# zbMATH — the first resource for mathematics

Forbidden configurations for distributive and modular ordered sets. (English) Zbl 0674.06003
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
Full Text:
##### References:
 [1] G. Grätzer (1978) General Lattice Theory, Birkhäuser, Basle, Stuttgart. [2] J. Larmerová, and J. Rachunek (1988) Translations of distributive and modular ordered sets, Acta Univ. Palack. Olomuc. Fac. Rerum. Natur. Math. (to appear). [3] K. Leutola, and J. Nieminen (1983) Posets and generalized lattices, Algebra Universalis 16, 344–354. · Zbl 0514.06003 · doi:10.1007/BF01191789 [4] J. Nieminen (1983) On distributive and modular $$\chi$$-lattices, Yokohama Math. J. 31, 13–20. · Zbl 0532.06002 [5] J. Rachunek (1981) Translations des ensembles ordonnés, Math. Slovaca 31, 337–340. · Zbl 0472.06002 [6] J. Rachunek (1985) Modal operators on ordered sets, Acta Univ. Palack. Olomuc. Fac. Rerum. Natur. Math. 82, 9–14.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.