Abstract
Following W. Taylor we define a hyperidentity ∈ to be formally the same as an identity (e.g.,F(G(x, y, z), G(x, y, z))=G(x, y, z)). However, a varietyV is said to satisfy a hyperidentity ∈, if whenever the operation symbols of ∈ are replaced by any choice of polynomials (appropriate forV) of the same arity as the corresponding operation symbols of ∈, then the resulting identity holds inV in the usual sense. For example, if a varietyV of type <2,2> with operation symbols ∨ and ∧ satisfies the hyperidentity given above, then substituting the polynomial (x∨y)∨z for the symbolG, and the polynomialx∧y forF, we see thatV must in particular satisfy the identity ((x∨y)∨z)∧((x∨y)∨z)=((x∨y)∨z). The set of all hyperidentities satisfied by a varietyV, will be denoted byH(V). We shall letH m (V) be the set of all hyperidentities hoiding inV with operation symbols of arity at mostm, andH n (V) will denote the set of all hyperidentities ofV with at mostn distinct variables. In this paper we shall show that ifV is a nontrivial variety of lattices or the variety of all semilattices, then for any integersm andn, there exists a hyperidentity ∈ such that ∈ holds inV, and ∈ is not a consequence ofH m (V)∪H n (V). From this it is deduced that the hyperidentities ofV are not finitely based, partly soling a problem of Taylor [7, Problem 3].
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The research of the author was supported by NSERC of Canada.
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Penner, P. Hyperidentities of lattices and semilattices. Algebra Universalis 13, 307–314 (1981). https://doi.org/10.1007/BF02483843
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DOI: https://doi.org/10.1007/BF02483843