Neutrality and betweenness in modular lattices

✍ Scribed by Juhani Nieminen

• Publisher: Elsevier Science
• Year: 1985
• Tongue: English
• Weight: 213 KB
• Volume: 55
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/s0012-365x(85)80012-1

✦ Synopsis

Two characterizations for neutral elements in modular lattices are given.

The purpose of this note is to characterize the neutrality in modular lattices by means of a betweenness relation. There are several betweenness relations in lattices and the following three can be found, e.g., in Padmanabhan's paper [6], where (a, z, b) always means that z is between a and b, and in addition B(a,z,b) means (ar\z)v(bAz)=z=(avz)A(bvz), C(a,z,b) means (aAz)v(bAz)=z=(aAb)vz, C*(a,z,b) means (avz)r(bvz)=z=(avb)Az. Padmanabhan proved that B(a, z, b) implies C(a, z, b) as well as C*(Q, z, b). Moreover the relations B, C and C* are equivalent in a lattice L if and only if L is modular [6, Theorem 41. Note that B(a, z, b) is equivalent to B(b, z, a); the same holds for the relations C and C*, too. An element n in a lattice L is neutral if and only if the equation (x A n)v o'An)V(XAY)=(