CUBO A Mathematical Journal Vol.20, No¯ 01, (65–78). March 2018 http: // dx. doi. org/ 10. 4067/ S0719-06462018000100065 On rigid Hermitean lattices, II Ana Cecilia de la Maza Departamento de Matemática y Estad́ıstica, Universidad de la Frontera, Temuco, Chile. Remo Moresi Cerfim, cp 1132, 6601 Locarno, Switzerland, anace.delamaza@ufrontera.cl, romicatj@yahoo.it ABSTRACT We study the indexed Hermitean lattice of type 0 generated by a single element a subjected to the relation a ≤ b⊥ ∧ bb⊥ = 0. We prove that it is finite, provided that two crucial indices are finite. We show that index relations imply algebraic relations and describe the lattice by means of its subdirectly irreducible factors. We finally use the results to confirm a conjecture appeared in 2000. RESUMEN Estudiamos el reticulado Hermitiano finito indexado de tipo 0 generado por un solo ele- mento a sujeto a la relación a ≤ b⊥ ∧bb⊥ = 0. Probamos que es finito, suponiendo que dos ı́ndices cruciales son finitos. Mostramos que las relaciones de ı́ndices implican rela- ciones algebraicas y describimos el reticulado a travs de sus factores subdirectamente irreductibles. Finalmente, usamos nuestros resultados para confirmar una conjetura aparecida el ao 2000. Keywords and Phrases: Lattices, semilattices, modular lattices, Hermitean lattices, orthogonal geometry. 2010 AMS Mathematics Subject Classification: 03G10,06A12, 06C05, 06B25. http://dx.doi.org/10.4067/S0719-06462018000100065 Ignacio Castillo Ignacio Castillo Ignacio Castillo Ignacio Castillo 66 Ana Cecilia de la Maza and Remo Moresi CUBO 20, 1 (2018) 1 Introduction The importance of lattices in infinite-dimensional orthogonal geometry was brought to attention by the pioneering work of Herbert Gross (1936-1989): see in particular [G1] and [G2]. All examples treated in origin are sublattices of some L(E), the subspace-lattice of an ℵ0-dimensional vector space E over an appropriate division ring k, together with the orthogonal operation induced by a Hermitean form φ (i.e. X 7→ X⊥ := {y ∈ E | φ(y, x) = 0 ∀x ∈ X}) and were used to study geometric invariants, for instance dimension of quotient spaces or intersections with the subspace E∗ of trace-valued vectors in E. The fact that E∗ 6= E only if char(k)= 2 was also playing some role. After some time of concrete investigations with subspace lattices (see e.g. [M1]), the natural idea to insert all considerations into an abstract setting gave rise to the following definitions (cf. [KKW], Ch. IV): A Hermitean lattice (HL for short) is an algebra (L, 0, 1, · , + , ⊥, b) of type 〈0, 0, 2, 2, 1, 0〉 such that i) (L, 0, 1, · , +) is a modular lattice with universal bounds 0 , 1; ii) ⊥ : L → L is a unary operation with 1⊥ = 0 and x ≤ (x⊥y)⊥ ∀ x, y ∈ L (1.1) iii) b ∈ L is a nullary operation with xx⊥ ≤ b ∀ x ∈ L. In case b is explicitly not trivial (i.e. b 6= 1), the modular law in i) is sometimes replaced by the stronger Fano identity (w + v)(y + z) ≤ (w + y)(v + z) + (w + z)(v + y). If we drop the operation “+”, then we obtain the notion of Hermitean semilattice (HSL for short). In the present paper we will endow HL L with a so-called index function of type 0 (IF for short), i.e a function δ from the set of quotients of L into the set of cardinals ≤ ℵo, with the following properties: δ(x/y) ≥ δ(xz/yz), (1.2) δ(x/y) ≥ δ(x + z/y + z), (1.3) δ(x/y) ≥ δ(y⊥/x⊥), (1.4) δ(x/y) + δ(y/z) = δ(x/z), (1.5) δ(x/y) = 0 ⇐⇒ x = y. (1.6) CUBO 20, 1 (2018) On rigid Hermitean lattices, II 67 We will speak about indexed Hermitean lattices (IHL). By dropping (1.3), we obtain the notion of indexed Hermitean semilattices (IHSL). A major task of the theory of H(S)L consists in describing the free objects S[a] and F[a], generated by a single element a in the varieties of HSL and HL, respectively. Since such objects are infinite, a more realistic project consists in studying appropriate presentations under (index) relations suggested by geometrical choice (see [G1], [M1], [M2] and also the bibliography in [KKW] for many known examples). One of these options is given by the relation a ≤ b⊥, which was intro- duced in [DM3] and gave rise to the concept of rigid H (S)L. Here we continue such investigation and consider rigid HL with the (somewhat complementary) property bb⊥ = 0. In the above work the HSL S := S[a; a ≤ b⊥ ∧ bb⊥ = 0] was already computed, but here we briefly reproduce its description, without proofs, to make this paper more self-contained. Since the corresponding HL is most probably infinite, we work with an IF δ and start our research with the following hypothesis: δ(a⊥/d⊥1 ) < ℵ0 ∧ δ(b ⊥/c⊥1 ) < ℵ0, (1.7) where c1 := d ⊥e⊥, d1 := c ⊥e⊥, and c := a⊥e⊥, d := b⊥e⊥, e := a⊥b⊥. (1.8) The algebraic relations forced by the index condition (1.7) are given below in (4.4), Theorem 4.1, and have the following important consequence: F := F[a; (1) ∧ (4.4)] is finite and has 23 subdirectly irreducible factors. The factors are listed in Tables II, III and IV, Section 7, together with the associated critical quotients. We will finally use these results to confirm conjecture 2 in [M2] and to suggest an application in orthogonal geometry. We conclude this introduction with two more remarks: - Without (1.7), F would be most probably infinite (cf. also the arguments given in [M2]). Thus we can recognize the importance of the intervals [d⊥1 , a ⊥] and [c⊥1 , b ⊥] in the above HL. Moreover, it is easy to prove that (1.7) is a weakening of the condition δ(1/b) < ℵ0, which has a natural interpretation in orthogonal geometry (cf. Section 6) and was used as hypothesis in many precedent investigations. - S appeared naturally as substructure in other works (see [M2] and [DM2]). This important fact was an additional motivation for the present study. 2 Preliminaries Lemma 2.1. Any countable HL is indexable. 68 Ana Cecilia de la Maza and Remo Moresi CUBO 20, 1 (2018) Proof. Each HL admits the trivial IF, defined to have value ℵ0 on each nontrivial quotient. Lemma 2.2. The class of IHL is closed with respect to subalgebras, homomorphic images and countable products. Proof. This is just a slight generalization of Proposition 21 in [KKW], Ch. IV. Clearly, the existence of a nontrivial IF on some HL is controlled by prime quotients. Our lattices do not present difficulties such as described in [S] because the subdirectly irreducible factors are finite and known. The next result represents the key to obtain algebraic relations from index relations (cf. proof of Theorem 4.1): Lemma 2.3. Let u/v be any finite quotient of an IHL. If v = v⊥⊥ then u = u⊥⊥. Proof. δ(u/v) ≥ δ(v⊥/u⊥) ≥ δ(u⊥⊥/v⊥⊥) = δ(u⊥⊥/v) ≥ δ(u/v). For the sake of precision we give also the following Definition 2.4. S[a : a ≤ b⊥] is the initial object of the class of rigid HSL. Similarly, F[a : a ≤ b⊥] is the initial object of the class of rigid HL. Thus any rigid H(S)L is a homomorphic image of the initial object. We could have been even more precise by saying that this is in fact the concept of a 1-generated rigid H(S)L, a special case of n-generated rigid H(S)L, but of course, for the moment, all this is not necessary. We conclude this section by remarking that the axiom (1.1) is equivalent with the following conditions: (i) x ≤ x⊥⊥; (ii) x ≤ y ⇒ y⊥ ≤ x⊥. This may facilitate some computations. 3 Description of S Theorem 3.1. The HSL S has 18 elements and its structure is given by the diagram depicted in Figure 1 (see Section 7). Proof. See [DM3]. CUBO 20, 1 (2018) On rigid Hermitean lattices, II 69 Since we are interested in indices, we consider an IF δ on S and put β1 := δ(a/0), β2 := δ(b/0), β3 := δ(e/0), β4 := δ(c/c1), β5 := δ(a ⊥⊥/a), β6 := δ(b ⊥⊥/b), β7 := δ(c1/b ⊥⊥), β8 := δ(d1/a ⊥⊥). (3.1) Theorem 3.2. (Relations among indices in S) (i) All other indices of S are determined by β1, · · · , β8 as is shown in Figure 2, Section 7. (ii) In particular, the following relations hold: a) β4 6= 0 implies β1 = β2 = ℵ0; b) β5 6= 0 implies β1 = ℵ0; c) β6 6= 0 implies β2 = ℵ0; d) β7 6= 0 or β8 6= 0 implies β1 = β2 = β3 = ℵ0. Proof. See [DM3]. Remark 3.3. Using the above Theorem, we find 8 subdirectly irreducible factors of S. They are reproduced in Tables I and II, Section 7. 4 Description of F Remembering (1.8), let us consider the two descending chains {a1, a2, a3} := {a ⊥, d⊥1 , d ⊥} and {b1, b2, b3} := {b ⊥, c⊥1 , c ⊥}. For 1 ≤ i, j ≤ 3 we define aij := ai(bj + e ⊥ ), bij := bj(ai + e ⊥ ), eij := e ⊥ (ai + bj). (4.1) Let I1, I2 and I3 be the modular sublattices of F generated by {a⊥, d⊥1 , d ⊥, c, b⊥31, b ⊥ 21, b ⊥ 11, b, e}∪{aij}, {b ⊥, c⊥1 , c ⊥, d, a⊥13, a ⊥ 12, a ⊥ 11, a, e}∪{bij} and {e ⊥, c, c1, b, b ⊥ 31, b ⊥ 21, b ⊥ 11, d, d1, a ⊥ 13, a {eij}, respectively. By the main result in [DM1], I1, I2 and I3 coincide with the principal ideals of F0 :=< I1 ∪ I2 ∪ I3 > generated by a ⊥, b⊥ and e⊥ respectively. Moreover, they are distributive and additively generate F0. We want to show that F0 = F. To this end it will be useful to define the following indices: 70 Ana Cecilia de la Maza and Remo Moresi CUBO 20, 1 (2018) αi := βi for i = 1, 2, 3, 4, 5, 6 and further α7 := δ(e ⊥/e11), α8 := δ(c ⊥/b13), α9 := δ(d ⊥/a31), α10 := δ(1/a ⊥ + b⊥ + e⊥), α11 := δ(b ⊥ 11/b ⊥⊥ ), α12 := δ(a ⊥ 11/a ⊥⊥ ), α13 := δ(d ⊥ 1 /a21 + d ⊥ ), α14 := δ(c ⊥ 1 /b12 + c ⊥ ), α15 := δ(b33/d1 + e), α16 := δ(d1/a ⊥ 13), α17 := δ(c1/b ⊥ 31), α18 := δ(a22/a23 + a32), α19 := δ(a ⊥ 12/a ⊥ 11), α20 := δ(a ⊥ 13/a ⊥ 12), α21 := δ(b ⊥ 31/b ⊥ 21), α22 := δ(b23/b33), α23 := δ(a32/a33). (4.2) Theorem 4.1. (Description of I1, I2 and I3 in F): 1) The plain structure of I1, I2 and I3 is represented by the diagrams depicted in Fig 3, Fig 4 and Fig 5 of Section 7. 2) The ideals are connected by the following relations between indices: α4 = δ(d/d1) = δ(c/c1), α11 = δ(b ⊥ 11/b ⊥⊥ ) = δ(b⊥/b⊥⊥11), α12 = δ(a ⊥ 11/a ⊥⊥ ) = δ(a⊥/a⊥⊥11), α15 = δ(b33/d1 + e) = δ(a33/c1 + e) = δ(e33/c1 + d1), α16 = δ(d1/a ⊥ 13) = δ(a13/a23) = δ(b13/b23) = δ(e13/e23), α17 = δ(c1/b ⊥ 31) = δ(a31/a32) = δ(b31/b32) = δ(e31/e32), α18 = δ(a22/a23 + a32) = δ(e22/e23 + e32), (4.3) α19 = δ(a ⊥ 12/a ⊥ 11) = δ(b ⊥ 21/b ⊥ 11) = δ(e11/e12 + e21), α20 = δ(a ⊥ 13/a ⊥ 12) = δ(a12/a13 + a22) = δ(b12/b13 + b22) = δ(e12/e13 + e22), α21 = δ(b ⊥ 31/b ⊥ 21) = δ(a21/a22 + a31) = δ(b21/b22 + b31) = δ(e21/e22 + e31), α22 = δ(b23/b33) = δ(a23/c + a33), α23 = δ(a32/a33) = δ(b32/b33 + d) = δ(e32/d + e33). 3) I1 ∪ I2 ∪ I3 is orthogonally closed in force of the following relations: a11 + d ⊥ 1 = a ⊥⊥ 11, a12 + d ⊥ 1 = a ⊥⊥ 12, a13 + d ⊥ 1 = a ⊥⊥ 13, b11 + c ⊥ 1 = b ⊥⊥ 11, b21 + c ⊥ 1 = b ⊥⊥ 21, b31 + c ⊥ 1 = b ⊥⊥ 31. (4.4) Proof. 1) This is routine verification. 2) δ(d/d1) ≥ δ(d ⊥ 1 /d ⊥) ≥ δ(d⊥1 e ⊥/d⊥e⊥) ≥ δ(c/c1) ≥ δ(c ⊥ 1 /c ⊥) ≥ δ(c⊥1 e ⊥/c⊥e⊥) ≥ δ(d/d1). This shows the first equality. The second and third ones are evident. As to the fourth, just consider the free modular lattice generated by the triple (d⊥, c⊥, e⊥). The other equalities are proved analogously. CUBO 20, 1 (2018) On rigid Hermitean lattices, II 71 3) We just show the first equality (the others follow in the same manner): δ(a11 + d ⊥ 1 /d ⊥ 1 ) = δ(a11/a11d ⊥ 1 ) = δ(a11/a21) ≤ δ(a ⊥/d⊥1 ) < ℵ0 (by (1.7)). Thus a11 + d ⊥ 1 = (a11 + d ⊥ 1 ) ⊥⊥ by Lemma 2.3, because d⊥1 = (d ⊥ 1 ) ⊥⊥. Since d⊥1 ≤ a ⊥⊥ 11 (because c + e ≤ a11), we obtain the desired equality. The rest is easy and it follows F = F0. Theorem 4.2. (Forced relations among indices): i) If α7 6= 0 then α1 = α2 = ℵ0; ii) If α8 6= 0 then α1 = α3 = ℵ0; iii) If α9 6= 0 then α2 = α3 = ℵ0; iv) For i ∈ {10, 11, 12, 15, 16, 17, 19}, if αi 6= 0 then α1 = α2 = α3 = ℵ0; v) For i ∈ {13, 14, 18, 20, 21, 22, 23}, if αi 6= 0 then α1 = α2 = α3 = α4 = ℵ0; vi) α11 + α12 + α16 + α17 + α19 + α20 + α21 < ℵ0 Proof. Each implication follows in a way as was shown in the proof of Theorem 3.2, possibly in conjunction with Lemma 2.3. The well known rule (x + y)⊥ = x⊥y⊥ may also be useful for computations. The last assertion is just the translation of (1.7) in terms of the indices αi. 5 The subdirectly irreducible factors of F. In order to discover the factors of F it is sufficient to work out I1, I2 and I3 at the same time, using the relations given in Theorem 4.1 and Theorem 4.2. The essence of the procedure consists in collecting all prime quotients that are connected with a given one via the algebraic operations: this will produce automatically the corresponding subdirectly irreducible factor, together with the associated relation. Observe how useful are indices in this procedure: on the one hand they are associated in natural way to congruences, on the other hand the forced relations among them give directly the non minimal congruences in the subdirectly irreducible factors. A little final caution is needed: there is a quotient which does not appear in the ideals, namely 1/(a⊥+b⊥+e⊥) (see the factor corresponding to α9 in Table III). Since (a ⊥+b⊥)⊥⊥ = (a⊥+e⊥)⊥⊥ = (e⊥ + b⊥)⊥⊥ = 1 this is the only exception. The factors are labelled from 1 to 23 in Tables II, III and IV. The last table contains all non distributive members. 72 Ana Cecilia de la Maza and Remo Moresi CUBO 20, 1 (2018) Remark 5.1. From all the above results we deduce in particular that Conjecture 2 in [M2] is true: in fact, the finite codimensions indicated in the conjecture correspond to the ones given by (1.7). Remark 5.2. There are plain lattice isomorphisms between different factors. Nevertheless we chose to give explicitly all diagrams, in order to facilitate visualization. It is worth noticing that the majority of this plain isomorphisms are induced by the map a 7→ b and b 7→ a, which defines an involution of S that extends naturally to F. More precisely, there are eight pairs of symmetric factors, namely (1,2), (5,6), (8,9), (11,12), (13,14), (16,17), (20,21) and (2,23), all other factors being self symmetric. 6 Remarks concerning applications to Hermitean spaces It is possible to prove that all factors of F are implemented by Hermitean models. Hence they can be used to describe the congruence class of a subspace A in a Hermitean space (E, φ) of denumerable dimension under the starting assumptions, where A, E, E∗ correspond to a, 1, b, respectively. In general, these IHL will not suffice to build a complete set of geometric invariants, but they constitute a very important part. Details on these aspects cannot be discussed in the present work. 7 Diagrams q q q q q q q q q q q q q q q q q q ✟ ✟ ✟ ✟ ✟ ✟✟✟ ✟ ✟ ✟ ✟ ✟✟ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ✟ ✟ ✟ ✟ ✟ ✟✟ ✟ ✟ ✟ ✟ ✟ ✟✟ ✟ ✟✟ c d⊥ 1 b⊥⊥ b c1 d⊥ a⊥ 0 e e⊥ 1 a a⊥⊥ b⊥ d1 c⊥ d c⊥ 1 S = S[a; a ≤ b⊥ ∧ bb⊥ = 0] Figure 1 q q q q q q q q q q q q q q q q q q ✟ ✟ ✟ ✟ ✟ ✟✟✟ ✟ ✟ ✟ ✟ ✟✟ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ✟ ✟ ✟ ✟ ✟ ✟✟ ✟ ✟ ✟ ✟ ✟ ✟✟ ✟ ✟✟β1 β1 β1 β1 β2 β2 β2 β2 β3 β3 β3 β3 β6 β5 β7 β7 β8 β8 β4 β4 β4 β4 S = S[a; a ≤ b⊥ ∧ bb⊥ = 0] Figure 2 CUBO 20, 1 (2018) On rigid Hermitean lattices, II 73 q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q � � � � � � � � ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍❍ � � � � � ❍ ❍ ❍ ❍❍ � � �� ❍ ❍ ❍❍ � � ❍ ❍ ❍ ❍❍ � � � � � � � � � � � � � � � �� � � � � � � � � � � �� � � � � � � � � � � ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍❍ c + e c a13 a12 a11 c1 a23 a⊥⊥ 11 a⊥ b⊥ 31 a22 a⊥⊥ 12 b⊥ 21 a33 a⊥⊥ 13 b⊥ 11 a32 d⊥ 1 = (c + e)⊥⊥ b b⊥⊥ 0 a31 d⊥ = (b + e)⊥⊥ e a21 b + e α6 α2 α3 α4 α9 α11 α12 α13 α15 α16 α17 α17 α18 α19 α19 α20 α21 α21 α22 α23 Figure 3: the ideal I1 in F q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q ❅ ❅ ✟ ✟✟ ✟ ✟✟ ✟ ✟✟ ❅ ❅ ✟ ✟✟ ✟ ✟ ✟ ✟✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟✟ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ✟ ✟ ✟ ✟✟ ✟ ✟ ✟ ✟✟ ✟ ✟ ✟ ✟✟ ✟ ✟ ✟ ✟✟ ✟ ✟ ✟✟❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅❅ ❅ ❅❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅❅ ❅ ❅ ❅ ❅ ❅❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ✟ ✟ ✟✟✟ ✟ ✟ ✟✟ ✟ ✟ ✟ ✟✟ ✟ ✟ ✟ ✟✟ ✟ ✟ ✟✟ ❅ ❅ e a + e 0 (a + e)⊥⊥ = c⊥ b13 b23 b33 (d + e)⊥⊥ = c⊥ 1 b12 b22 b32 b⊥ b⊥⊥ 11 b⊥⊥ 31 b⊥⊥ 21 b11 b21 b31 d a⊥⊥ a a⊥ 11 a⊥ 12 a⊥ 13 d1 d + e α5 α1α3 α4 α8 α12 α11 α14 α15 α16 α16 α17 α18 α19 α19 α22 α23 α20 α20 α21 Figure 4: the ideal I2 in F 74 Ana Cecilia de la Maza and Remo Moresi CUBO 20, 1 (2018) r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r rr r ✟✟ ✟ ✟ ✟ ✟ ✟✟ ❍❍ ❍ ❍ ❍ ❍ ❍❍ ✟✟ ❍❍ ❍ ❍ ❍❍ ✟ ✟ ✟✟ ✟✟ ❍❍ ❍❍ ✟✟ ✟ ✟ ✟✟ ❍ ❍ ❍❍ ✟ ✟ ✟✟ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍❍ ✟ ✟ ✟ ✟ ✟✟ ✟✟❍❍ ❍❍✟✟ ❍ ❍ ❍❍ ✟ ✟ ✟✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟✟ ✟✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟✟ ✟✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟✟ ✟✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟✟ ✟✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟✟ ✟✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟✟ ✟✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟✟ ✟✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟✟ ✟✟❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍❍ ❍❍ ❍❍ ❍❍ ❍❍ ❍❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍❍ c c1 b⊥ 31 b⊥ 21 b⊥ 11 b⊥⊥ b 0 a⊥⊥ a a⊥ 11 a⊥ 12 a⊥ 13 d1 d e13 e23 e12 e33 e22 e11 e⊥ = (a + b)⊥⊥ e32 e21 e31 α1 α5α6 α2 α7 α4 α4 α12α11 α15 α16 α16 α17 α17 α18 α19α19 α19 α22 α23 α20 α20α21 α21 Figure 5: the ideal I3 in F Table I r r r r r r b a b⊥a ⊥ � � �� � � �� ❅ ❅ ❅❅ ❅ ❅ ❅❅ β4 = δ ( c/c1 ) β4 β2 β1 β1 β4 β2 r r r r r r r r r r b c c⊥ e a b⊥a⊥ e⊥� � � � ❅ ❅ ❅❅ ❅ ❅ ❅❅ � � � � ❅ ❅ ❅❅ ❅ ❅ ❅❅ β7 = δ ( c1/b ⊥⊥ ) β3 β7 β1 β1 β2 β1 β3 β3 β2 β2 β1 β1 β3 β7 r r r r rr r r r r b d⊥ d e a⊥ a e⊥ b⊥ � � �� � � �� � � �� � � �� ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ β8 = δ ( d1/a ⊥⊥ ) β3 β2 β2 β8 β1 β3 β1 β1 β1 β3 β8 β2 β3 β2 Table II CUBO 20, 1 (2018) On rigid Hermitean lattices, II 75 r r β1 = α1 = δ (a/0) a b α1 r r β2 = α2 = δ (b/0) b a α2 r r β3 = α3 = δ (e/0) e ba α3 r r r β5 = α5 = δ ( a ⊥⊥ /a ) a⊥ b a⊥⊥ a α1 α5 α1 = ℵ0 r r r β6 = α6 = δ ( b ⊥⊥ /b ) b⊥ a b⊥⊥ b α2 α6 α2 = ℵ0 76 Ana Cecilia de la Maza and Remo Moresi CUBO 20, 1 (2018) Table III r r r r r r r r r α4 = δ ( c/c1 ) e b = c1 a = d1 b⊥a⊥ � � � � � � � � � ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ α1 α4 α4 α2 α1 = α2 = ℵ0 r r r r r α7 = δ ( e ⊥ /e11 ) a = b⊥b = a⊥�� ��❅❅ ❅❅ α7 α1 α2 α1 = α2 = ℵ0 r r r r r α8 = δ ( c ⊥ /b13 ) b⊥ b a = a⊥⊥a⊥ �� ��❅❅ ❅❅ α8 α1 α3 α1 = α3 = ℵ0 r r r r r α9 = δ ( d ⊥ /a31 ) a a⊥ b⊥b⊥⊥ = b�� ��❅❅ ❅❅ α9 α3 α2 α2 = α3 = ℵ0 r r r r r r r r r r α10 = δ ( 1 a⊥ + b⊥ + e⊥ ) b⊥e ⊥a⊥ e ab �� ��❅❅ ❅❅ �� ��❅❅ ❅❅ α1 α2 α3 α10 α1 = α2 = α3 = ℵ0 r r r r r r r r r r r r r r r α11 = δ ( b ⊥ 11/b ⊥⊥ ) b⊥⊥ = b c c ⊥ e a = d b⊥a ⊥ e⊥� � � � � � � � � ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ �� �� ��❅ ❅ ❅ ❅ ❅ ❅ α1 α11 α11 α2 α3 α1 = α2 = α3 = ℵ0 r r r r r r r r r r r r r r r α12 = δ ( a ⊥ 11/a ) c = b d⊥ = a11 d = a ⊥ 11 e a = a⊥⊥ e⊥ b⊥a⊥ � � � � � � � � � � � � � � � ❅❅ ❅❅ ❅❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ α1 α12 α12 α2 α3 α1 = α2 = α3 = ℵ0 r r r r r r r r r r r r r r r r r r r r r α13 = δ   d⊥ 1 a21 + d ⊥   b = c1 d⊥ e a = d1 c⊥ b⊥ a⊥ e⊥ c d � � � � � � � � � � � � ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ � � � � � � � � � ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ α1 α4 α13 α4 α2 α3 α1 = α2 = α3 = α4 = ℵ0 r r r r r r r r r r r r r r r r r r r r r α14 = δ   c⊥ 1 b12 + c ⊥   b = c1 d⊥ e a = d1 c⊥ b⊥e⊥ a⊥ c d � � � � � � � � � � � �❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ � � � � � � � � � ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ α1 α4 α14 α4 α2 α3 α1 = α2 = α3 = α4 = ℵ0 CUBO 20, 1 (2018) On rigid Hermitean lattices, II 77 Table IV q q q q q qq q q q q q qq q α15 = δ ( b33 d1 + e ) c = b e a = d = a⊥ 11 b⊥ = b33a ⊥ e⊥ �� ��� �� � ��� ��� �� ❅❅ ❅❅❅ ❅❅ ❅❅ ❅ ❅❅❅❅❅ ❅❅ α1 = α2 = α3 = ℵ0 α1 α15 α15 α2 α3 q q q q q qq q q q q q qq q q qq q q α16 = δ ( d1/a ⊥ 13 ) c = b e a = a⊥ 13 d = d1 b33 d⊥ b⊥ = b13 a⊥ e⊥ ❅❅ ❅❅ ❅❅ ❅ ❅❅ ❅ ❅❅ ❅ ❅❅ ❅ ❅❅ ❅❅ � ��� � �� � � �� � � �� � � �� α1 = α2 = α3 = ℵ0 α1 α16 α16 α16 α2 α3 q q q q q q q q q q q q q q q q q q q q α17 = δ ( c1/b ⊥ 31 ) b c⊥ b⊥ a⊥ c e a = d e⊥ ❅❅ ❅ ❅ ❅❅ ❅ ❅ ❅❅ ❅ ❅❅ ❅ ❅❅ ❅ ❅ ❅❅ � ��� ��� �� �� � �� �� � �� α1 = α2 = α3 = ℵ0 α1 α17 α17 α17 α2 α3 q q q q q q q q q q q q q q q q q q q q q q q q q q q α18 = δ ( a22 a23 + a32 ) b = c1 d⊥ e a = d1 c⊥ b⊥a⊥ c d e⊥ � � �� � � �� � � �� � � �� ❅❅ ❅ ❅ ❅❅ ❅ ❅ ❅❅ ❅ ❅ ❅❅ � � �� � �� � �� ❅ ❅❅ ❅ ❅❅ ❅ ❅❅ ❅ ❅ ❅❅ α1 = α2 = α3 = α4 = ℵ0 α1 α4 α18 α18 α4 α2 α3 q q q q q q q q q q q q q q q q q q q q q q q q q q q α19 = δ ( a ⊥⊥ 11/a ⊥⊥ 12 ) b e a b⊥a⊥ d⊥ c1 = c d = d1 e⊥ c⊥ � � �� � � �� � � �� � � �� � �� � � �� � �� ❅ ❅ ❅❅❅ ❅❅ ❅ ❅ ❅❅❅❅ ❅ ❅ ❅❅ ❅ ❅❅ ❅ ❅❅ ❅ ❅ ❅❅ α1 = α2 = α3 = ℵ0 α1 α19 α19 α19 α19 α3 α2 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q α20 = δ ( a ⊥ 13/a ⊥ 12 ) c1 = b d⊥ e a b⊥ a⊥ c d1 c⊥ d e⊥ d⊥ 1 � � � �� � � � �� � � � �� � � � �� � � � �� � � �� � � ��❅ ❅ ❅❅ ❅ ❅ ❅❅❅ ❅❅ ❅ ❅ ❅❅ ❅❅ ❅ ❅ ❅❅ ❅ ❅❅ ❅ ❅❅ ❅ ❅❅ ❅ ❅ ❅❅ α1 = α2 = α3 = α4 = ℵ0 α4 α20 α20 α3 α1 α4 α2α20 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q α21 = δ ( b ⊥ 31/b ⊥ 21 ) c1 d⊥ b e a = d1 c⊥ b⊥ a⊥ c d e⊥ c⊥ 1 � � �� � � �� � � �� � � �� � �� � � �� � �� � ��� � �� ❅ ❅ ❅ ❅❅ ❅ ❅ ❅ ❅❅ ❅❅ ❅ ❅ ❅ ❅❅ ❅ ❅ ❅❅ ❅ ❅ ❅❅ ❅ ❅ ❅❅ ❅ ❅ ❅ ❅❅ α1 = α2 = α3 = α4 = ℵ0 α4 α21α21 α4 α3 α21 α2 α1 q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q α22 = δ ( b23/b33 ) a = d1 b⊥a⊥ c⊥ d⊥ ec1 = b dc e⊥ ✑ ✑ ✑✑❅ ❅❅ ❅ ❅❅ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍❍❍❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍❍ ✑✑ ✑✑❍ ❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍ ❍ ❍❍ ✑✑ ✑✑ ✑ ✑ ✑✑ ✑ ✑ ✑✑ ✑ ✑ ✑✑✑ ✑ ✑✑ ✑ ✑ ✑✑ ✑ ✑ ✑✑✑ ✑ ✑✑ ✑✑ ✑✑ α1 = α2 = α3 = α4 = ℵ0 α1 α4 α22 α4 α2 α3 α22 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q α23 = δ ( a32/a33 ) a = d1 b⊥ a⊥ c⊥ d⊥ ec1 = b dc e⊥✪ ✪✪ ✪ ✪✪ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍❍ ❍ ❍❍ ❍ ❍ ❍ ❍❍ ❍ ❍❍ ❍ ❍ ❍ ❍❍ ❍ ❍ ❍ ❍❍ ❍ ❍❍ ❍ ❍❍ ❍ ❍ ❍ ❍❍ ❍ ❍❍ ❍ ❍❍ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑✑✑ ✑ ✑ ✑ ✑✑ ✑ ✑ ✑ ✑ ✑✑ ✑ ✑ ✑ ✑ ✑✑ ✑ ✑ ✑ ✑✑ ✑ ✑ ✑✑✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑✑ α1 = α2 = α3 = α4 = ℵ0 α1 α4 α23 α4 α2 α23 α3 References [DM1] A.C. de la Maza, R.Moresi, On modular lattices generated by chains, Algebra Universalis 54 (2005), 475-488. [DM2] A.C. de la Maza, R.Moresi, Hermitean (semi) lattices and Rolf’s lattice, Algebra Universalis 66 (2011), 49-62. [DM3] A.C. de la Maza, R.Moresi, On rigid Hermitean lattices, I, Preprint. [G1] H. Gross, Quadratic forms in infinite dimensional vector spaces, Birkäuser, Boston, 1979. [G2] H. Gross, Lattices and infinite-dimensional forms. “The lattice method”, Order 4 (1987), 233-256. 78 Ana Cecilia de la Maza and Remo Moresi CUBO 20, 1 (2018) [KKW] H. A. Keller, U.-M. Künzi, M. Wild (eds), Orthogonal geometry in infinite dimensional vector spaces, Heft 53, Bayreuther Mathematische Schriften, Bayreuth, 1998. [M1] R. Moresi, Modular lattices and Hermitean forms, Algebra Universalis 22 (1986), 279-297. [M2] R. Moresi, A test-example of a quadratic lattice, Order 17 (2000), 215-226. [R] H. L. Rolf, The free lattice generated by a set of chains, Pacific J. Math. 8 (1958), 585-595. [S] E. T. Schmidt, On finitely generated simple modular lattice, Periodica Mathematica Hungarica 6(3) (1975), 213-216. Introduction Preliminaries Description of S Description of F The subdirectly irreducible factors of F. Remarks concerning applications to Hermitean spaces Diagrams