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Relation products of congruences and factor congruences. (English) Zbl 0790.08001
Let \(A_ 1\), \(A_ 2\) be algebras of the same type. Denote by \(\pi_ i\) the canonical projection from the Cartesian product \(A_ 1\times A_ 2\) onto \(A_ i\), \(i=1,2\). Then \(\Pi_ i= \text{Ker }\pi_ i\), \(i=1,2\), are called factor congruences on \(A_ 1\times A_ 2\). A congruence \(\Theta\) on \(A_ 1\times A_ 2\) is called a subfactor congruence whenever \(\Theta\subseteq \Pi_ 1\) or \(\Theta\subseteq \Pi_ 2\) holds.
Factor congruences and subfactor congruences were introduced by J. Hagemann [“Congruences on products and subdirect products of algebras”, Preprint No. 219, TH Darmstadt (1975)]; see also H.-P. Gumm [“Geometrical methods in congruence modular algebras”, Mem. Am. Math. Soc. 286 (1983; Zbl 0547.08006)] for further information. The present paper studies four congruence properties on the product \(A_ 1\times A_ 2\) which are related to permutability and “3 1/2- permutability” of congruences on \(A_ 1\times A_ 2\) with factor congruences \(\Pi_ 1\) and \(\Pi_ 2\).
08A30 Subalgebras, congruence relations
08B25 Products, amalgamated products, and other kinds of limits and colimits
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