Lattice Matrices

This paper is a revised version of part of a previous report (Give'on, 1962) which was prepared for the U. S. Office of Naval Rsearch, Information Systems Branch, under contract No. 62558-2214, at the Applied Logic Branch of the Hebrew University in Jerusalem, Israel (February 1962). This revision was done for the U. S. Office of Naval Research, Information Systems Branch, under contract No. Nonr-1224(21), NR 049-i14. We discuss in this paper the particular properties of the algebra of square matrices over an arbitrary distributive lattice with 0 and 1 (2~-matrices). Due to these properties, 2n-matrices in various special cases become useful tools in various domains like the theory of switching nets, automata theory, and the theory of finite graphs. In addition to this, we develop the theory of invertible 2n-matrices, thus generalizing R. D. Luce's (1952) discussion on invertible Boolean matrices.