Basic questions for general algebras
This paper gives a simple method of generating inherently non-finitely based finite algebras, that is, finite algebras with the property that any locally finite variety which contains such an algebra does not have a finite basis for its laws. The algebras in question are groupoids so the laws are of the form \(t=u\) where the terms t and u are monomials. With each such term can be associated a binary tree with labelled leaves. A set P of terms is said to be closed under subtree free labelling if it contains every term obtained from one of its terms by taking a subtree and labelling the leaves arbitrarily with distinct variables. A term is basic if every subterm is a variable or of the form (v.s) or (s.v) where v is a variable. It is linear if no variable occurs more than once. The main result is: Theorem: Let A be a groupoid which generates a locally finite variety. Let P be a set of linear basic terms with unbounded lengths which is closed under subtree free labelling. If \(t\in P\) and \(t=u\) a law of A imply \(u\in P\) and \(\{xy=yx\}\vdash t=u\) then A is not contained in any finitely based, locally finite variety.- Some examples of such sets of terms and small groupoids with the required properties are given.
scientific article; zbMATH DE number 3867405