Slim Groupoids

Slim groupoids are groupoids satisfying x(yz) ≈ xz. We find all simple slim groupoids and all minimal varieties of slim groupoids. Every slim groupoid can be embedded into a subdirectly irreducible slim groupoid. The variety of slim groupoids has the finite embeddability property, so that the word problem is solvable. We introduce the notion of a strongly nonfinitely based slim groupoid (such groupoids are inherently nonfinitely based) and find all strongly nonfinitely based slim groupoids with at most four elements; up to isomorphism, there are just two such groupoids. We are going to investigate groupoids (algebras with one binary operation) satisfying the equation x(yz) ≈ xz. Since every term operation of such a groupoid can be represented by a slim term (a term that is a product of a finite sequence of variables with all parentheses grouped to the left), these groupoids are called slim. Similarly as in the case of semigroups, a free object in the variety of slim groupoids is the set of words over a given set of generators; only the multiplication of words differs from that in a free semigroup. One can expect that the variety of slim groupoids will have similar properties as the variety of semigroups. In some cases it is true. We will see, however, that the variety of slim groupoids has solvable word problem and has the strong amalgamation property. The purpose of this paper is to introduce and investigate basic properties of the variety of slim groupoids. We are particularly interested in the existence of finite, nonfinitely based slim groupoids. It has been shown by McKenzie [2] that the finite basis problem for equations of finite algebras is unsolvable: there is no algorithm deciding for an arbitrary finite algebra, or a finite groupoid, whether it has a finite basis for its equations. For many varieties, like those of groups or lattices, the problem is solvable in a trivial way: every finite algebra in such a variety has a finite basis. So, it is desirable to look for (natural) examples of varieties with the finite basis problem solvable but in a nontrivial way. Such a variety should be in some sense reasonably small and in another sense reasonably large. Perhaps the variety of slim (or idempotent slim) groupoids could be a good candidate. We introduce the notion of a strongly nonfinitely based slim groupoid (such 1991 Mathematics Subject Classification. 20N02.