Structure and Decomposition Theory of Lattices
R. P. Dilworth · Birkhäuser Boston eBooks · 1990
One of the most natural problems which arise in the investigation of an abstract algebraic system is that of representing the elements of the system in terms of a canonical subset by means of the operations of the system. Thus for a polynomial domain over a field with the operation that of ordinary polynomial multiplication it is the problem of representing polynomials as products of irreducible polynomials. For lattices there are two operations with respect to which we may consider the representation of the elements of the lattice. Since the operations are dual it suffices to consider representations with respect to one of the operations. Thus we shall treat only meet representations. Now an element which cannot be expressed as the meet of elements distinct from itself clearly has only trivial representations. Furthermore these elements must surely be included in any reasonable canonical set. Thus we shall be particularly concerned with meet representations in terms of meet irreducible elements.