A partial function f on a k-element set k is a partial Sheffer function if every partial function on k is definable in terms of f. Since this holds if and only if f belongs to no maximal partial clone on k, a characterization of partial Sheffer functions reduces to finding families of minimal coverings of maximal partial clones on k. We present, for each k ≥ 2, some members of every minimal covering of maximal partial clones on k. Furthermore, we describe the minimal coverings of maximal partial clones on k for k = 2 and k = 3 and deduce criteria for partial Sheffer functions on a 2-element and a 3-element set.
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