Consistent algebras
In the following let \({\mathbf N}\) denote the set of all positive integers, let \({\mathcal A}\) be an algebra with two constants \(0\) and \(1\) and for every \(n\in{\mathbf N}\) let \(\text{P}_n({\mathcal A})\) denote the set of all \(n\)-ary polynomial functions over \({\mathcal A}\). \({\mathcal A}\) is called consistent if for every \(\Theta\in\text{Con}{\mathcal A}\) and every subuniverse \(B\) of \({\mathcal A}\) the inclusion \([0]\Theta\subseteq B\) is equivalent to \([1]\Theta\subseteq B\). \({\mathcal A}\) is said to have balanced congruences (BC) if for every \(\Theta,\Phi\in\text{Con}{\mathcal A}\) the equality \([0]\Theta=[0]\Phi\) is equivalent to \([1]\Theta=[1]\Phi\). \({\mathcal A}\) is said to have allied constant classes (ACC) if for every \(\Theta\in\text{Con}{\mathcal A}\), every subuniverse \(B\) of \({\mathcal A}\), every \(n\in{\mathbf N}\) and every \(p\in\text{P}_n({\mathcal A})\) the following holds: If \([0]\Theta\subseteq B\) and \(p(0,\dots,0)=1\) then \(p(([0]\Theta)^n)\subseteq B\), and if \([1]\Theta\subseteq B\) and \(p(1,\dots,1)=0\) then \(p(([1]\Theta)^n)\subseteq B\). \({\mathcal A}\) is said to be permutable at \(0\) and \(1\) if \([x](\Theta\circ\Phi)=[x](\Phi\circ\Theta)\) for all \(x\in\{0,1\}\) and all \(\Theta,\Phi\in\text{Con}{\mathcal A}\). \({\mathcal A}\) is said to have polynomially linked constant classes (PLCC) if for every \(\Theta\in\text{Con}{\mathcal A},\) every subuniverse \(B\) of \({\mathcal A}\), every \(n\in{\mathbf N}\), every \(p_1,\dots,p_n\in\text{P}_1({\mathcal A})\) and every \(a_1,\dots,a_{n-1}\in[0]\Theta\) the following holds: If \([0]\Theta\subseteq B\), \(p_1(0)=1\) and \(p_i(a_i)=p_{i+1}(0)\) for all \(i=1,\dots,n-1\) then \(p_n(0)\in B\), and if \([1]\Theta\subseteq B\), \(p_1(1)=0\) and \(p_i(a_i)=p_{i+1}(1)\) for all \(i=1,\dots,n-1\) then \(p_n(1)\in B\). A variety with constant unary terms \(0\) and \(1\) is said to have one of the mentioned properties if each of its members has this property. Consistent varieties are characterized by a Mal'tsev type condition and the following results are proved: Every consistent variety has BC and ACC. Every algebra having BC and ACC and being permutable at \(0\) and \(1\) is consistent. An algebra having BC need not be consistent. Every algebra having BC and PLCC is consistent. The conditions of having PLCC resp. BC are independent. A variety is consistent if and only if it has BC and PLCC.NEWLINENEWLINEFor the entire collection see [Zbl 0970.00014].
scientific article; zbMATH DE number 1683834