An algebra \(A\) with a constant 0 is 0-permutable if \([0]R\circ S=[0]S\circ R\) for \(R, S\) congruences; is 0-distributive if \([0](R\cup S)\cap T=[0](R\cap S)\cup (R\cap T)\) for \(R, S, T\) congruences; is 0-arithmetical if it is 0-permutable and 0-distributive. Similarly for varieties. A variety is 0-arithmetical iff there is a binary term \(b\) satisfying in \(V: b(x,x)=b(0,x)=0\), \(b(x,0)=x\). An \(n\)-ary polynomial \(u\) is a 0-unanimity if \(u(x,\ldots,x)=x\) and \(u(x_1,\ldots,x_n)\) is 0 whenever at most one \(x_j\) is not 0. For a variety, the existence of a 0-unanimity is equivalent to a form of the Chinese remainder theorem (``localized at 0''), and together with 0-permutability is equivalent to 0-arithmeticity. There are also some remarks in the case of 0-regular (\([0]R=[0]S\) implies \(R=S\)) and in the case of ideal determined varieties.