The theory of abelian totally ordered groups has a model completion. We show that the theory of abelian lattice-ordered groups has no model companion. Indeed, the Archimedean property can be captured by a first order V3V sentence for existentially complete abelian lattice-ordered groups, and distinguishes between finitely generic abelian lattice-ordered groups and infinitely generic ones. We th...

(See e.g. [6] for background.) When X is equipped with a distinguished basis D for its topology, closed under finite meets and joins, one can investigate situations where D is also closed under the implication (2), i.e., where D is a Heyting subalgebra of O (X). Recall that X is a spectral space if it is compact and T0, its collection D of compact open subsets forms a basis which is closed unde...

(See e.g. [6] for background.) When X is equipped with a distinguished basis D for its topology, closed under finite meets and joins, one can investigate situations where D is also closed under the implication (2), i.e., where D is a Heyting subalgebra of O (X). Recall that X is a spectral space if it is compact and T0, its collection D of compact open subsets forms a basis which is closed unde...

(Day 1): References (for general background) →Varieties of lattice-ordered groups, N. R. Reilly, in Lattice-Ordered Groups, Advances and Techniques, A. M. W. Glass and W. C. Holland (eds.), Kluwer Academic Publishers, 1989. Lattice-Ordered Groups, an Introduction, M. Anderson and T. Feil, D. Reidel Pub. Co., 1988. Theory of Lattice-Ordered Groups, M. Darnel, Marcel Dekker, 1995. Partially Order...

At present, the determination of crystal structures from data that have been acquired from twinned crystals is routine; however, with the increasing number of crystal structures additional crystal lattice disorders are being discovered. Here, a previously undescribed partial rotational order-disorder that has been observed in crystals of stefin B is described. The diffraction images revealed no...

a group $g$ is called metahamiltonian if all its non-abelian subgroups are normal‎. ‎the aim of this paper is to provide an updated survey of research concerning certain classes of generalized metahamiltonian groups‎, ‎in various contexts‎, ‎and to prove some new results‎. ‎some open problems are listed‎.

Derivative lattices associated with partially ordered linear spaces over skew fields are considered. Properties of the convex projective geometry $$ \mathcal{L} for a space FV field F investigated. The convexity subspaces means Abelian (ab-convexity), which is based on definition subgroup group. It shown that ab-convex directed plays theory same role as subgroups groups. We obtain element-wise ...