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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 $$\Pi_ 2$$.
##### MSC:
 08A30 Subalgebras, congruence relations 08B25 Products, amalgamated products, and other kinds of limits and colimits
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##### References:
  Chajda I.: The egg-box property of congruences. Math. Slovaca 38 (1988), 243-247. · Zbl 0649.08002  Duda J.: On two schemes applied to Maľcev type theorems. Ann. Univ. Sci. Budapest, Sectio Math. 26 (1983), 39-45. · Zbl 0518.08002  [31 Fraser G. A., Horn A.: Congruence relations in direct products. Proc. Amer. Math. Soc. 26 (1970), 390-394. · Zbl 0241.08004  Gumm H.-P.: Geometrical methods in congruence modular algebras. Memoirs Amer. Math. Soc. Nr. 286-Providence 1983. · Zbl 0547.08006  Hagemann J.: Congruences on products and subdirect products of algebras. Preprint Nr. 219, TH-Darmstadt 1975.
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