Lattices with involution
John Arnold Kalman · Transactions of the American Mathematical Society · 1958
Introduction.By a "lattice with involution," or "i-lattice," we shall mean a lattice A together with an involution [l, p. 4] x->x' in A. A distributive *'lattice in which xC\x' Sy\Jy' for all x and y will be called a "normal" i-lattice.The underlying lattice of an /-group becomes a normal i-lattice when x' is defined as the group inverse of x; also a Boolean algebra becomes a normal t-lattice when x' is defined as the complement of x.In this paper /-groups and Boolean algebras will always be understood to have the involutions defined above.§1 of the paper contains subdirect decomposition theorems for distributive and normal t-lattices, with applications; in §2, as a contribution to the study of nondistributive i-lattices, modular and nonmodular ^-lattices are classified with respect to certain laws each of which, for distributive ilattices, is equivalent to normality; and § §3 and 4 contain some extension and embedding theorems concerning normal z'-lattices.