A CONGRUENCE IDENTITY SATISFIED BY m-PERMUTABLE VARIETIES

We present a new and useful congruence identity satisfied by m-permutable varieties. It has been proved in [L1] that every m-permutable variety satisfies a non-trivial lattice identity (depending only on m). In [L2] we have found another interesting identity: Theorem 1. For m ≥ 3, every m-permutable variety satisfies the congruence identity αβh = αγh, for h = m[ m+1 2 ]− 1 Here, [ ] denotes integer part, and βh, γh are defined as usual: β0 = γ0 = 0, βn+1 = β + αγn, γn+1 = γ + αβn. The proof of Theorem 1 consists of two steps. As for the second step, it is an easy application of commutator theory, but in the present note we shall be concerned only with the first step. The first step is a commutator-free proof of the following Theorem 2. If every subalgebra of A is m-permutable then A satisfies (Xm). More generally: (i) If every congruence of A, thought of as a subalgebra of A, is m-permutable then A satisfies (Xm). (ii) If every subalgebra of A generated by m + 1 elements is mpermutable then A satisfies (Xm); actually, A satisfies the stronger version of (Xm) in which α is only supposed to be a compatible relation on A, and δ is any relation on A. In the statement of Theorem 2 we have used: 2000 Mathematics Subject Classification. 08A30, 08B05.