## Congruence distributivity in varieties with constants.(English)Zbl 0615.08005

An algebra $${\mathfrak A}$$ with a nullary operation $$c$$ is $$c$$-distributive if $$[c]\Phi \wedge (\Theta \vee \Psi)=[c](\Phi \wedge \Theta)\vee (\Phi \wedge \Psi)$$ for any $$\Phi, \Theta,\Psi\in \text{Con}\, A$$. A variety $${\mathcal V}$$ with a nullary operation $$c$$ in its type is $$c$$-distributive provided each $${\mathfrak A}\in {\mathcal V}$$ has this property.
Theorem. Let $${\mathcal V}$$ be a variety of algebras with a nullary operation $$c$$. The following conditions are equivalent: (1) $${\mathcal V}$$ is $$c$$-distributive; (2) there exist binary polynomials $$d_ 0,\ldots,d_ n$$ in $${\mathcal V}$$ such that $$d_ 0(x,y)=c$$, $$d_ n(x,y)=x$$, $$d_ i(c,y)=c$$ for $$i<n$$, $$d_ i(x,c)=d_{i+1}(x,c)$$ for $$i<n$$, $$i$$ even, and $$d_ i(x,x)=d_{i+1}(x,x)$$ for $$i<n$$, $$i$$ odd.

### MSC:

 08B10 Congruence modularity, congruence distributivity 08B05 Equational logic, Mal’tsev conditions 08A30 Subalgebras, congruence relations

### Keywords:

nullary operation; c-distributive; binary polynomials
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