Abstract-For a 3-generated free modular lattice we obtain a set of 11 defining relations and prove that this set is minimal. DOI: 10.3103/S1066369X13100071Keywords and phrases: free modular lattices, defining relations.Recall that the rank of free algebra from some manifold is the cardinal number of the set of its free generators. We concentrate our attention on a free lattice of the rank 3 in the manifold of modular lattices; we denote it by A. Let F be a free lattice of the rank 3 in the manifold of all lattices; let f , g, and h be its free generators, and let ϕ be a homomorphism from F to A. By standard considerations of a universal algebra, elements a = ϕ(f ), b = ϕ(g), and c = ϕ(h) are free generators of the lattice A. Relations defining this lattice in the manifold of all lattices were considered in [1] and [2]. In the paper [1] one has particularly shown that A can be defined by 21 relations. In [2] one has proved that this set of defining relations is not minimal; namely, it was shown there that 15 relations among those mentioned above define the lattice A, moreover, they form the minimal set of defining relations for A. Note that in [1] one has described a set of seven defining relations for a free distributive lattice of the rank 3, and in [3] this set was proved to be minimal.The following assertion is the main result of this paper: There exists a set of 11 defining relations for the lattice A. Note that this set is not a subset of the set of defining relations indicated in [1]. Let us enumerate these relations:*

No abstract

We construct the system of 11 defining relations for the 3-generated free modular lattice. This system is proved to be minimal. Systems of defining relations for lattices close to modular one are studied.

No abstract

No abstract