Permanence of Identities on Algebras

In our previous paper [1] we investigated under what circumstances the identities of a partially ordered (universal) algebra remain valid when the algebra is completed to a suprema preserving superior completion ([1], (2.6), (3.7), (5.11)). In general we use the terminology and notation of [1], with some minor exceptions that will be noted later. We there showed that if both sides of an identity contain no repeated variables then the identity remains valid under the completion. Gautam [7] handled the unordered case of the problem in [1] where T is the complex algebra including the empty set, which causes certain changes. Fuchs' work [5], [6] also has considerable overlap with [1], although our papers were written independently. (See also Clifford [3].) In this paper we consider the converse problem; namely, if the identities on a (finitary) algebra and its completion are the same what can be said about the identities of the original algebra. From another point of view we give conditions under which the identities on an algebra can be determined by looking at the identities on appropriate sub algebras. We maintain the former point of view in this paper. In §2 we introduce the notions of split words and split pairs of words and give a quasi ordering to the words of a free algebra (called polynomial algebra by Gratzer [8]), in which the maximal words are the split words except when the cardinality of the generating set is finite and the word is long. At the suggestion of the referee we now use the term 'linear' where 'split' was used in [1]. In § 3, we prove two technical lemmas about homomorphisms and suprema of a certain special collection of points in the completion T of an algebra S. In §4, we prove the main results. These are simplest to describe when S is trivially ordered (unordered) and free over itself with an infinity of generators and T is the complex algebra. We show (4.17) that the identities on Sand T are the same if and only if the identities on S considered as a fully invariant congruence on a free infinitely generated (polynomial) algebra are generated by linear pairs of words. As a corollary we obtain (4.10) that an equational class is closed under taking complex algebras if and only if the smallest fully invariant congruence on an infinitely generated free algebra containing the defining identities is generated by pairs of linear words. When the order is non-trivial or the number of free generators of S is finite, then more