Abstract
Let (P,⩽)=(P,∧,∨) be a lattice, let S={x1,x2,…,xn} be a meet-closed subset of P and let f:P→Z+ be a function. We characterize the matrix divisibility of the join matrix [S]f=[f(xi∨xj)] by the meet matrix (S)f=[f(xi∧xj)] in the ring Zn×n in terms of the usual divisibility in Z, and we present two algorithms for constructing certain classes of meet-closed sets S such that (S)f divides [S]f. As an example we present the lattice-theoretic structure of all meet-closed sets with at most five elements possessing the matrix divisibility property. Finally, we show that our methods solve some open problems in the divisor lattice, concerning the divisibility of GCD and LCM matrices.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
