In this paper we formalized some theorems concerning the cyclic groups of prime power order. We formalize that every commutative cyclic group of prime power order is isomorphic to a direct product of family of cyclic groups [1], [18]. Let G be a finite group. The functor Ordset(G) yielding a subset of N is defined by the term (Def. 1) the set of all ord(a) where a is an element of G. One can ch...

For a fairly general reductive group G/Qp , we explicitly compute the space of locally algebraic vectors in the Breuil–Herzig construction (ρ)ord , for a potentially semistableBorel-valued representationρ ofGal(Q̄p/Qp). The point beingwedealwith thewhole representation, not just its socle—and we go beyond GLn(Qp). In the case of GL2(Qp), this relation is one of the key properties of the p-adic l...

A [0, m]-space is a linear space with the following property: For any point-line pair (x, G) there are at most m lines through x which are coplanar with G and which have no point in common with G. For every [0, m]-space (M, 90l) we define an order o r d M in a natural way. For dimM~>3 and o r d M ~ > 3 m + 2 , every [0, m]-space (M, 991) can be embedded in a projective space (P, £) with dim P =...

Write OK for the valuation ting, MK for the maximal ideal of K and kK for the residue field. Let us fix $ some uniformizer of K. We denote by acm : K → OK/(MK) the map sending some nonzero x ∈ K to x$−ord(x) mod MK , and sending zero to zero. This is a definable map. We denote by RV the union of K×/(1 +MK) and {0} and by rv : K → RV the quotient map. More generally, if m ∈ N∗, we set RVm = K×/(...

For a fairly general reductive group G/Qp , we explicitly compute the space of locally algebraic vectors in the Breuil-Herzig construction Π(ρ)ord, for a potentially semistable Borel-valued representation ρ of Gal(Q̄p/Qp). The point being we deal with the whole representation, not just its socle – and we go beyond GLn(Qp). In the case of GL2(Qp), this relation is one of the key properties of the...

This paper continues Nebel and B urckert's investigation of Allen's interval algebra by presenting nine more maximal tractable subclasses of the algebra (provided that P 6 = NP), in addition to their previously reported ORD-Horn subclass. Furthermore, twelve tractable subclasses are identiied, whose max-imality is not decided. Four of these can express the notion of sequentiality between interv...