Lattices with involution

Abstract

Introduction. By a with involution, or i-lattice, we shall mean a lattice L together with an involution [1, p. 4] xx' in L. A distributive ilattice in which xnx' <yY)y' for all and y will be called a i-lattice. The underlying lattice of an i-group becomes a normal i-lattice when x' is defined as the group inverse of x; also a Boolean algebra becomes a normal i-lattice when x' is defined as the complement of x. In this paper i-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 i-lattices, with applications; in ?2, as a contribution to the study of nondistributive i-lattices, modular and nonmodular i-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 i-lattices. 1. Subdirect decomposition of distributive and normal i-lattices. Every i-lattice is an algebra [1, p. vii] with operations C, ' which satisfy the identities xny=ynx, x(T(ynz) = (xny)Cnz, x' =x'T(xny)', and x =x; it may be proved that these identities are independent postulates for i-lattices. We shall apply the usual terminology of abstract algebra (cf. [1, pp. viif.]) to i-lattices, except that we shall use the terms i-sublattice, i-homomorphism, and i-isomorphism instead of subalgebra, homomorphism, and isomorphism. Let L be a distributive i-lattice. For elements x, y, p of L we shall set x=y(C(p)) if and only if xnp=ynp and x'np=y'np. It is easily verified that this defines a congruence relation C(p) on L, and that