Locally coherent algebras. (English) Zbl 0958.08003

Let \({\mathbb A}\) be an algebra with a constant unary term function \(0\) and \({\mathbb V}\) a variety with a constant unary term \(0\). \({\mathbb A}\) is called locally coherent if every subalgebra of \({\mathbb A}\) which contains a class of some \(\Theta\in\text{ Con}{\mathbb A}\) also contains \([0]\Theta\). \({\mathbb A}\) is called locally regular if every single class of every \(\Theta\in\text{Con}{\mathbb A}\) determines \([0]\Theta\). \({\mathbb A}\) is said to have \(\text{LCUT}\) if for every subalgebra \((B,F)\) of \({\mathbb A}\), every \(b\in B\), every \(\Theta\in\text{Con}{\mathbb A}\) with \([b]\Theta\subseteq B\), every positive integer \(n\) and every \(n\)-ary polynomial function \(p\) over \({\mathbb A}\) with \(p(b,\dots,b)=0\), there holds \(p([b]\Theta,\dots,[b]\Theta)\subseteq B\). \({\mathbb A}\) is called permutable at \(0\) if \([0](\Theta\circ\Phi)=[0](\Phi\circ\Theta)\) for all \(\Theta,\Phi\in\text{ Con}{\mathbb A}\). \({\mathbb V}\) is called locally coherent, locally regular or permutable at \(0\) or it is said to have \(\text{LCUT}\) if every of its members has the corresponding property. Locally coherent varieties are characterized by a Mal’tsev condition. Further it is proved that if \({\mathbb V}\) is locally coherent then it both is locally regular and has \(\text{LCUT}\), and that the converse holds if \({\mathbb V}\) is permutable at \(0\).


08B05 Equational logic, Mal’tsev conditions
08A30 Subalgebras, congruence relations
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