## Non-modular and non-distributive primitive ordered subsets of lattices.(English)Zbl 0796.06009

Let $$L$$ be a lattice and $$P$$ a finite non-void subset of $$L$$. Then $$P$$ is called a primitive subset of $$L$$ if $$\bigwedge(\theta(x,x\lor y)$$; $$x,y\in P$$, $$x\neq x\lor y)\neq \omega$$, where $$\omega$$ is the trivial congruence relation on $$L$$ and $$\theta(x,y)$$ denotes the principal congruence relation on $$L$$ generated by the pair $$(x,y)$$.
Lattices not containing an isomorphic copy of a member from a given set of finite ordered sets as a primitive subset form a variety of lattices. The author characterizes the variety of distributive lattices by some collections of primitive ordered sets. Finally, two small non- distributive varieties of lattices are described similarly.
Reviewer: B.F.Šmarda (Brno)

### MSC:

 06B20 Varieties of lattices 06A99 Ordered sets

### References:

 [1] Chajda I., Rachůnek J.: Forbidden configurations for distributive and modular ordered sets. Order 5 (1989), 407-429. · Zbl 0674.06003 [2] Grätzer G.: General Lattice Theory. Akademie-Verlag, Berlin, 1978. · Zbl 0436.06001 [3] Jónsson B., Rival I.: Lattice varieties covering the smallest non-modular variety. Pacific J. Math. 82 (1979), 463-478. · Zbl 0424.06004 [4] Larmerová J., Rachůnek J.: Translations of modular and distributive ordered sets. Acta UPO, Fac. Rer. Nat., Math. 91(1988), 13-23. · Zbl 0693.06003 [5] McKenzie R.: Equational bases and nonmodular lattice varieties. Trans. Amer. Math. Soc. 174 (1972), 1-43. · Zbl 0265.08006 [6] Wille R.: Primitive subsets of lattices. Alg. Univ. 2 (1972), 95-98. · Zbl 0269.06001
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.