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One-sided orthogonality, orthomodular spaces, quantum sets, and a class of Garside groups

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Abstract

The classification of orthomodular lattices by means of their structure group is extended to non-symmetric orthogonality relations. Spaces with a non-hermitian sesquilinear form, quantum sets, and a new class of Garside groups are given as examples.

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... On the other hand, among the various classes of KL-algebras X we didn't find an example where Proposition 4(c) does not hold if the meet is taken in X. For CKL-algebras, see the remark after Proposition 6. Definition 4. For an L-algebra X, two elements x, y ∈ X are said to be orthogonal [23] if x·y = y and y · x = x. By δ : X × X → X × X we denote the map δ(x, y) := (x · y, y · x). ...
... (b) ⇒ (c): For x, y ∈ X, Eqs. (23) show that y ≤ yx −1 ∧ 1 = x · y. Hence X is an L-subalgebra of G − . ...
... For x, y, z ∈ X, Eqs. (23) ...
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... If the distinctness relation x ‰ y is replaced by orthogonality, ordinary sets turn into quantum sets [61,22]. More precisely, let K be a symmetric irreflexive binary relation on a set S. For a subset A of S, define A K :" tx P A|@ a P A : aKxu, and let P pSq (the "power set" of S) be the set of subsets A K of S. Since P pSq is closed under intersection, it is a complete lattice. ...
... An L-algebra is said to be sharp if it satisfies the equation x Ñ px Ñ yq " x Ñ y. This concept arose in connection with unsharp quantum measurement [41,32], formalized by means of a class of L-algebras [22] which become orthomodular lattices if and only if they are sharp [62]. A sharp discrete L-algebra has the property that for distinct points x and y, the point x Ñ y belongs to the connecting line. ...
... Quantum sets with a non-symmetric orthogonality K were defined in [22]. We relate them to symmetric quantum sets by introducing a one-sided version of orthomodular lattices (Definition 10). ...
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The relationship of discrete L -algebras to projective geometry is deepened and made explicit in several ways. Firstly, a geometric lattice is associated to any discrete L -algebra. Monoids of I-type are obtained as a special case where the perspectivity relation is trivial. Secondly, the structure group of a non-degenerate discrete L -algebra X is determined and shown to be a complete invariant. It is proved that X ∖ {1} is a projective space with an orthogonality relation. A new definition of non-symmetric quantum sets, extending the recursive definition of symmetric quantum sets, is provided and shown to be equivalent to the former one. Quantum sets are characterized as complete projective spaces with an anisotropic duality, and they are also characterized in terms of their complete lattice of closed subspaces, which is one-sided orthomodular and semimodular. For quantum sets of finite cardinality n > 3, a representation as a projective space with duality over a skew-field is given. Quantum sets of cardinality 2 are classified, and the structure group of their associated L -algebra is determined.
... Much earlier, n implicitly occurs in Artin's representation [2] of the braid group by the Hurwitz action [35]. We will show that besides its relationship to braid groups, n belongs to a class of L-algebras which connects one-sided orthogonality relations [17] with generalized effect algebras [20-22, 27, 54] and quantum sets [17]. ...
... Much earlier, n implicitly occurs in Artin's representation [2] of the braid group by the Hurwitz action [35]. We will show that besides its relationship to braid groups, n belongs to a class of L-algebras which connects one-sided orthogonality relations [17] with generalized effect algebras [20-22, 27, 54] and quantum sets [17]. ...
... One-sided orthogonality naturally arises in connection with non-hermitian sesquilinear forms [17]. The general concept is given by a lattice with a non-symmetric orthogonality relation, satisfying natural conditions introduced by Janowitz [36] in the symmetric case. ...
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Etingof et al. introduced a concept of structure group and applied it to involutive set-theoretic solutions to the Yang-Baxter equation. Other types of structure groups arise as Artin-Tits groups, lattice-ordered groups or groups associated to projection lattices of von Neumann algebras. In many cases, the structure group is generated by an L-algebra, with the exception of knot groups, where an L-algebra has been found only for torus knots. With a concept of group representation for L-algebras, a construction similar to that of Etingof et al. is shown to exist for structure groups of general L-algebras. As an application, new results on semidirect products of L-algebras are obtained. An approach toward the unsolved case of knot groups is made by investigating a lattice-ordered L-algebra Θn where the braid group Bn acts freely and transitively on the maximal chains, which are all of length n. It is shown that Θn has a peculiar geometry, equipped with a one-sided orthogonality, the set of points being closely related to quantum sets, and lines having an intrinsic orientation coming from automorphisms of Θn. Orthogonal pairs of points are interpreted as crossings of strands in the braid group.
... In this paper, we define a (symmetric) quantum set to be a set with an (irreflexive symmetric) orthogonality relation satisfying two simple axioms (Definition 1) which allow its enumeration as a quantum set. Each stage of enumeration is given by an (orthogonally) closed subset A. The first axiom states that singletons are closed (allowing to start the counting process), while the second one assures that any x R A can be made orthogonal with respect to A. A slightly more general concept of quantum set with non-symmetric orthogonality has been introduced in [14]. Here we stick to symmetric quantum sets, mostly because of their pithy description in Definition 1. ...
... During the last decade, L-algebras occurred in various contexts, including algebraic logic, combinatorial group theory, operator algebras, and number theory (see, e. g., [56,58,61,60,14,63]). In particular, they provide a succinct description of orthomodular lattices [61], which give the semantics of quantum logic [3]. ...
... Note. Quantum sets with non-symmetric orthogonality K were defined previously in [14]. ...
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... An important class of L-effect algebras arises from (one-sided) orthogonality lattices [14], which occur, e. g., in connection with Frobenius algebras. For a vector space H over a skew-field with involution, endowed with a non-degenerate sesquilinear form σ, the one-dimensional subspaces form an L-algebra which determines the space pH, σq. ...
... For a vector space H over a skew-field with involution, endowed with a non-degenerate sesquilinear form σ, the one-dimensional subspaces form an L-algebra which determines the space pH, σq. The associated projective space PpHq is a quantum set [14], and every quantum set is an L-effect algebra. Finite quantum sets generate a Garside group. ...
... (d) ñ (a): By Proposition 7, we infer that y ď px Ñ yq Ñ y. With (14), this yields x _ y ď px Ñ yq Ñ y. Hence px Ñ yqx " xpx Ñ yq ď px _ yqpx Ñ yq ď y, and thus px Ñ yqpx _ yq " px _ yqpx Ñ yq ď y. ...
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L-effect algebras are introduced as a class of L-algebras which specialize to all known generalizations of effect algebras with a \wedge -semilattice structure. Moreover, L-effect algebras X arise in connection with quantum sets and Frobenius algebras. The translates of X in the self-similar closure S(X) form a covering, and the structure of X is shown to be equivalent to the compatibility of overlapping translates. A second characterization represents an L-effect algebra in the spirit of closed categories. As an application, it is proved that every lattice effect algebra is an interval in a right \ell -group, the structure group of the corresponding L-algebra. A block theory for generalized lattice effect algebras, and the existence of a generalized OML as the subalgebra of sharp elements are derived from this description.
... Apart from derived operations, basic concepts of algebra, topology, and analysis can be defined in the general context of L-algebras. For example, x · y = y gives an orthogonality relation between x and y which is non-symmetric, in general (see [26]). Here we only provide the concepts which are closest to our present purpose. ...
... By Theorem 3, S( X) ∼ = N (X) . Hence, Eq. (26) implies that as a lattice, G( X) ∼ = Z (X) . For any x ∈ X, let x ′ denote the inverse in G( X). ...
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... In other cases, like our present occasion, it is more natural to understand the operation¨as a difference x y :" y¨x, the simplest example being the power set PpM q; ˘o f a set M . So the partial order has to be replaced by the opposite one, which turns the logical unit into the smallest element 0. Therefore, we speak of a positive L-algebra, or simply an L`-algebra [24]. The axioms of an L`-algebra are thus ...
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... However, there are already various notions of 'quantum set theory' in the literature; e.g. [12,16,31,32,36,37]. Later on, we were told that the objects that we are studying in this article have recently been studied under the name 'orthogonality spaces' in [28,38,39]. ...
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... However, there are already various notions of 'quantum set theory' in the literature; e.g. [12,16,31,32,36,37]. Later on, we were told that the objects that we are studying in this article have recently been studied under the name 'orthogonality spaces' in [28,38,39]. ...
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This is an expository introduction to tropical algebraic geometry based on my lectures at the Workshop on Tropical Geometry and Integrable Systems in Glasgow, July 4-8, 2011, and at the ELGA 2011 school on Algebraic Geometry and Applications in Buenos Aires, August 1-5, 2011.
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Define a Garside monoid to be a cancellative monoid where right and left lcm's exist and that satisfy additional finiteness assumptions, and a Garside group to be the group of fractions of a Garside monoid. The family of Garside groups contains the braid groups, all spherical Artin-Tits groups, and various generalizations previously considered.2 Here we prove that Garside groups are biautomatic, and that being a Garside group is a recursively enumerable property, i.e., there exists an algorithm constructing the (infinite) list of all small Gaussian groups. The latter result relies on an effective, tractable method for recognizing those presentations that define a Garside monoid.
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A Garside monoid is a cancellative monoid with a finite lattice generating set; a Garside group is the group of fractions of a Garside monoid. The family of Garside groups contains all Artin–Tits groups of spherical type. We show that the well-known notion of a parabolic subgroup of an Artin–Tits group can be extended to the framework of Garside groups so that most of the standard properties known in the Artin–Tits groups case are preserved. The extension is not trivial and it requires a new approach. We also define the more general notion of a Garside subgroup of a Garside group that nicely extends the notion of an LCM-homomorphism between Artin–Tits groups.
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Non-degenerate cycle sets are equivalent to non-degenerate unitary set-theoretical solutions of the quantum Yang–Baxter equation. We embed such cycle sets into generalized radical rings (braces) and study their interaction in this context. We establish a Galois theory between ideals of braces and quotient cycle sets. Our main result determines the relationship between two square-free cycle sets operating transitively on each other.
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Examples suggest that there is a correspondence between L-spaces and 3-manifolds whose fundamental groups cannot be left-ordered. In this paper we establish the equivalence of these conditions for several large classes of such manifolds. In particular, we prove that they are equivalent for any closed, connected, orientable, geometric 3-manifold that is non-hyperbolic, a family which includes all closed, connected, orientable Seifert fibred spaces. We also show that they are equivalent for the 2-fold branched covers of non-split alternating links. To do this we prove that the fundamental group of the 2-fold branched cover of an alternating link is left-orderable if and only if it is a trivial link with two or more components. We also show that this places strong restrictions on the representations of the fundamental group of an alternating knot complement with values in Homeo_+(S^1).
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We are concerned with mapping class groups of surfaces with nonempty boundary. We present a very natural method, due to Thurston, of finding many different left orderings of such groups. The construction involves equipping the surface with a hyperbolic structure, embedding the universal cover in the hyperbolic plane, and extending the action of the mapping class group on it to its limit points on the circle at infinity. We classify all orderings of braid groups which arise in this way. Moreover, restricting to a certain class of ``nonpathological'' orderings, we prove that there are only finitely many conjugacy classes of such orderings.
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In 1992 V.Drinfeld formulated a number of problems in quantum group theory. In particular, he suggested to consider ``set-theoretical'' solutions to the quantum Yang-Baxter equation, i.e. solutions given by a permutation R of the set X×XX\times X, where X is a fixed set. In this paper we study such solutions, which in addition satisfy the unitarity and nondegeneracy conditions. We discuss the geometric and algebraic interpretations of such solutions, introduce several constructions of them, and make first steps towards their classification.
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Maria Pia Sol\`er has recently proved that an orthomodular form that has an infinite orthonormal sequence is real, complex, or quaternionic Hilbert space. This paper provides an exposition of her result, and describes its consequences for Baer \ast-rings, infinite-dimensional projective geometries, orthomodular lattices, and Mackey's quantum logic. Comment: 30 pages
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We investigate the orderability properties of fundamental groups of 3-dimensional manifolds. Many 3-manifold groups support left-invariant orderings, including all compact P^2-irreducible manifolds with positive first Betti number. For seven of the eight geometries (excluding hyperbolic) we are able to characterize which manifolds' groups support a left-invariant or bi-invariant ordering. We also show that manifolds modelled on these geometries have virtually bi-orderable groups. The question of virtual orderability of 3-manifold groups in general, and even hyperbolic manifolds, remains open, and is closely related to conjectures of Waldhausen and others.
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We study left orderable groups by using dynamical methods. We apply these techniques to study the space of orderings of these groups. We show for instance that for the case of (non-Abelian) free groups, this space is homeomorphic to the Cantor set. We also study the case of braid groups (for which the space of orderings has isolated points but contains homeomorphic copies of the Cantor set). To do this we introduce the notion of the Conradian soul of an order as the maximal subgroup which is convex and restricted to which the original ordering satisfies the so called conradian property, and we elaborate on this notion. Comment: Final version, with updated references. Ann. Int. Fourier (to appear)
Algebraic analysis of many valued logics
  • Chang
Sheaves in geometry and logic
  • Mac Lane