Series-Parallel Posets and Polymorphisms

Abstract

We examine various aspects of the poset retraction problem for series-parallel posets. In particular we show that the poset retraction problem for series-parallel posets that are already solvable in polynomial time are actually also solvable in nondeterministic logarithmic space (assuming P 6= NP). We do this by showing that these series-parallel posets when expanded by constants have bounded path duality. We also give a recipe for constructing members of this special class of series-parallel poset analogous to the construction of all series-parallel posets. Piecing together results from [5],[15],[14] and [12] one can deduce that if a relational structure expanded by constants has bounded path duality then it admits SD-join operations. We directly prove the existence of SD-join operations on members of this class by providing an algorithm which constructs them. Moreover, we obtain a polynomial upper bound to the length of the sequence of these operations. This also proves that for this class of series-parallel posets, having bounded path duality when expanded by constants is equivalent to admitting SD-join operations. This equivalence is not yet known to be true for general relational structures; only the forward direction is proven. However the reverse direction is known to be true for structures that admit NU operations. Zádori has classified in [26] the class of series-parallel posets admitting an NU operation and has shown that every such poset actually admits a 5-ary NU operation. We give a recipe for constructing series-parallel posets of this class analogous to the one mentioned before. Then we show an alternative proof for Zádori's result.