weak direct product decomposition of algebras. Contributions to General Algebra 5. Proc. of the Salzburg Conference, May 29-June 1, 1986, Wien, 1987, pp. 105-121.
[2] HASHIMOTO J.: Direct, subdirect decompositions and congruence relations. Osaka J. Math. 9 (1957), 87-112. · Zbl 0078.01805
[3] JAKBÍK J.: Weak product decompositions of discrete lattices. Czechoslovak Math. J. 21 (96) (1971), 399 -412. · Zbl 0224.06003
[4] McKENZIE R., McNULTY G., TAYLOR W.: Algebras, Lattices, Varieties. Vol. I. Wadsworth & Brooks, Monterey, 1987. · Zbl 0611.08001
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.