Abstract: | We deal with varieties with one basic operation f(x1,...,xn) and one defining identity f(x1,..., xn) = f(xπ(1),...,xπ(n)), where π is a permutation whose cyclic set consists of distinct primes p1,...,pr, with the sum p1+...+pr = n. Their interpretability types, together with the greatest element 1 in a lattice int, are said to be arithmetic. It is proved that the arithmetic types constitute a distributive lattice ar, which is dual to a lattice Sub fΠ of finite subsets of the set Π of all primes. It is shown that for n ⩾ 2, the poset ar( n) of arithmetic types defined by permutations in n, for n fixed, is a lattice iff n = 2, 3, 4, 6, 8, 9, 11. __________ Translated from Algebra i Logika, Vol. 44, No. 5, pp. 622–630, September–October, 2005. |