Matrices over orthomodular lattices
Jean H. Bevis · Glasgow Mathematical Journal · 1969
In this paper elementary properties are established for matrices whose coordinates are elements of a lattice L. In particular, many of the results of Luce [4] are extended to the case where L is an orthomodular lattice, a lattice with an orthocomplementation denoted by in which a ≦ b ⇒ a ∨(a′ ∧ b) = b. Originally, these were called orthocomplemented weakly modular lattices, Foulis [2]. In Theorem 1 a characterization is given of the nucleus with respect to matrix multiplication, which is in general nonassociative. Matrices with A-1 = transpose (A) are characterized in Lemma 8. Theorems 3 and 4 respectively, give partial characterizations of zero divisors and inverses.