Interpolation on Semilattices (original) (raw)

Semigroups, 1980

Abstract

Abstract (An) Let be a variety, A an algebra in , and n > 0 an integer; A ( A n ) is an algebra in . Let P n ( A ) be the subalgebra of A ( A n ) generated by the n projections from A n to A and the constants; then P n ( A ) is called the algebra of n -place polynomial functions on A. A map ϕ from A n to A is said to be a k -local polynomial map if, for any k elements a 1 , …, a k e A n , there exists a polynomial function p such that p ( a i ) = ϕ( a i ), i = 1, …, k . The set of k -local polynomial maps from A n to A is denoted by L k P n ( A ). The behaviour of the chain L 1 P n ( A ) ⊇ L 2 P n ( A )⊇ … ⊇ L k P n ( A )⊇ … has been investigated by various authors for a number of varieties, e.g. for any abelian group A and any n , L 4 P n ( A ) = L k P n ( A ), for all k ⩾ > 4, and L 3 P 1 ( A ) = L k P 1 ( A ) for all k ⩾ 3 (Hule and Nobauer [1977]). It will be shown that, for any semilattice S, L n +2 P n ( S ) = L n P n ( S ), for k ⩾ n + 2.

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