On the relations “semi-between” and “parallel” in lattices
Yatarō Matsushima · Proceedings of the Japan Academy Series A Mathematical Sciences · 1958
In a recent paper 2, we hve studied the concept of B-covers in lattices as a generalization of the metric-betweeness in a normed lattice which is investigated by L.M. Kelley 1, and discussed some geometrical properties of lattices by means of B-covers and B*-covers in [3, 4_.At first we shall introduce the concept of J-cover and CJ-cover which will be considered as semi-between sets in lattices since the B-covers are treated as between sets in lattices.For any two elements a and b of a lattice L, we shall define as follows:) is called the J-cover of a and b, and if x e J(a,b), then we shall write J(axb).Similarly we shall define CJ-cover and CJ(axb).Further, we define J*(a,b)-[ J(abx)}, CJ*(a,b)-{x CJ(abx)}, J(a,J*(a,b))- {y J(ayx) for all x e J*(a,b)}, etc. B(a,b)--J(a,b)CJ(a,b) is called the B-cover of a and b; and we write axb when x eB(a,b) (cf.[2-4).Next we shall define the notion of "parallel" as follows: ab//cd means that B*(a,b)B*(c,d)-O, where B*(a,b)--{xlabx}.In 1 we shall give characterizations of modular or distributive lattices by means of "semi-between", and in 2 we shall consider the geometrical properties of lattice polygons by the notion of "parallel".1. "Semi-between".Lemma 1. (a(b_ -J(a,b)(ab, a)b) --CJ(a,b)ab), where (-{zlzx}, x)-{zlzx}, AB--{xy xeA, yeB} if A,BL.The proof is found in 2.Lemma 2. J(axb) implies x(ab)-(ax)(bx)-x; CJ(axb) implies x(ab)-(a)(bx)-x.Lemma 3. J(abc), CJ(axb) and CJ(byc) imply J(xby).CJ(abc), J(axb) and J(byc) imply CJ(xby).Proof.We have bbab, bbybc by CJ(axb), CJ(byc), and hence b (bx)(by) (ab)(bc)--b by J(abc); thus we have (bx)(by)-b, that is, J(xby).Lemma 4. J(axb) and J(ayb) imply J(a(xy)b).CJ(axb) and CJ(ayb) imply CJ(a(xy)b).J(axb) and J(ayb) do not necessarily imply J(a(xy)b).