We show that a O-l law holds for propositional modal logic, both for structure validity and frame validity. In the case of structure validity, the result follows easily from the well-known O-l law for first-order logic. However, our proof gives considerably more information. It leads to an elegant axiomatization for almost-sure structure validity and to sharper complexity bounds. Since frame va...

In the pointfree topological context of locales and frames, real functions on a locale L are represented as localic morphisms S(L)→ L(R) (i.e. frame homomorphisms L(R) → S(L)) where S(L) stands for the frame of all sublocales of L and L(R) denotes the frame of reals. This is reminiscent of dealing with (not necessarily continuous) real functions X → R as with continuous functions D(X)→ R where ...

A sectionally complemented modular lattice L is coordinatizable if it is isomorphic to the lattice L(R) of all principal right ideals of a von Neumann regular (not necessarily unital) ring R. We say that L has a large 4-frame if it has a homogeneous sequence (a0, a1, a2, a3) such that the neutral ideal generated by a0 is L. Jónsson proved in 1962 that if L has a countable cofinal sequence and a...

It is shown that some familiar properties of epimorphisms in the category of frames cary over to the category of uniform frames. This is achieved by suitably enriching certain frame homomorphisms to uniform frame homomorphisms. This note deals with the natural question, apparently never considered so far, whether certain familiar facts concerning epimorphisms, and specifically epi-extensions, o...

for all f ∈ H . The constant A (respectively, B) is a lower (resp. upper) frame bound for the frame. One of the most important frames for applications, especially signal processing, are the Weyl-Heisenberg frames. For g ∈ L(R) we define the translation parameter a > 0 and the modulation parameter b > 0 by: Embg(t) = e , Tnag(t) = g(t− na). For g ∈ L(R) and a, b > 0, we say for short that (g, a,...

If a modal logic L is finitely axiomatisable, then it is of course decidable whether a finite frame is a frame for L: one just has to check the finitely many axioms in it. If L is not finitely axiomatisable, then this might not be the case. For example, it is shown in [7] that the finite frame problem is undecidable for every L between the product logics K ×K ×K and S5 × S5 × S5. Here we show t...

‎In this paper‎, ‎we define and study the notion of zero elements in topoframes; a topoframe is a pair‎ ‎$(L‎, ‎tau)$‎, ‎abbreviated $L_{ tau}$‎, ‎consisting of a frame $L$ and a‎ ‎subframe $ tau $ all of whose elements are complemented elements in‎ ‎$L$‎. ‎We show that‎ ‎the $f$-ring $ mathcal{R}(L_tau)$‎, ‎the set of $tau$-real continuous functions on $L$‎, ‎is uniformly complete‎. ‎Also‎, ‎t...

It is considered the procedure of definition of the elements of length and time by means of photon. In order to define the element of length one needs an a priori given element of time. In order to define the element of time one needs an a priori given element of length. It is taken the laboratory time and length as a priori given time and length respectively. It is defined the elements of leng...

The excess of a sequence in a Hilbert space H is the greatest number of elements that can be removed yet leave a set with the same closed span. This paper proves that if F is a frame for H and there exist infinitely many elements gn ∈ F such that F \ {gn} is complete for each individual n and if there is a uniform lower frame bound L for each frame F \ {gn}, then for each ε > 0 there exists an ...