On the little secondary Bruhat order

Let $R$ and $S$ be two sequences of positive integers in nonincreasing order having the same sum. We denote by ${\cal A}(R,S)$ the class of all $(0,1)$-matrices having row sum vector $R$ and column sum vector $S$. Brualdi and Deaett (More on the Bruhat order for $(0,1)$-matrices, Linear Algebra Appl., 421:219--232, 2007) suggested the study of the secondary Bruhat order on ${\cal A}(R,S)$ but with some constraints. In this paper, we study the cover relation and the minimal elements for this partial order relation, which we call the little secondary Bruhat order, on certain classes ${\cal A}(R,S)$. Moreover, we show that this order is different from the Bruhat order and the secondary Bruhat order. We also study a variant of this order on certain classes of symmetric matrices of ${\cal A}(R,S)$.

Relative formal topology: the binary positivity predicate comes first

In this paper we introduce relative formal topology so that we can deal constructively with the closed subsets of a topological space as well as with the open ones. In fact, within standard formal topology, the cover relation allows the definition of the formal opens, which are supposed to act as the formal counterparts of the open subsets, and within balanced formal topology, the binary positivity predicate allows the definition of the formal closed subsets, which are supposed to act as the formal counterparts of the closed subsets. However, these approaches are only fully satisfactory (according to the criterion introduced by the author in Valentini (2005)) if we can provide an adequate formalisation of the cover relation and the positivity predicate. But current formalisations fail in this respect since some intuitionistically valid relations between the open and closed subsets of a concrete topological space cannot be expressed. Our central result is that we can solve this problem through a generalisation of the standard cover relation together with a binary positivity predicate satisfying the positivity axiom.

Sublocales in formal topology

The paper studies how the localic notion of sublocale transfers to formal topology. For any formal topology (not necessarily with positivity predicate) we define a sublocale to be a cover relation that includes that of the formal topology. The family of sublocales has set-indexed joins. For each set of base elements there are corresponding open and closed sublocales, boolean complements of each other. They generate a boolean algebra amongst the sublocales. In the case of an inductively generated formal topology, the collection of inductively generated sublocales has coframe structure.Overt sublocales and weakly closed sublocales are described, and related via a new notion of “rest closed” sublocale to the binary positivity predicate. Overt, weakly closed sublocales of an inductively generated formal topology are in bijection with “lower powerpoints”, arising from the impredicative theory of the lower powerlocale.Compact sublocales and fitted sublocales are described. Compact fitted sublocales of an inductively generated formal topology are in bijection with “upper powerpoints”, arising from the impredicative theory of the upper powerlocale.

A structural investigation on formal topology: coreflection of formal covers and exponentiability

We present and study the category of formal topologies and some of its variants. Two main results are proven. The first is that, for any inductively generated formal cover, there exists a formal topology whose cover extends in the minimal way the given one. This result is obtained by enhancing the method for the inductive generation of the cover relation by adding a coinductive generation of the positivity predicate. Categorically, this result can be rephrased by saying that inductively generated formal topologies are coreflective into inductively generated formal covers.The second result is that unary formal covers are exponentiable in the category of inductively generated formal covers and hence, thanks to the coreflection, unary formal topologies are exponentiable in the category of inductively generated formal topologies.From a localic point of view the exponentiability of unary formal topologies means that algebraic dcpos are exponentiable in the category of open locales. But, the coreflection theorem states that open locales are coreflective in locales and hence, as a consequence of well-known impredicative results on exponentiable locales, it allows to prove that locally compact open locales are exponentiable in the category of open locales.

The Hahn-Banach theorem in type theory

We present the basic concepts and definitions needed in a pointfree approach to functional analysis via formal topology. Our main results are the constructive proofs of localic formulations of the Alaoglu and Helly-Hahn-Banach theorems. Earlier pointfree formulations of the Hahn-Banach theorem, in a topos-theoretic setting, were presented by Mulvey and Pelletier (1987, 1991) and by Vermeulen (1986). A constructive proof based on points was given by Bishop (1967). In the formulation of his proof, the norm of the linear functional is preserved to an arbitrary degree by the extension and a counterexample shows that the norm, in general, is not preserved exactly. As usual in pointfree topology, our guideline is to define the objects under analysis as formal points of a suitable formal space. After this has been accomplished for the reals, we consider the formal topology ℒ(A) obtained as follows. To the formal space of mappings from a normed vector space A to the reals, we add the linearity and norm conditions in the form of covering axioms. The linear functional of norm ≤1 from A to the reals then correspond to the formal points of this formal topology. Given a subspace M of A, the classical Helly-Hahn-Banach theorem states that the restriction mapping from the linear functionals on A of norm ≤1 to those on M is surjective. In terms of covers, conceived as deductive systems, it becomes a conservativity statement (cf. Mulvey and Pelletier 1991): whenever a is an element and U is a subset of the base of the formal space ℒ(M) and we have a derivation in ℒ(A) of a ⊲ U, then we can find a derivation in ℒ(M) with the same conclusion. With this formulation it is quite natural to look for a proof by induction on covers. Moreover, as already pointed out by Mulvey and Pelletier (1991), it is possible to simplify the problem greatly, since it is enough to prove it for coherent spaces of which ℒ(A) and ℒ(M) are retracts. Then, in a derivation of a cover, we can assume that only finite subsets occur on the right-hand side of the cover relation.

Contrasting metamorphic terranes near Chéticamp, Cape Breton Highlands, Nova Scotia

Two contrasting metamorphic terranes can be recognized in northwestern Cape Breton Island. One terrane (Pleasant Bay complex) consists of biotite gneiss and quartzite with minor calc-silicate lenses that were metamorphosed in Late Precambrian time (about 550 Ma) and were subsequently intruded by Silurian salic and mafic plutons that were, in turn, deformed and intruded by granite in Devonian time. The other terrane (Jumping Brook complex) consists of volcanogenic and sedimentary schists of probable Silurian age that were metamorphosed in Devonian time. P–T estimates indicate that the older parts of the Pleasant Bay complex were metamorphosed at about 790 °C and 7 kbar (1 kbar = 100 MPa) at low to moderate water fugacities during a major intrusive episode. The Jumping Brook complex exhibits a single progressive metamorphic sequence now disrupted by faulting. P–T conditions during this Devonian (370–390 Ma) metamorphism varied from greenschist (300 °C at <3 kbar) to amphibolite (650 °C at 4 kbar) facies. Metamorphism probably occurred in a thermal dome. The data suggest a moderately deformed basement–cover relation between the Pleasant Bay and Jumping Brook complexes.