Some properties of row-adjusted meet and join matrices

摘要

Let (P, ≤) be a lattice, S a finite subset of P and f 1, f 2,..., f n complex-valued functions on P. We define row-adjusted meet and join matrices on S by (S) _(f1,...,fn)= (f _i(x _i ∧ x _j)) and [S] __(f1,...,fn)= (f _i(x _i ∧ x _j)). In this article, we determine the structure of the matrix (S) _(f1,...,fn) in general case and in the case when the set S is meet closed we give bounds for rank(S) _(f1,...,fn) and present expressions for det(S) _(f1,...,fn) and (S) ~(-1) _(f1,...,fn). The same is carried out dually for row-adjusted join matrix of a join-closed set S.