## $$(\mathcal L$$, $$\mathcal L')$$-products of algebras.(English)Zbl 0959.08003

Let $$I$$ be a nonempty set. If $$B$$ is an algebra, $$\theta_i$$, $$i\in I$$, a system of congruence relations on $$B$$ and $$M$$ a subset of $$I$$, the symbol $$\theta (M)$$ is used to denote the congruence relation on $$B$$ given as $$\bigwedge \{\theta_j\:j\in I-M\}$$. $$0_B$$ denotes the smallest congruence on $$B$$ and $$1_B$$ denotes the greatest congruence on $$B$$. Let $$A_i$$, $$i\in I$$, be a system of algebras of the same type. $$\prod A_i$$ denotes the direct product of algebras $$A_i$$. If $$x,y\in \prod A_i$$, let us denote $$I(x,y)=\{i\in I\:x(i)\neq y(i)\}$$. Let $$\mathcal L$$, $$\mathcal L'$$ be ideals of $$\mathcal P (I)$$, the power set of $$I$$.
The author defines an $$(\mathcal L,\mathcal L')$$-product of algebras $$A_i$$ as follows. Let $$A$$ be a subdirect product of algebras $$A_i$$. The algebra $$A$$ is an $$(\mathcal L,\mathcal L')$$-product of algebras $$A_i$$ if (i) for all $$x,y\in A$$, $$I(x,y)\in \mathcal L$$, and (ii) if $$x\in A$$, $$y\in \prod A_i$$ and $$I(x,y)\in \mathcal L'$$, then $$y\in A$$. Another definition is the following one. If $$B$$ is an algebra then a system $$\theta_i$$, $$i\in I$$, of congruences on $$B$$ is an $$(\mathcal L,\mathcal L')$$-representation of $$B$$ if the map $$f\:B\to \prod (B/\theta_i)$$, $$\rightarrow f(x)$$, defined by the rule $$f(x)(i)=x/\theta_i$$ $$(x/\theta_i$$ is the congruence class of $$\theta_i$$ containing x), is injective and $$f(B)$$ is an $$(\mathcal L,\mathcal L')$$-product of algebras $$B/\theta_i$$. Special cases of $$(\mathcal L,\mathcal L')$$-representations of algebras are direct, full subdirect and weak direct representations of algebras.
The main result is a characterization of $$(\mathcal L,\mathcal L')$$-representations of algebras: If $$B$$ is an algebra, then a system $$\theta_i$$, $$i\in I$$, of congruence relations on $$B$$ is an $$(\mathcal L,\mathcal L')$$-representation of $$B$$ if and only if (i) $$0_B=\bigwedge \{\theta_i\: i\in I\}$$, (ii) $$1_B=\bigvee \{\theta (M)\:M\in \mathcal L\}$$, and (iii) if $$M\in \mathcal L'$$ and if $$x,y_i\in B (i\in I)$$ are such that $$(x,y_i)\in \theta_i$$ for all $$i\in I-M$$ then there exists $$z\in B$$ satisfying $$(z,y_i)\in \theta_i$$ for all $$i\in I$$.

### MSC:

 08B25 Products, amalgamated products, and other kinds of limits and colimits 08A30 Subalgebras, congruence relations
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### References:

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