Automorphism Groups Of Orthomodular Lattices

In this paper, the characterization of automorphism groups of orthomodular lattices have been studied.The following theorems and propositions are the main results.About the automorphism groups of products of orthomodular lattices and the products of the automorphism groups of orthomodular lattices, we have:Theorem 3.15 If < L_i : i ∈I > is a system of orthomodular lattices,the index set I is finite or countable,thenIf I is uncountable,this theorem is not true even though < L_i : i∈I > is a system of Boolean algebras, so that J.Donald Monk added the so-called condition "pairwise totally different" to Boolean algebras and proved this theorem was true. In this paper,under the condition "index set I is finite or countable",we not only drop the condition "pairwise totally different" away but also generalize the theorem from Boolean algebras to orthomodular lattices.We present the automorphism groups of subdirectly irreducible modular ortholattices and obtain the theorem as follows.Theorem 4.2 Let MO_k,k ≥ 2 be subdirectly irreducible modular ortholattices, then the automorphism groups Aut(MO_k) can be generated by {r₁,r₂,…,r_k}. where MO_k = {0,a₁,a₂,…,a_k,a'₁,a'₂… ,a'_k,,1},r₁ = (a₁a'₁),r₂ = (a₁a₂)(a'₁a'₂),r₃ = (a₁a₃)(a'₁a'₃), … ,r_k = (a₁a_k)(a'₁a'_k).Proposition 4.5 Φ_σ is a coset of Φ_Id relative to Aut(MO_k).Theorem 4.7 Aut(MO_k) = (?)Φσ.By Theorem 3.15 and Theorem 4.2,we can get the following theorem,thus we solute the problem of the structures of AutF_MOk(n), which can be stated as follows.Theorem 5.1 p=\Theorem 5.2 AutF_Mok{n can be generated by {r₁¹, r₁², r₂², ? ? ?, r₁^p, r₂^p, ? ? ?, r_p^p,…, r₁^k, r₂^k, ? ? ?, r_k^k}. where {r₁^p, r₂^p, …, r_p^p} are p genenrators of Aut(MO_p) in Theorem 4.2.We imply Theorem 3.1 to block-finite orthomodular lattices,and get the important theorem as follows.Theorem 6.3 Let L be an orthomodular lattices with finitely many blocks,thenwhere B₀ is a Boolean algebra and L₁, L₂, ? ? ?, L_n are irreducible orthomodular lattices with at least two blocks each.For the sevral important orthomodular lattices,we have the results as follow.