66 research outputs found
On scattered convex geometries
A convex geometry is a closure space satisfying the anti-exchange axiom. For
several types of algebraic convex geometries we describe when the collection of
closed sets is order scattered, in terms of obstructions to the semilattice of
compact elements. In particular, a semilattice , that does not
appear among minimal obstructions to order-scattered algebraic modular
lattices, plays a prominent role in convex geometries case. The connection to
topological scatteredness is established in convex geometries of relatively
convex sets.Comment: 25 pages, 1 figure, submitte
On structures in hypergraphs of models of a theory
We define and study structural properties of hypergraphs of models of a
theory including lattice ones. Characterizations for the lattice properties of
hypergraphs of models of a theory, as well as for structures on sets of
isomorphism types of models of a theory, are given
On scattered convex geometries
A convex geometry is a closure space satisfying the anti-exchange axiom. For several
types of algebraic convex geometries we describe when the collection of closed sets is order scattered, in terms of obstructions to the semilattice of compact elements. In particular, a semilattice ( ), that does not appear among minimal obstructions to order-scattered algebraic modular lattices, plays a prominent role in convex geometries case. The connection to topological scatteredness is established in convex geometries of relatively convex set
Distributive lattice models of the type C one-rowed Weyl group symmetric functions
We present two families of diamond-colored distributive lattices – one known and one new – that we can show are models of the type C one-rowed Weyl symmetric functions. These lattices are constructed using certain sequences of positive integers that are visualized as filling the boxes of one-rowed partition diagrams. We show how natural orderings of these one-rowed tableaux produce our distributive lattices as sublattices of a more general object, and how a natural coloring of the edges of the associated order diagrams yields a certain diamond-coloring property. We show that each edge-colored lattice possesses a certain structure that is associated with the type C Weyl groups. Moreover, we produce a bijection that shows how any two affiliated lattices, one from each family, are models for the same type C one-rowed Weyl symmetric function. While our type C one-rowed lattices have multiple algebraic contexts, this thesis largely focusses on their combinatorial aspects
Independence and totalness of subspaces in phase space methods
YesThe concepts of independence and totalness of subspaces are introduced
in the context of quasi-probability distributions in phase
space, for quantum systems with finite-dimensional Hilbert space.
It is shown that due to the non-distributivity of the lattice of
subspaces, there are various levels of independence, from pairwise
independence up to (full) independence. Pairwise totalness,
totalness and other intermediate concepts are also introduced,
which roughly express that the subspaces overlap strongly among
themselves, and they cover the full Hilbert space. A duality between
independence and totalness, that involves orthocomplementation
(logical NOT operation), is discussed. Another approach to independence
is also studied, using Rota’s formalism on independent
partitions of the Hilbert space. This is used to define informational
independence, which is proved to be equivalent to independence.
As an application, the pentagram (used in discussions on contextuality)
is analysed using these concepts
Congruences on lattices (with application to amalgamation)
Bibliography: pages 124-128.We present some aspects of congruences on lattices. An overview of general results on congruence distributive algebras is given in Chapter 1 and in Chapter 2 we examine weak projections; including Dilworth's characterization of congruences on lattices and a finite basis theorem for lattices. The outstanding problem of whether congruence lattices of lattices characterize distributive algebraic lattices is discussed in Chapter 3 and we look at some of the partial results known to date. The last chapter (Chapter 6) characterizes the amalgamation class of a variety B generated by a B-lattice, B, as the intersection of sub direct products of B, 2-congruence extendible members of B and 2-chain limited members of B. To this end we consider 2-congruence extendibility in Chapter 4 and n-chain limited lattices in Chapter 5. Included in Chapter 4 is the result that in certain lattice varieties the amalgamation class is contained in the class of 2-congruence extendible members of the variety. A final theorem in Chapter 6 states that the amalgamation class of a B-lattice variety is a Horn class
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