A geometric characterization of the Monster

Abstract

Let M be the monster model of a complete first-order theory T. If D is a subset of M, following D. Zambella we consider e(D) = {D ′ | (M, D) ≡ (M, D ′)} and o(D) = {D ′ | (M, D) ∼ = (M, D ′)}. The general question we ask is when e(D) = o(D) ? The case where D is A-invariant for some small set A is rather straightforward: it just means that D is definable. We investigate the case where D is not invariant over any small subset. If T is geometric and (M, D) is an H-structure (in the sense of A. Berenstein and E. Vassiliev) we get some answers. In the case of SU-rank one, e(D) is always different from o(D). In the o-minimal case, everything can happen, depending on the complexity of the definable closure. We also study the case of lovely pairs of geometric theories. 1 Subsets of the monster model We will consider complete first-order theories T having an infinite model. By a monster model of T we understand a model M of T whose universe is a proper class and realizes all types over all subsets. Every theory has a monster model and it is unique up to isomorphism. Alternatively, one can assume the existence of a strongly inaccesible cardinal λ and take a saturated model of cardinality λ as the monster model of T. In this last case only set of cardinality less than λ will be considered. A set is small if its cardinality is smaller than |M|, which means that it is just a set (not a proper class) if we understand M as a proper class model and that it has cardinality < λ if we consider M as a saturated model of cardinality λ. The requirements on λ for the existence of a saturated model of cardinality λ are weaker than strong inaccesibility (see Theorem VIII.4.7 in [10]), but for some results we will need to use the assumption that if A is small, its power set is small too. The language of T is L. If ϕ is a formula, ϕ ∈ L means that the symbols of ϕ belong to L. We consider n-ary relations D on M and corresponding expansions (M, D) to the language L ∪ {D}, where D is a new n-ary predicate symbol. There are two classes of relations naturally associated to D, the class o(D) of all images of D under automorphisms of M and the class e(D) of all relations with expansion elementarily equivalent to (M, D). If the relation D consists of a single n-tuple a, then e(D) is the class of all n-tuples having the *

Deflating infinite Coxeter groups to finite groups

“…The proofs of these theorems are similar those given in this paper. The principal complication is that the root enumeration would involve |M| (almost 10 54 ) root elements and therefore much less elementary methods [8,11,9] are required to identify the group.…”

A 7-Local Identification of the Monster

“…These results are included in Section 2. Finally we mention that the monster has been characterized from the amalgam of its 2-local subgroups by Ivanov [15] and from the amalgam of its 3-local subgroups by Ivanov and Meierfrankenfeld [16]. These two results actually show that the universal completion of the amalgam under investigation is isomorphic to the monster.…”

Structure of the maximal -local geometry point-line collinearity graph

“…Some of the papers which have as their major aim the uncovering of the structure of the point-line collinearity graph are Rowley [23], Rowley and Walker [25][26][27][28][29][30][31][32] and Segev [35]. Those which have other aims are Buekenhout [3], Buekenhout et al [4], Hall and Shpectorov [8], Ivanov [9][10][11], Ivanov and Shpectorov [12][13][14][15][16], Ivanov and Wiedorn [17], Mason and Smith [18], Rowley [21,22], Rowley and Walker [24], Shpectorov [36,37], Smith [38], Stroth [39,40], Weiss and Yoshiara [44], Weiss [43] and Yoshiara [45]. This is only a partial list; for further references, consult the bibliographies of the above-cited papers.…”