We present Kripke semantics for some substructural logics with weakening, known as logics with BCK implication. This work is a continuation of the work of Allwein and Dunn on Kripke semantics for Linear Logic, which in turn rested on Dunn’s Gaggle Theory and on Urquhart’s Representation Theory for nondistributive lattices. The basic idea in the representation theory is to use maximally disjoint...

Let k be a finite field with q elements, and let k̄ be an algebraic closure of k. We consider the field L := k̄((ǫ)) and its subfield F := k((ǫ)). We write σ : x 7→ x for the Frobenius automorphism of k̄/k, and we also regard σ as an automorphism of L/F in the usual way, so that σ( ∑ anǫ ) = ∑ σ(an)ǫ . We write o for the valuation ring k̄[[ǫ]] of L. Let G be a split connected reductive group over k...

we derive the formulas of the maximal andminimal ranks of four real matrices $x_{1},x_{2},x_{3}$ and $x_{4}$in common solution $x=x_{1}+x_{2}i+x_{3}j+x_{4}k$ to quaternionmatrix equations $a_{1}x=c_{1},xb_{2}=c_{2},a_{3}xb_{3}=c_{3}$. asapplications, we establish necessary and sufficient conditions forthe existence of the common real and complex solutions to the matrixequations. we give the exp...

For certain classes of Prüfer domains A, we study the completion Â,T ofA with respect to the supremum topology T = sup{Tw|w ∈ Ω}, where Ω is the family of nontrivial valuations on the quotient field which are nonnegative on A and Tw is a topology induced by a valuation w ∈ Ω. It is shown that the concepts ‘SFT Prüfer domain’ and ‘generalized Dedekind domain’ are the same. We show that if E is t...

In a temporal database, each data tuple is accompanied by a time interval during which its attribute values are valid. In this paper, we consider the null time intervals, that is, time intervals not intersected by any time intervals in the temporal database. We deal with the problem of computing temporal aggregates over null time intervals with length constraints. By interval folding, we transf...

Let R be a discrete valuation ring with maximal ideal πR and residue field k. We study obstructions to lifting a Lie algebra L over R/πR to one over R/πR. If L̄ is the Lie algebra over k obtained from reducing L then we show there exists a well-defined class in H(L̄, ad) which vanishes if and only if L lifts. Furthermore, if L lifts, the lifts are shown to be in one to one correspondence with H(L̄...

Let F be a local complete field with discrete valuation, and let G be a quasisplit group over F which splits over some unramified extension of F . Let P be a parahoric subgroup of the group G(F ) of F -points of G; the open normal pronilpotent subgroups of P can be classified using the standard normal filtration subgroups of Prasad and Raghanathan. More precisely, we show that if G is quasisimp...

We consider dynamical systems arising from iterating a morphism of a projective variety defined over the field of fractions of a discrete valuation ring. Our goal is to obtain information about the dynamical system over the field of fractions by studying the dynamical system over the residue field. In particular, we aim to bound the possible primitive periods for a periodic point. This is discu...

Let R be an n-dimensional regular local domain essentially of finite type over a ground field k of characteristic zero, and let ν be a rank one valuation centered on R. Recall that this is equivalent to asking that ν be an R-valued valuation on the fraction field K of R, taking non-negative values on R and positive values on the maximal ideal m ⊆ R. A theorem of Zariski and Abhyankar states tha...

The notion and applicability of ethics in the public sphere have become a relatively obscure matter with prevailing incursion political secularism. functioning model contemporary polity leads us to believe that play no role politics, whether at national or international levels. Ethics is not predominant concept leading relations theories, such as realism liberalism. Pacing these for which power...