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BASES FOR THE k-NORMALIZATIONS OF VARIETIES OF BANDS

The usual depth measurement on terms of a fixed type type r assigns to each term a non-negative integer called its depth. For k > 1, an identity s « t of type r is said to be Abnormal (with respect to the depth measurement) if either s = t or both s and t have depth > k. Taking k = 1 gives the well-known property of normality of identities. A variety is called Abnormal (with respect to the depth measurement) if all its identities are A;-normal. For any variety V, there is a least fc-normal variety Nk(V) containing V, the variety determined by the set of all fc-normal identities of V. In this paper we produce for every subvariety V of the variety B of bands (idempotent semigroups) a finite equational basis for Nk(V), for k > 1.