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Problem 17 of Gr~itzer and Kisielewicz



BURRIS AND R. WILLARD If A is an algebra and n < co then pn(A) denotes the number of n-ary term operations of A that depend on all n variables. Let ~r denote the class of all algebras of type (2, 2) in which both basic operations are commutative and idempotent. In [6, Problem 17] G. Gr/itzer and A. Kisielewicz ask whether there exists A~Cg satisfying p2(A) = 2 and such that for all B ~ cg, ifpz(B) = 2 then p~(B)> pn(A) for all n. J. Dudek [4] has shown that if the answer is yes, then the algebra A can be taken to be the two-element distributive lattice D2; and more recently has shown [5] that to answer the question it suffices to compare the pn-sequence of D2 to the pn-sequences of two specific four-element members of cg. One of these 'test members ' is the idempotent commutative groupoid N2 whose universe is {0, 1, 2, 3} and whose binary operation o is given by the following table: o 0 1 2

Year: 2013
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