On the structure of a generalization of weakly associative lattices

The concept of weakly associative lattices (i.e. relational systems with a reflexive and antisymmetric relation ≤, in which for each pair of elements there exist a least upper and a greatest lower bound) was introduced in [3] and [5]. In [4] WU-systems are defined, i.e. weakly associative lattices with the unique bound property, and their equivalence with projective planes is described. In this paper we introduce WUλ -systems, and discuss their relation to symmetric 2−(v, k, λ) designs equipped with a special “loop-free” mapping. The concept of weakly associative lattices (i.e. relational systems with a reflexive and antisymmetric relation ≤, in which for each pair of elements there exist a least upper and a greatest lower bound) was introduced in [3] and [5]. Ervin Fried and Vera T. Sós defined WU-systems in [4]; i.e. weakly associative lattices with the unique bound property; their equivalence with projective planes is described in [4]. In this paper we introduce WUλ systems (here ‘U’ stands for ‘uniform’), and discuss their relation to symmetric 2−(v, k, λ) designs equipped with a special “loop-free” mapping. Definition. Let V = 〈V,≤〉 be a system with a reflexive and antisymmetric relation ≤. Let U(v) = {w ∈ V : v ≤ w} and L(v) = {w ∈ V : w ≤ v} for each v ∈ V . Given a positive integer λ, we call V a WUλ -system if for each v1 6= v2 ∈ V both of the sets U(v1) ∩ U(v2) and L(v1) ∩ L(v2) has size λ. Note that for λ = 1 we get WU-systems of [4]. Remark. The Paley tournaments for (prime power) q = 4k − 1 give a series of examples for WUλ -systems with λ = (q + 1)/4. (V = GF (q), and for a, b ∈ GF (q) define a ≤ b iff (b− a) is a square element of GF (q).) Definition. A 2 − (v, k, λ) design (sometimes they are denoted by Sλ(2, k, v)) is an incidence structure D = (P,B), where the elements of P are called points, the elements B ∈ B are subsets of P called blocks, and |P| = v, |B| = k for all B ∈ B, and for any pair of distinct points there are exactly λ blocks containing both of them. A design is called symmetric, if |P| = |B|; in this case v = k(k−1) λ + 1 holds ([2]). ∗Research was partially supported by OTKA and Eötvös grants.