Layer-projective lattices. I (Q1272272)

A finite modular lattice \(L\) is primary, if its join respectively meet-irreducibles form a tree under order respectively inverse order and if no element has exactly 2 upper covers. The numbers \(k_i\) of join-irreducibles (=cycles) of height \(i\) in a direct sum decomposition of the top element are uniquely determined. All maximal complemented subintervals are irreducible projective geometries of the same order \(q\) and dimension \(\sum k_i-1\). Thus, each such lattice has a unique `type' \((k_1,\dots, k_n;q)\) with \(k_n\neq 0\). Examples are the submodule lattices of finite modules over completely primary uniserial rings -- and for each type there is at least one such lattice. According to Baer, Inaba, Jónsson, and Monk [\textit{B. Jónsson} and \textit{G. S. Monk}, Pac. J. Math. 30, 95-139 (1969; Zbl 0186.02204)] these are the only examples with \(k_n\geq 4\) resp. \(k_n\geq 3\) and \(L\) Arguesian. The authors address the problem under which circumstances equality of type suffices to yield isomorphism. Of course, in the cases just mentioned it suffices to require, in addition, that the associated coordinate rings are isomorphic -- and if \(n>1\) one has to do so in view of the rings \(Z/(p^2)\) resp. \((Z/(p))[x]/(x^2)\), \(Z\) the ring of integers. Also, in the case \(\sum k_i= 3\) one has to assume that all complemented intervals are Arguesian. \textit{H. Ribeiro} [Comment. Math. Helv. 23, 1-17 (1949; Zbl 0037.15702)] has claimed that type suffices in case \(q\) is a prime, but his proofs amount to the case \(\sum k_i\leq 2\). The authors require the existence of an involutory dual automorphism and call such lattices `layer-projective'. All primary submodule lattices over commutative rings provide examples (via symmetric forms). But, non-Arguesian examples always exist in case \(\sum k_i= 3\), \(n>1\) [cf. \textit{A. Day}, \textit{C. Herrmann}, \textit{B. Jónsson}, \textit{J. B. Nation} and \textit{D. Pickering}, Algebra Univers. 34, 66-94 (1994; Zbl 0808.06007)]. On the other hand, according to \textit{J. B. Nation} and \textit{D. A. Pickering} [Algebra Univers. 24, 91-100 (1987; Zbl 0633.06003)], types of the form \(k_n= 1\), \(k_i= 0\) for \(1< i< n\), and \(k_1\geq 3\) (the latter can be omitted under the Arguesian law) suffice for all primary lattices to grant isomorphism -- which includes Theorem 4 of the paper read under this proviso. A layer-projective lattice is called a `regular C-lattice' if every cycle is isotropic (this concept derives from the first author's earlier study of lattices of centralizers). In such lattices each \(k_i\) has to be even -- commutative rings with skew-symmetric forms provide examples. The main result of the paper is that regular C-lattices of the same type \((k_1,2; q)\) are isomorphic.