Subdirect Decompositions of Lattice Effect Algebras

We prove a theorem about subdirect decompositions of lattice effect algebras. Further, we show how, under these decompositions, blocks, sets of sharp elements and centers of those effects algebras are decomposed. As an application we prove a statement about the existence of subadditive state on some block-finite effect algebras.     

Open Problems for Sequential Effect Algebras

A sequential effect algebra (SEA) is an effect algebra on which a sequential product with certain natural properties is defined. In such structures, we can study combinations of simple measurements that are series as well as parallel. This article presents some open problems for SEAs together with background material, comments and partial results. Two examples of open problems are the following: is A° B = A ^1/2 BA ^1/2 the only sequential product on a Hilbert space SEA? It is known that the sharp elements of a SEA form an orthomodular poset. Is every orthomodular poset isomorphic to the set of sharp elements for some SEA?     

Lattice Uniformities on Effect Algebras

Let L be a lattice ordered effect algebra. We prove that the lattice uniformities on L which make uniformly continuous the operations − and + of L are uniquely determined by their system of neighborhoods of 0 and form a distributive lattice. Moreover we prove that every such uniformity is generated by a family of weakly subadditive [0,+∞]-valued functions on L.     

Tensor Product of Distributive Sequential Effect Algebras and Product Effect Algebras

A distributive sequential effect algebra (DSEA) is an effect algebra on which a distributive sequential product with natural properties is defined. We define the tensor product of two arbitrary DSEA’s and we give a necessary and sufficient condition for it to exist. As a corollary we obtain the result (see Gudder, S. in Math. Slovaca 54:1–11, 2004, to appear) that the tensor product of a pair of commutative sequential effect algebras exists if and only if they admit a bimorphism. We further obtain a similar result for the tensor product of a pair of product effect algebras.     

Ideal Topology on Effect Algebras

The ideals of effect algebras induce a topology on effect algebras. The operations and of effect algebras are continuous with respect to this topology.     

Two Variables Operation Continuity of Effect Algebras

In this paper, we study two variables operation continuity of lattice effect algebra with respect to its order topology.PACS: 02.10 By, 02.10 Gd     

Congruences and States on Pseudoeffect Algebras

We study congruences on pseudoeffect algebras, which were recently introduced as a non-commutative generalization of effect algebras. We introduce ideals for these algebras and give a sufficient condition for an ideal to determine a congruence. Furthermore, states on pseudoeffect algebras are considered. It is shown that any interval pseudoeffect algebra maps homomorphically into an effect algebra whose states are in a one-to-one correspondence to the states of the original algebra.     

Existence of States on Pseudoeffect Algebras

Pseudoeffect (PE) algebras have been introduced as a noncommutative generalization of effect algebras. We study in this paper PE algebras with the special property of having a nonempty state space. To this end, we consider PE algebras which are po-group intervals and which are, in a certain sense, noncommutative only in the small. Such a PE algebra is shown to possess a nontrivial commutative homomorphic image from which then follows that there exist states. A typical example is given by an interval of the lexicographical product of two po-groups the first of which is abelian.     

The Brooks-Jewett Theorem on Effect Algebras with the Sequential Completeness Property

In this paper, we show that the Brooks-Jewett theorem on effect algebras with the sequential completeness property is valid.     

Central Elements of Effect Algebras

Central elements of an effect algebra can be characterized by means of a weak form of distributivity and a maximality property. We give examples where both conditions are fulfilled.     

De Morgan Property for Effect Algebras of von Neumann Algebras

It is shown that the unit interval of a von Neumann algebra is a Sum Brouwer–Zadeh algebra when equipped with another unary operation sending each element to the complement of its range projection. The main result of this Letter says that a von Neumann algebra is finite if and only if the corresponding Brouwer–Zadeh structure is de Morgan or, equivalently, if the range projection map preserves infima in the unit interval. This provides a new characterization of finiteness in the Murray–von Neumann structure theory of von Neumann algebras in terms of Brouwer–Zadeh structures.     

Braided m-Lie Algebras

Braided m-Lie algebras induced by multiplication are introduced, which generalize Lie algebras, Lie color algebras and quantum Lie algebras. The necessary and sufficient conditions for the braided m-Lie algebras to be strict Jacobi braided Lie algebras are given. Two classes of braided m-Lie algebras are given, which are generalized matrix braided m-Lie algebras and braided m-Lie subalgebras of End _F M, where M is a Yetter–Drinfeld module over B with dimB < . In particular, generalized classical braided m-Lie algebras sl _{q, f} (GM _G (A), F) and osp _{q, t} (GM _G (A), M, F) of generalized matrix algebra GM _G (A) are constructed and their connection with special generalized matrix Lie superalgebra sl _{s, f} (GM _{Z_2}(A ^s ), F) and orthosymplectic generalized matrix Lie super algebra osp _{s, t} (GM _{Z_2}(A ^s ), M ^s , F) are established. The relationship between representations of braided m-Lie algebras and their associated algebras are established.This revised version was published online in March 2005 with corrections to the cover date.     

Uniqueness and Order in Sequential Effect Algebras

A sequential effect algebra (SEA) is an effect algebra on which a sequential product is defined. We present examples of effect algebras that admit a unique, many and no sequential product. Some general theorems concerning unique sequential products are proved. We discuss sequentially ordered SEAs in which the order is completely determined by the sequential product. It is demonstrated that intervals in a sequential ordered SEA admit a sequential product.     

Irreducibility and Compositeness in q-Deformed Harmonic Oscillator Algebras

A study of the reducibility of the Fock space representation of the q-deformed harmonic oscillator algebra for real and root of unity values of the deformation parameter is carried out by using the properties of the Gauss polynomials. When the deformation parameter is a root of unity, an interesting result comes out in the form of a reducibility scheme for the space representation which is based on the classification of the primitive or nonprimitive character of the deformation parameter. An application is carried out for a q-deformed harmonic oscillator Hamiltonian, to which the reducibility scheme is explicitly applied.On leave from     

Central Elements of Atomic Effect Algebras

Various conditions ensuring that an atomic effect algebra is a Boolean algebra are presented. PACS: 02.10.-v.     

Irreducible Representations of a Class of Current Algebras of Etingof and Frenkel

A class of representations is described for the central extensions, found by Etingof and Frenkel of current algebras over Riemann surfaces. Their irreducibility is proved. The possibility/impossibility to obtain integrable representations within that class is discussed briefly.     

Pre-Poisson Algebras

A definition of pre-Poisson algebras is proposed, combining structures of pre-Lie and zinbiel algebra on the same vector space. It is shown that a pre-Poisson algebra gives rise to a Poisson algebra by passing to the corresponding Lie and commutative products. Analogs of basic constructions of Poisson algebras (through deformations of commutative algebras, or from filtered algebras whose associated graded algebra is commutative) are shown to hold for pre-Poisson algebras. The Koszul dual of pre-Poisson algebras is described. It is explained how one may associate a pre-Poisson algebra to any Poison algebra equipped with a Baxter operator, and a dual pre-Poisson algebra to any Poisson algebra equipped with an averaging operator. Examples of this construction are given. It is shown that the free zinbiel algebra (the shuffle algebra) on a pre-Lie algebra is a pre-Poisson algebra. A connection between the graded version of this result and the classical Yang–Baxter equation is discussed.     

Idempotents of Clifford Algebras

A classification of idempotents of Clifford algebras C ^p,q is presented. It is shown that using isomorphisms between Clifford algebras C ^p,q and appropriate matrix rings, it is possible to classify idempotents in any Clifford algebra into continuous families. These families include primitive idempotents used to generate minimal one-sided ideals in Clifford algebras. Some low-dimensional examples are discussed.     

Product Effect Algebras

We introduce a product on an effect algebra. We prove that every product effect algebra with the Riesz decomposition property (RDP), is an interval in an Abelian unital interpolation po-ring, and we show that the category of product effect algebras with the RDP is categorically equivalent with the category of unital Abelian interpolation po-rings. In addition, we show that every product effect algebra with the RDP and with 1 as a product unity is a subdirect product of antilattice product effect algebras with the RDP.