On eigenvalues of meet and join matrices associated with incidence functions

Let (P,⪯,∧)(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 PP. Then the n×nn×n matrix (S)f(S)f, where((S)f)ij=f(xi∧xj),((S)f)ij=f(xi∧xj),is called the meet matrix on SS with respect to ff. The join matrix on SS with respect to ff 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 ff on any set SS. We also estimate eigenvalues of meet matrices respect to any ff on meet closed set SS and with respect to semimultiplicative ff on join closed set SS. The same is carried out dually for join matrices.