Upper and lower complementation in a modular lattice
S. P. Avann · Proceedings of the American Mathematical Society · 1960
In this paper we define upper and lower complements of aEL, always a finite modular lattice, such that these become an ordinary complement of a when L is complemented (Theorem 8). Uniqueness of upper or of lower complements for all aEL implies L is distributive (Theorem 5). Our principal result is a set of 8 equivalent conditions for uniqueness of upper or lower complement of a particular element aEL (Theorem 4). We shall employ D, D, > for proper inclusion, inclusion, and covering respectively, and C, C, < for their duals. Unit and zero of L will be u and z respectively. Otherwise, notation and terminology of Birkhoff's Lattice theory [2 ] will be adhered to. We shall denote by A the set A = {xEPIxCa} where P is the set of join-irreducibles of L partially ordered by the ordering relation of L. Elementary properties enjoyed by these sets A are: (1) a = UXEA x; (2) A =B if and only if a=b; (3) ADB if and only if aDb; (4) c=a'Ub and d=anb imply CDA+B and D=A.B where (+) and (.) are point-set sum and product. We denote by a* the join of all elements covering a and by a* the meet of all elements covered by a. We define u * = u and z* = z. The quotient c/a is an upper transpose of b/d and b/d is a lower transpose of c/a if and only if c = a'Ub and d = aCb. In Lemma 1, Theorems 5 and 8, and Corollary to Theorem 9 use of parenthesized words yields the dual theorem.