## 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 $$0$$.

### MSC:

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

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