\((\mathcal L\), \(\mathcal L')\)-products of algebras (Q2702773)

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 \(I\). NEWLINENEWLINENEWLINEThe 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. NEWLINENEWLINENEWLINEThe 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 \(i\in I\).