On meet and join matrices on A-sets and related sets
Ismo Korkee · 2004
Let (P;^) be a meet-semilattice and let S = fx1;x2;:::;xng be a subset of P . We say that S is an A-set if A =fxi^xjj xi6= xjg is a chain. For example, chains and a-sets (with A =fag) are known trivial A-sets. The meet matrix (S)f on S with respect to a function f : P ! C is deflned as ((S)f )ij = f(xi^xj). We present a recursive structure theorem for meet matrices on A-sets and thus obtain a recursive formula for det(S)f and for (S) i1 on A-sets. The recursive formulae also yield explicit formulae, e.g. the known determinant and inverse formulae on chains anda-sets. We also present the dual forms of our results, i.e. the determinant formulae and the inverse formulae for join matrices on join-semilattices. Finally, we suggest how our results can be generalized to more complicated cases. As special cases these results hold also for GCD and LCM matrices and for their unitary analogies GCUD and LCUM matrices.