On the Mutual Connectedness of Elements in Lattices
Takayuki Nôno · Hiroshima Mathematical Journal · 1953
In this paper we shall first in § 1 extend the notion of the connected sets in topological spaces to the case for the lattices with a binary relation, next in § § 2 and 3 prove the fundamental theorems concerning this notion, and finally in § 4 discuss the relations between this binary relation and a mapping in a complete lattice. Definitions and axiomsLet L be a lattice.Given in L a binary relation ry, we shall write x ry y to express the fact that the elements x and y of L are in the ry-relation in this order.We shall not assume that x ry y implies y r1 x.And x p y means that x ry y and y ry x simultaneously.Next we shall define p-irreducible elements of L as follows.Definition.An element a of L is said to be p-irreducible, if a is not expressible as• a=X'--"Y, x p y and x,y =l= a.This notion is an extension of that of connected sets in topological spaces.As the axioms concerning this ry-relation, we shall consider the following conditions :n> (I) If z < X'--"Y and x py, then (xvy) ,-,.z=(X"""Z)'--"(Y"""Z).(II) If x ry y, Xi < x and Yi < Y, then Xi ry'yi.(IIJ1) If x ry Yi and x ry Y2, then x ry (Yi'--" Yz).(III2) If Xi'YY and X2'YY, then (X1'--"X2)ryy.(IV) If x p x, then x=O.(O denotes the zero element of L).(IV*) If x < y and x p y, then x=O.Remark.It is obvious that (IV) implies (IV*).Under the axiom (II), (IV*) implies (IV).For, by (I[) we have (X"""Y) p (x,-,.y); and since x < y, i.e., 1) This concept corresponds to that of "Verkettung" which has been proposed by F. Riesz as a primitive concept of abstract space.