Algebra Universalis, 28 (1991) 245-273 0002-5240/91/020245-29501.50 +0.20/0 9 1991 Birkh~.user Verlag, Basel Hyperidentities for some varieties of commutative semigroups SHELLY L. WISMATH ~ Abstract. Hyperidentities and hypervarieties have been defined by Taylor in [5]. A hypervariety is a class of varieties closed under the formation of equivalent, product, reduct and subvarieties. Hyperidentities are used to define hypervarieties, in the same way that ordinary identities define varieties. This paper produces some hyperidentities satisfied by various varieties of commutative semigroups, and identifies some restrictions as to what kind of hyperidentities such varieties can satisfy. It also continues the study, begun in [6], of the closure and hypervariety operators defined there, as they apply to varieties of commutative semigroups. 1. Introduction A hyperidentity is formally the same as an identity, and is made up of operation symbols and variables. A hyperidentity H is said to be satisfied by a variety V if whenever the operation symbols of H are replaced by terms of V of the appropriate arity, the identity which results is true in V. The identities produced in this way, by a choice of V-terms to be used in H, will be called (V-)instances of H. For example, the idempotent hyperidentity F(x, x) = x is satisfied by any variety all of whose binary terms are idempotent. Another well-known hyperidentity is called the medial hyperidentity, F(G(x, y), G(z, w)) = G(F(x, z), F(y, w)), based on the medial iden- tity xyzw = xzyw; it is known to be satisfied by the variety A of all commutative semigroups, and hence by all the subvarieties of A. A hypervariety is a collection of varieties closed under the formation of equivalent, product, reduct and subvarieties. Hyperidentities and hypervarieties are related by the following Birkhoff-type result: THEOREM 1.1 (Taylor [5]). Every hyperidentity defines a hypervariety, and conversely every hypervariety is definable by a set of hyperidentities. Presented by Walter Taylor. Received September 27, 1988 and in final form July 25, 1989. ~The results described in this paper form part of the author's Ph.D thesis, submitted to Simon Fraser University, Burnaby, Canada. The author is grateful for the help of her supervisor, Dr. N. R. Reilly, and for the financial support received from Simon Fraser University. 245