Centrally orthocomplete effect algebras

Motivated by the theory of Loomis dimension lattices, we generalize the notion of a hull mapping to an arbitrary effect algebra (EA). Using hull mappings, we identify certain special types of elements in an EA, including generalizations of the invariant elements and of the simple elements in a dimension lattice. We introduce and study a new class of effect algebras, called centrally orthocomplete effect algebras (COEAs), satisfying the condition that every family of elements that is dominated by an orthogonal family of central elements has a supremum. We show that COEAs admit a central cover mapping and we develop the basic theory of direct decomposition of COEAs.     

Musings about the Triple Representation Theorem for Effect Algebras

The aim of our paper is to analyze the assumptions under which the Triple Representation Theorem for effect algebras holds.     

效应代数的表示理论

The aim of this paper is to characterize representable and weak representable effect algebras and establish a representation theory of effect algebras. An effect algebra E is said to be representable if there exists a Hilbert space H and a monomorphism π from E into the Hilbert space effect algebra ε(H) and it is said to be weakly representable if there exists an injective morphism from E into some ε(H). It is proved that an effect algebra E with the nonempty state space S(E) is representable if and only if x, y ∈ E, f(x)+f(y) ≤ 1 implies x⊕y is defined; it is weakly representable if and only if the state space S(E) separates the points of E. Some operational properties of representable effect algebras are established, and some applications of the obtained results are listed.     

Representation theorems for Banach algebras

For a space X   denote by _Cb(X)$C_b(X)$ the Banach algebra of all continuous bounded scalar-valued functions on X   and denote by _C0(X)$C_0(X)$ the set of all elements in _Cb(X)$C_b(X)$ which vanish at infinity.     

V-quantale的表示定理

研究了V-quantale上的(+)和→运算, 得到了强德摩根律的等价刻画; 最后, 给出了V-quantale的表示定理.     

Serre Theorem for involutory Hopf algebras

We call a monoidal category C a Serre category if for any C, D ∈ C such that C ⊗ D is semisimple, C and D are semisimple objects in C. Let H be an involutory Hopf algebra, M, N two H-(co)modules such that M ⊗ N is (co)semisimple as a H-(co)module. If N (resp. M) is a finitely generated projective k-module with invertible Hattory-Stallings rank in k then M (resp. N) is (co)semisimple as a H-(co)module. In particular, the full subcategory of all finite dimensional modules, comodules or Yetter-Drinfel’d modules over H the dimension of which are invertible in k are Serre categories.     

Representation and duality for Hilbert algebras

In this paper we introduce a special kind of ordered topological spaces, called Hilbert spaces. We prove that the category of Hilbert algebras with semi-homomorphisms is dually equivalent to the category of Hilbert spaces with certain relations. We restrict this result to give a duality for the category of Hilbert algebras with homomorphisms. We apply these results to prove that the lattice of the deductive systems of a Hilbert algebra and the lattice of open subsets of its dual Hilbert space, are isomorphic. We explore how this duality is related to the duality given in [6] for finite Hilbert algebras, and with the topological duality developed in [7] for Tarski algebras.      

Representation Theorem for Mild C-existence Families

As we know,the representation theorem is one of the most important theorems foroperator semigroup theory.The representation theorem for exponentially bounded C-semigroup with dense range R(C) was given in [1 ] ,which was generalized to the casethat C does notnecessarily have dense range(see[2 ] ) .In this paper,more general caseis considered.We get a representation theorem for exponentially bounded mild C-exis-tence families.According to this,a simple proof of a construction theorem for C-s…     

Chain homogeneous Souslin algebras

Assuming Jensen's principle ?⁺ we construct Souslin algebras all of whose maximal chains are pairwise isomorphic as total orders, thereby answering questions of Koppelberg and Todor?evi?.     

A Representation Stability Theorem for VI-modules

Let VI be the category whose objects are the finite dimensional vector spaces over a finite field of order q and whose morphisms are the injective linear maps. A VI-module over a ring is a functor from the category VI to the category of modules over the ring. A VI-module gives rise to a sequence of representations of the finite general linear groups. We prove that the sequence obtained from any finitely generated VI-module over an algebraically closed field of characteristic zero is representation stable - in particular, the multiplicities which appear in the irreducible decompositions eventually stabilize. We deduce as a consequence that the dimension of the representations in the sequence {V _n } obtained from a finitely generated VI-module V over a field of characteristic zero is eventually a polynomial in q ⁿ . Our results are analogs of corresponding results on representation stability and polynomial growth of dimension for FI-modules (which give rise to sequences of representations of the symmetric groups) proved by Church, Ellenberg, and Farb.     

A Representation Theorem for Geometric Morphisms

It it shown that geometric morphisms between elementary toposes can be represented as certain adjunctions between the corresponding categories of locales. These adjunctions are characterized by (i) they preserve the order enrichment and the Sierpiński locale, and (ii) they satisfy Frobenius reciprocity.     

Principal homogeneous spaces for arbitrary Hopf algebras

LetH be a Hopf algebra over a field with bijective antipode,A a rightH-comodule algebra,B the subalgebra ofH-coinvariant elements and can:A ⊗ _B A →A ⊗H the canonical map. ThenA is a faithfully flat (as left or rightB-module) Hopf Galois extension iffA is coflat asH-comodule and can is surjective (Theorem I). This generalizes results on affine quotients of affine schemes by Oberst and Cline, Parshall and Scott to the case of non-commutative algebras. The dual of Theorem I holds and generalizes results of Gabriel on quotients of formal schemes to the case of non-cocommutative coalgebras (Theorem II). Furthermore, in the dual situation, a normal basis theorem is proved (Theorem III) generalizing results of Oberst-Schneider, Radford and Takeuchi.     

The Birkhoff Variety Theorem for continuous algebras

Varieties of continuous algebras, i.e., classes presented by (in)equalities for terms, are characterized as precisely the HSP classes. Terms here are the usual algebraic terms enriched by iterated join symbols. A discussion of varieties which require only finitely many variables, or only the usual terms, is also presented.Presented by W. Taylor.     

Infinite homogeneous algebras are anticommutative

A (non-associative) algebra , over a field , is called homogeneous if its automorphism group permutes transitively the one dimensional subspaces of . Suppose is a nontrivial finite dimensional homogeneous algebra over an infinite field. Then we prove that for all in , and so for all .

     

Representation Theorem for Generators of Quadratic BSDEs

In this paper, we establish a general representation theorem for generator of backward stochastic differential equation (BSDE), whose generator has a quadratic growth in z. As some applications, we obtain a general converse comparison theorem of such quadratic BSDEs and uniqueness theorem, translation invariance for quadratic g-expectation.