QUASISHEFFER OPERATIONS AND k-PERMUTABLE ALGEBRAS

A well known theorem of Murskiı̆’s asserts that almost every finite, nonunary algebra is idemprimal. We derive an analagous result under the assumption that all basic operations are idempotent. If the algebra contains a basic l-ary idempotent operation with l > 2 then the algebra is idemprimal with probability 1. However, for an algebra with a single basic binary operation, the probability of idemprimality is only e−2 ≈ 0.14. These observation are applied to show that for any k > 2, a finite algebra generating a congruence k-permutable variety will, with probability 1, generate a congruence 2-permutable variety. This investigation arose from the observation that we know few examples of varieties that are congruence 3-permutable but not 2-permutable, 4-permutable but not 3-permutable, etc. (Definitions of these terms appear in Sections 1 and 2.) On the face of it, it ought to be easy to create such varieties. The Hagemann-Mitschke terms provide a recipe for constructing a k-permutable variety. By choosing the remaining values of those operations randomly, one would expect that the resulting variety would fail to satisfy any other identities (such as (k − 1)-permutability). Aswe discovered, that turns out not to be the case. A random, finite, k-permutable algebra almost surely generates a 2-permutable variety. In fact, as we shall demonstrate below, a random, idempotent, ternary operation on a finite set will be quasisheffer with probability 1. The resulting algebra is not only 2-permutable, but also rigid, and quasiprimal, so it generates a variety that is congruence distributive, semisimple, and equationally complete. These results are not surprising in light of Murskiı̆’s theorem: almost every nonunary operation on a finite set is quasisheffer. The only new wrinkle here is the requirement that the operation be idempotent. But note that Murskiı̆’s theorem applies even to binary operations. As we show in Theorem 2.13, the probability that an idempotent binary operation be quasisheffer is quite small. 1. BACKGROUND AND DEFINITIONS Basic concepts of universal algebra as well as explanations for the notation used in this paper can be found in [1]. Let A be a set, , binary relations on A. The Date: August 2017. 2010 Mathematics Subject Classification. Primary: 08A40; Secondary: 08B05, 60A99.