Forbidden configurations for distributive and modular ordered sets. (English) Zbl 0674.06003

Chajda, Ivan; Rachůnek, Jiří

Order 5, No. 4, 407-423 (1989).

Let A be an ordered set, \(a,b\in A\). Denote by L(a,b) (U(a,b)) the set of all lower (upper, respectively) bounds of a, b. A is called distributive whenever \(L(U(a,b),c)=L(U(L(a,c),L(b,c))\) for all a,b,c\(\in A\). A is called modular if \(a\leq c\) implies \(L(U(a,b),c)=L(U(a,L(b,c)))\) for all a,b,c\(\in A\). It is clear that every distributive ordered set is modular and, moreover, if A is a lattice, then it is modular (distributive) iff it is modular (distributive, respectively) as an ordered set. The authors study the natural question whether modular ordered sets and distributive ordered sets can be characterized by some “forbidden configurations” similarly as in the case of lattices. The paper presents such configurations in the form of so-called strong subsets and LU subsets.

Reviewer: J.Duda

MSC:

06A06

Partial orders, general

Keywords:

distributive ordered set; modular ordered sets; forbidden configurations; strong subsets; LU subsets

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