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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
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