Projective layer-lattices of small geometric dimension (Q1599446)

According to \textit{B. Jónsson} and \textit{G. S. Monk} [Pac. J. Math. 30, 95-139 (1969; Zbl 0186.02204)] a finite modular lattice \({\mathcal L}\) is called primary if (1) each element of \({\mathcal L}\) is a join of cycles and a meet of cocycles and (2) each interval of \({\mathcal L}\) that is not a cycle contains at least three atoms. An arbitrary element of a primary lattice \({\mathcal L}\) can be represented as a join of independent cycles. If \( L = A_1 + A_2 + \dots + A_n \) is such a representation of the greatest element then \(n\) is called the dimension of the lattice \({\mathcal L}\). If in this case \(d\) denotes the greatest number having the property \( l(A_1) = l(A_2) = \cdots = l(A_d)\), then \(d\) is called the geometric dimension of the primary lattice \({\mathcal L}\). If \(X\) is an element of \({\mathcal L}\) and \(X^{\ast}\) denotes the join of all elements covering \(X\), then the interval \( [X,X^{\ast}]\) is a complemented lattice. Such an interval will be called a layer of the lattice \({\mathcal L}\). A primary lattice is called layer-projective if each of its layers is an arguesian projective geometry over a fixed Galois field \(\text{GF}(p^k)\). A projective geometry of dimension at least 3 is arguesian and therefore any primary lattice of dimension at least 4 is layer-projective. This paper is a continuation of the investigation by \textit{V. A. Antonov} and the present author on layer-projective lattices of the same type [Mat. Zametki 63, 150-160 (1998); translation in Math. Notes 63, 170-182 (1998; Zbl 0916.06009)]. The following results are known: (1) If \( d \geq 3 \) and \({\mathcal L}_1 \) and \({\mathcal L}_2 \) are arguesian, then \({\mathcal L}_1 \cong {\mathcal L}_2 \). (This follows from the basic result of Jónsson-Monk.) (2) If \( d \geq 4 \), then the lattices \({\mathcal L}_1\) and \( {\mathcal L} \) are arguesian [\textit{G. S. Monk}, Pac. J. Math. 30, 175-186 (1969; Zbl 0186.02301)] and consequently \({\mathcal L}_1 \cong {\mathcal L}_2 \). In order to solve the isomorphism problem for layer-projective lattices of the same type it is therefore sufficient to restrict oneself to lattices having a small geometric dimension. For such lattices the following is known: (3) If \( n = 2 \), then \({\mathcal L}_1 \cong {\mathcal L}_2 \) (proved by \textit{S. A. Anishchenko} [``On the representation of some modular lattices by subgroup lattices'' (Russian), Mat. Zap. Krasnojarsk. Gos. Ped. In-ta 1965, No. 1, 1-21 (1965)] in the case \( k = 1 \), but the proof is also true for \( k > 1 \)); (4) If \( m_2 = m_3 = \cdots = m_n \) then \({\mathcal L}_1 \cong {\mathcal L}_2 \) (see Antonov-Nazyrova [loc. cit.]). In general the isomorphism problem of layer-projective lattices of the same type has no positive solution: Monk [loc. cit.] constructed an example showing the existence of nonisomorphic layer-projective lattices of type \((2,2,1,p^k)\). The present paper aims at a clarification of isomorphism conditions for lattices of type \( (2,2,1, \ldots, 1, p^k)\). The following condition \((\ast)\) plays a crucial role: \((\ast)\) There exists an isomorphism \[ \varphi: [0, A_1^2 + A_2^2 + \cdots + A_n ] \rightarrow [A_1^2 + A_2^2, L] \] such that \( \varphi (X) = A_1^2 + A_2^2 + X \) for all atoms \( X \leq A_3 + \cdots + A_n \) and such that any atom \( X \leq A_1^2 + A_2^2 \) satisfies the equation \(\varphi (X) = Y + (A_1^2 + A_2^2)\) where \( Y \) is a cycle of height 2 below \( A_1 + A_2 \) and covering \(X\). The following results are proved: Theorem 1.: Let \( {\mathcal L}_1 \) and \( {\mathcal L}_2 \) be layer-projective lattices having the same type \(( 2,2,2, \ldots, 1, p^k)\). If the dimension of \( {\mathcal L}_i \) is greater than three and if the lattices \( {\mathcal L}_1 \) and \( {\mathcal L}_2 \) satisfy condition \((\ast)\) , then \({\mathcal L}_1 \cong {\mathcal L}_2 \). A layer-projective lattice \( {\mathcal L}\) is a \(C\)-lattice in the sense of Antonov and the author [loc. cit.] if and only if it has an involutary anti-isomorphism \( \varphi \) such that \( A \leq \varphi (A) \) holds for an arbitrary cycle \( A \in {\mathcal L} \). Theorem 2.: A layer-projective \(C\)-lattice of type \((2,2,1,1, \ldots, 1,1, p^k)\) satisfies condition \((\ast)\). This implies the following result proved with other means by Antonov-Nazyrova [loc. cit.]: Corollary: All layer-projective \(C\)-lattices of the same type \((2,2,1,1, \ldots, 1,1, p^k)\) are isomorphic.