Let G be a group and D be a subgroup of G. In this paper we are interested in the description of the lattice L(D; G) = fH j D H Gg of subgroups of G which contain D. We call subgroups H 2 L(D; G) subgroups intermediate between D and G. We are primarily interested in the parametrization of intermediate subgroups and their behaviour with respect to set-theoretic and group-theoretic operations lik...

A CAT(−1)-space is a metric geodesic space in which every geodesic triangle is thinner than its associated comparison triangle in the hyperbolic plane ([B], [BriHa], [Gr]). The CAT(−1)-property is one among many possible generalizations to singular spaces of the notion of negative curvature. Important examples of CAT(−1)-spaces include Riemannian manifolds of sectional curvature k ≤ −1 and thei...

Lattice games are real-valued functions de...ned on a ...nite lattice L. The basic players are the nonzero join-irreducible elements of the lattice and the coalitions are its elements. If L is the Boolean algebra 2 then we obtain a n-person game. Gilboa and Lehrer introduced the global games, which are lattice games where L = ¦n, the lattice of all partitions of N ordered by re...nement. Faigle...

This note aims to clarify what conditions on a connected Lie group G imply that its maximal connected normal solvable subgroup R intersects each lattice of G as a lattice in R.

Freese and J6nsson [8] showed that the congruence lattice of a (universal) algebra in a congruence modular variety is always arguesian. On the other hand J6nsson [16] constructed arguesian lattices which cannot be embedded into the normal subgroup lattice of a group. These lattices consist of two arguesian planes of different prime order glued together over a two element sublattice (cf. Dilwort...

The concepts of $L$-convex systems and Scott-hull spaces are proposed on frame-valued setting. Also, we establish the categorical isomorphism between  $L$-convex systems and Scott-hull spaces. Moreover, it is proved that the category of $L$-convex structures is  bireflective in the category of $L$-convex systems. Furthermore, the quotient systems of $L$-convex systems are studied.

An rj-normal lattice is a distributive lattice with 0 such that each prime ideal contains at most n minimal prime ideals. A relatively ra-normal lattice is a distributive lattice such that each bounded closed interval is an /¡.-normal lattice. The main results of this paper are: (1) a distributive lattice L with 0 is 71,-normal if and only if for any %n, x,,*•', X e L such that x Ax. = 0 for an...

For a partially ordered set P , we denote by Co(P ) the lattice of order-convex subsets of P . We find three new lattice identities, (S), (U), and (B), such that the following result holds: Theorem. Let L be a lattice. Then L embeds into some lattice of the form Co(P ) iff L satisfies (S), (U), and (B). Furthermore, if L has an embedding into some Co(P ), then it has such an embedding that pres...

By extracting combinatorial structures in well-solved nonlinear combinatorial optimization problems, Murota (1996,1998) introduced the concepts of M-convexity and L-convexity to functions defined over the integer lattice. Recently, Murota–Shioura (2000, 2001) extended these concepts to polyhedral convex functions and quadratic functions defined over the real space. In this paper, we consider a ...

In Part I of this paper, we introduced a method of making two isomorphic intervals of a bounded lattice congruence equivalent. In this paper, we make one interval dominate another one. Let L be a bounded lattice, let [a, b] and [c, d] be intervals of L, and let φ be a homomorphism of [a, b] onto [c, d]. We construct a bounded (convex) extension K of L such that a congruence Θ of L has an extens...