ON DIRECT DECOMPOSITIONS OF CERTAIN ORTHOMODULAR LATTICES

Abstract. Let L be an orthomodular lattice. For a, b ∈ L define a↔cb if either a and b both belong to the centre C(L) of L or if {a, b} ∩C(L) = ∅ and a↔b (i.e. a is compatible with b). Let R be the transitive closure of the relation ↔c. Then there exist at least three equivalence classes of the relation R in L if and only if either L is a horizontal sum (if C(L) = {0, 1}) or L is a direct product of a Boolean algebra and a horizontal sum. In the theory of orhomodular lattices (abbreviated: OML) a classical theorem states that every finitely generated OML decomposes into a direct product of a Boolean algebra and an OML without nontrivial Boolean factor [7, 1]. This classical decomposition theorem has obtained several generalizations [4, 6, 8]. In the present paper, we will characterize certain type of OML’s admitting the above direct decomposition by means of so-called c-compatibility. Basic definitions and facts about orthomodular lattices can be found in [1] and [2]. We recall that an orthomodular lattice L(0, 1, ′,∨,∧) is a horizontal sum of orthomodular lattices Li, i ∈ I, if L = ∪i∈ILi, Li ∩ Lj = {0, 1} for i 6 = j, i, j ∈