On eigenvalues of meet and join matrices associated with incidence functions

AbstractLet (P,⪯,∧) be a locally finite meet semilattice. LetS={x1,x2,…,xn},xi⪯xj⇒i⩽j,be a finite subset of P and let f be a complex-valued function on P. Then the n×n matrix (S)f, where((S)f)ij=f(xi∧xj),is called the meet matrix on S with respect to f. The join matrix on S with respect to f is defined dually on a locally finite join semilattice.In this paper, we give lower bounds for the smallest eigenvalues of certain positive definite meet matrices with respect to f on any set S. We also estimate eigenvalues of meet matrices respect to any f on meet closed set S and with respect to semimultiplicative f on join closed set S. The same is carried out dually for join matrices