Free lattices over halflattices. (English) Zbl 0696.06004

Ježek, Jaroslav; Slavík, Václav

Commentat. Math. Univ. Carol. 30, No. 2, 203-211 (1989).

Let P be a partial lattice in which \(x\wedge y\) is defined for all pairs \(x,y\in P\). Then P is called a halflattice, if \(x\vee y\) is defined whenever \(x,y\in P\) have a common upper bound. F(P) denotes a free lattice generated by P and preserving all partial operations \(\wedge\) and \(\vee\) defined in P. The authors prove the following result: Let P be a finite halflattice. If F(P) is finite, then the set of the elements \(x\vee y\in F(P)-P\) with \(x,y\in P\) form a chain of at most four elements. [See also R. Wille, Contrib. Universal Algebra, Szeged 1975, Colloq. Math. Soc. János Bolyai 17, 581-593 (1977; Zbl 0366.06007).]

Reviewer: T.Katriňák

MSC:

06B25

Free lattices, projective lattices, word problems

Keywords:

partial lattice; halflattice

Citations:

Zbl 0366.06007