Identity element

Abstract: Let ℜ be an expansion of a dense linear order (R, <) without endpoints having the intermediate value property, that is, for all a, b ∈ R, every continuous (parametrically) definable function f: [a, b] → R takes on all values in R between f(a) and f(b). Every expansion of the real line (ℝ, <), as well as every o-minimal expansion of (R, <), has the intermediate value property. Conversely, some nice properties, often associated with expansions of (ℝ, <) or with o-minimal structures, hold for sets and functions definable in ℜ. For example, images of closed bounded definable sets under continuous definable maps are closed and bounded (Proposition 1.10).Of particular interest is the case that ℜ expands an ordered group, that is, ℜ defines a binary operation * such that (R, <, *) is an ordered group. Then (R, *) is abelian and divisible (Proposition 2.2). Continuous nontrivial definable endo-morphisms of (R, *) are surjective and strictly monotone, and monotone nontrivial definable endomorphisms of (R, *) are strictly monotone, continuous and surjective (Proposition 2.4). There is a generalization of the familiar result that every proper noncyclic subgroup of (ℝ, +) is dense and codense in ℝ: If G is a proper nontrivial subgroup of (R, *) definable in ℜ, then either G is dense and codense in R, or G contains an element u such that (R, <, *, e, u, G) is elementarily equivalent to (ℚ, <, +, 0, 1, ℤ), where e denotes the identity element of (R, *) (Theorem 2.3).Here is an outline of this paper. First, we deal with some basic topological results. We then assume that ℜ expands an ordered group and establish the results mentioned in the preceding paragraph. Some examples are then given, followed by a brief discussion of analytic results and possible limitations. In an appendix, an explicit axiomatization (used in the proof of Theorem 2.3) is given for the complete theory of the structure (ℚ, <, +, 0, 1, ℤ).

Abstract: We introduce and study a family of operators which act in the group algebra of a Weyl group W and provide a multiparameter solution to the quantum Yang-Baxter equations of the corresponding type. These operators are then used to derive new combinatorial properties of W and to obtain new proofs of known results concerning the Bruhat order of W. The paper is organized as follows. Section 2 is devoted to preliminaries on Coxeter groups and associated Yang-Baxter equations. In Theorem 3.1 of Section 3, we describe our solution of these equations. In Section 4, we consider a certain limiting case of our solution, which leads to the quantum Bruhat operators. These operators play an important role in the explicit description of the (small) quantum cohomology ring of G/B. Section 5 contains the proof of Theorem 3.1. Section 6 is devoted to combinatorial applications of our operators. For an arbitrary element u ∈W,we define a graded partial order onW called the tilted Bruhat order; this partial order has unique minimal element u. (The usual Bruhat order corresponds to the special case where u = e, the identity element.) We then prove that tilted Bruhat orders are lexicographically shellable graded posets whose every interval is Eulerian. This generalizes the well-known results of D.-N. Verma, A. Bjorner, M. Wachs, and M. Dyer.

Abstract: In this paper we will study a class of groups of even order which satisfy the following condition: (TI): two different Sylow 2-groups contain only the identity element in common. There are three series of finite non-abelian simple groups known to satisfy the above condition (TI). Let q denote a power of 2 greater than 2. The linear fractional group in 2 variables over the field of q elements, denoted here by L2(q), satisfies the condition (TI). The projective unitary groups U3(q) provide another series of groups satisfying (TI). The third series consists of groups G(q) defined by the author in [7]. The main result of this paper is the converse of the above statement. We will prove the following theorem.