In this paper we extend these formulae by providing calculation formulae for and in terms of meet-closed subsets and supersets of S. We compare the effectivity of these methods based on reduction and extension of the set S to the effectivity of row-reduction.
As applications we first combine our reduction and extension methods for calculating and by exchanging the "difficult" elements of S for "easy" ones. Second, we obtain known formulae for and new formulae for where S is an a-set (i.e., for all Next we give new formulae for the determinant and the inverse of the meet matrix on two sets X and Y, where The methods of this paper are also appropriate for join matrices on join-semilattices. As special cases these results hold also for GCD and LCM matrices.
