Some results involving multilattice ideals and distributivity

✍ Scribed by Ivor J. Johnston

• Book ID: 103056118
• Publisher: Elsevier Science
• Year: 1990
• Tongue: English
• Weight: 505 KB
• Volume: 83
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(90)90218-7

✦ Synopsis

Several definitions

of 'ideal' for multilattices are considered, as are several definitions of 'distributivity':

the result "If a lattice (L, =G) is distributive then (9, E), the lattice of its ideals, is implicative" is extended to multilattices; it is also shown that one of the distributivity conditions is satisfied by all directed multilattice groups. A relationship is established between two of the sets of ideals for directed multilattices of width 2.

Terminology

Let (P, c) be a poset. If X E P, then by L(X) we shall mean {y E P ( y < ~V~EX}, andbyM(X), {y~Ply>xVx~X}. BY L(x) (or (4) we shall mean L({x}), and by M(x) (or Lx)), M({x]). Similarly M(a, b) and L(a, b) mean M({a, b}) and L({a, b}).