Strong compact elements in multiplicative lattices
C. Jayaram, E. W. Johnson · Czechoslovak Mathematical Journal · 1997
Throughout we assume that L is a C-lattice.It is well known that a Noether lattice is a principal element lattice if and only if every maximal element is weak meet principal (see Theorem 5 of [6]).Also it is known that if L is principally generated, then L is a principal element lattice if and only if L is an M-lattice satisfying the ascending chain condition (see Theorem 6 of [4]).In this paper, we introduce strong compact elements, P-weak meet principal elements and P-principal elements and using them, principal element lattices and almost principal elements lattices are characterized. ooFor any a G L, we define a w by a w = f\ a n .The reader is referred to [1] and [3] n=l for general background and terminology.We begin with the following definitions.Definition 1.An element a G L is said to be a strong compact element if both a and a w are compact elements.