A dynamic Effect Algebras with Dual Operation

We study remarkable sub-lattice effect algebras of Archimedean atomic lattice effect algebras E, namely their blocks M, centers C(E), compatibility centers B(E) and sets of all sharp elements S(E) of E. We show that in every such effect algebra E, every atomic block M and the set S(E) are bifull sub-lattice effect algebras of E. Consequently, if E is moreover sharply dominating then every atomic block M is again sharply dominating and the basic decompositions of elements (BDE of x) in E and in M coincide. Thus in the compatibility center B(E) of E, nonzero elements are dominated by central elements and their basic decompositions coincide with those in all atomic blocks and in E. Some further details which may be helpful under answers about the existence and properties of states are shown. Namely, we prove the existence of an (o)-continuous state on every sharply dominating Archimedean atomic lattice effect algebra E with $$B(E)\not =C(E).$$ Moreover, for compactly generated Archimedean lattice effect algebras the equivalence of (o)-continuity of states with their complete additivity is proved. Further, we prove “State smearing theorem” for these lattice effect algebras.

In [Riečanová Z, Zajac M.: Hilbert Space Effect-Representations of Effect Algebras] it was shown that an effect algebra E with an ordering set M of states can by embedded into a Hilbert space effect algebra E(l<sub>2</sub>(M)). We consider the problem when its effect algebraic MacNeille completion Ê can be also embedded into the same Hilbert space effect algebra E(l<sub>2</sub>(M)). That is when the ordering set M of states on E can be be extended to an ordering set of states on Ê. We give an answer for all Archimedean MV-effect algebras and Archimedean atomic lattice effect algebras.

The lattice CongD of all dynamic congruences on a given dynamic algebra D is presented. Whenever D is separable with zero we define dynamic ideal on D, given rise to the lattice IdeD. The notions of kernel of a dynamic congruence and the congruence generated by a dynamic ideal are introduced to describe a Galois connection between CongD and IdeD. We study conditions under which a dynamic congruence is determined by its kernel.

The present paper is an attempt to suggest and scrutinize tense operators in the dynamic logic $textbf{B}$ which is regarded as a set of propositions about the general fuzzy automaton $ tilde{F} $, in which its underlying structure has been a bounded poset. Here, the operators $ T_{delta}, P_{delta}, H_{delta}$ and $ F_{delta} $ are proposed regardless of what propositional connectives the logic comprises. For this purpose, the axiomatization of universal quantifiers is applied as a starting point and these axioms are modified. In this study, firstly, we demonstrate that the operators can be identified as modal operators and the pairs $ (T_{delta},P_{delta}) $ are examined as the so-called dynamic pairs. In addition, constructions of these operators are attained in the corresponding algebra and in the following a transition frame is suggested. Besides, the problem of finding a transition frame is solved in the case when the tense operators are given. Specifically, this study shows that the tense algebra $ textbf{B} $ is representable in its Dedekind-MacNeille completion. Representation theorems for dynamic and tense algebra are explicated in details in the related given theorems.

An effect algebra is a partial algebraic structure, originally formulated as an algebraic base for unsharp quantum measurements. In this article we present an approach to the study of lattice effect algebras (LEAs) that emphasizes their structure as algebraic models for the semantics of (possibly) non-standard symbolic logics. This is accomplished by focusing on the interplay among conjunction, implication, and negation connectives on LEAs, where the conjunction and implication connectives are related by a residuation law. Special cases of LEAs are MV-algebras and orthomodular lattices. The main result of the paper is a characterization of LEAs in terms of so-called Sasaki algebras. Also, we compare and contrast LEAs, Hajek's BL-algebras, and the basic algebras of Chajda, Halas, and Kuhr.

We solve a characterization problem for dual Hardy-space Toeplitz operators on the unit sphere \({\mathbb{S}_{n}}\) in \({\mathbb{C}^{n}}\) posed by Guediri (Acta Math Sin (English series) 29(9):1791–1808, 2013). Our proof relies on the observation that dual Toeplitz operators on the orthogonal complement \({H^{2}(\mathbb{S}_{n})^{\bot}}\) of the Hardy space in L2 can be viewed as Toeplitz operators with respect to a suitable spherical isometry. This correspondence also allows us to determine the commutator ideal of the dual Toeplitz C*-algebra.

Pseudoeffect (PE-) algebras are partial algebras differing from effect algebras in that they need not satisfy the commutativity assumption. PE-algebras typically arise from intervals of po-groups; this applies in particular to all those which satisfy a certain Riesz property.

Erik M. Alfsen and Frederic W. Shultz had recently developed the characterisation of state spaces of operator algebras. It established full equivalence (in the mathematical sense) between the Heisenberg and the Schrodinger picture, i.e. given a physical system we are able to construct its state space out of its observables as well as to construct algebra of observables from its state space. As an underlying mathematical structure they used the theory of duality of ordered linear spaces and obtained results are valid for various types of operator algebras (namely C*, von Neumann, JB and JBW algebras). Here, we show that the language they developed also admits a representation of an effect algebra.

G. Gr¨atzer’s gives the following example in his monograph «Universal algebra». Let A be a universal algebra (with some family of operations Σ). Let us take an arbitrary set B ⊆ A. For all of the operations f ∈ Σ (let n be the arity of f) let us look how f transformas the elements of Bn. It is not necessary that f(B) ⊆ B, so in the general case B is not a subalgebra of A. But if we define partial operation as mapping from a subset of the set Bn into the set B. then B be a set with a family of partial operations defined on it. Such sets are called partial universal algebras. In our example B will be a partial universal subalgebra of the algebra A, which means the set B will be closed under all of the partial operations of the partial algebra B. So, partial algebras can naturally appear when studying common universal algebras. The concept of congruence of universal algebra can be generalized to the case of partial algebras. It is well-known that the congruences of a partial universal algebra A always from a lattice, and if A be a full algebra (i.e. an algebra) then the lattice of the congruences of A is a sublattice of the lattice of the equivalence relations on A. The congruence lattice of a partial universal algebra is its important characteristics. For the most important cases of universal algebra some results were obtained which characterize the algebras A without any congruences except the trivial congruences (the equality relation on A and the relation A2). It turned out that in the most cases, when the congruence lattice of a universal algebra is trivial the algebra itself is definitely not trivial. And what can we say about the algebras A whose equivalence relation is, vice versa, contains all of the equivalence relations on A? It turns out, in this case any operation f of the algebra A is either a constant (|f(A)| = 1) or a projection (f(x1, ..., xi, ..., xn) ≡ xi). Kozhukhov I. B. described the semigroups whose equivalence relations are one-sided congruences. It is interesting now to generalize these results to the case of partial algebras. In this paper the partial n-ary groupoids G are studied whose operations f satisfy the following condition: for any elements x1, ..., xk−1, xk+1, ..., xn ∈ G the value of the expression f(x1, ..., xk−1, y, xk+1, ..., xn) is defined for not less that three different elements y ∈ G. It will be proved that if any of the congruence relations on G is a congruence of the partial n-ary groupoid (G, f) then under specific conditions for G the partial operation f is not a constant.

Perturbation of least squares problem of dual linear operator in dual-Hilbert spaces

Let U be any nontrivial primitive class of partial algebras, i.e. there existsA ∈ U with |A|≥2, and U is closed with respect to homomorphic images (in the weak sense), subalgebras (on closed subsets) and cartesian products of U-algebras, and let U f denote the—also nontrivial and primitive—class of all full U-algebras. Then every U-algebra with at least two elements is a relative algebra of some U f -algebra. For any U-algebraAsetU A =U i eI({i}×(A K i—domf i A )), where (K i) i eI is the type under consideration. Furthermore let F(N, U) denote any U-algebra U-freely generated by some setN (and let F (M, U f ) be similarly defined). Then for every nonempty setM there exists a setN satisfyingM ⊆N such that there exists a bijective mapping σ:U F(N, U) →N −M satisfying σ((i, α)) ∉ α(K i ) for all (i, α) ∈U F (N, U), and, for the structureg=(g i)ieI defined by ,g i : =f i F(N, U) ∪ {(α, σ((i, α))) | (i, α ∈U F(N, U)} id M induces an isomorphism betweenF(M, U f ), and (F(N, U)g).

An algebra is effective if its operations are computable under some numbering. When are two numberings of an effective partial algebra equivalent? For example, the computable real numbers form an effective field and two effective numberings of the field of computable reals are equivalent if the limit operator is assumed to be computable in the numberings (theorems of Moschovakis and Hertling). To answer the question for effective algebras in general, we give a general method based on an algebraic analysis of approximations by elements of a finitely generated subalgebra. Commonly, the computable elements of a topological partial algebra are derived from such a finitely generated algebra and form a countable effective partial algebra. We apply the general results about partial algebras to the recursive reals, ultrametric algebras constructed by inverse limits, and to metric algebras in general. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

In [2]2 we classified the isometric mappings of one C*-algebra (uniformly closed, self-adjoint operator algebra) onto another. It was remarked in that paper that the results obtained were a non-commutative extension of results of Banach [1] and Stone [7]. While this was true in spirit, we were well aware that it was not accurate to the letter. Banach and Stone deal with the algebra of real continuous functions on a compact-Hausdorff space, and our results concerning C*-algebras are actually the non-commutative analogue of results concerning the complex function algebra. The strict non-commutative analogue of the real function algebra is the Jordan algebra of self-adjoint elements in a C*-algebra (Jordan C*-algebra). The complex and real theorems follow very easily from one another in the commutative case, so that one might justifiably consider the C*-algebra theorem an extension of both of the function algebra theorems. Despite such trifling considerations, two questions still remain: what are the isometries of one C*-algebra onto another, and what are the isometries of one Jordan C*-algebra onto another? At the time [21 was written, the C*-algebra seemed the more natural object to consider. In view of the results obtained, answering the Jordan C*-algebra questions appeared to be an unnecessary decoration to the theory. We felt that the Jordan C*-algebra results could be obtained from the C*-algebra results in the same way that the real function algebra theorem follows from the complex function algebra theorem (viz., by showing that the complexified linear map is everywhere isometric). Subsequent investigations have changed our attitude in this matter. An important application of these considerations requires a Jordan C*-algebra theorem for one thing, and our attempts to derive this theorem directly from the C*-algebra theorem failed for another. The result in question is contained in Theorem 2 of ?2 and states (in normalized form) that an isometry between two Jordan C*-algebras which carries the identity into the identity is a C* (Jordan) -isomorphism. This theorem was eventually proved with the aid of a Generalized Schwarz Inequality (cf. Theorem 1 of ?2). In effect, an alternative ending has been given to the proof of [Theorem 7; 2]. This ending is by no means simpler or shorter than the one given in [2] (though it is, perhaps, less contrived), but it is flexible enough to allow us to draw the desired Jordan C*-algebra conclusion. The critical application of these results is contained in Corollary 3. A discussion accompanies Corollaries 3 and 4, but a few additional remarks are in order.

Geometry of C⁎-algebras, and the bidual of their projective tensor product

We prove an analogue of the Poincare'-Birkhoff-Witt theorem and of the Cartier-Milnor-Moore theorem for non-cocommutative Hopf algebras. The primitive part of a cofree Hopf algebra is a B-infini-algebra. We construct a universal enveloping functor U2 from B-infini-algebras to 2-associative algebras, i.e. algebras equipped with two associative operations. We show that any cofree Hopf algebra H is of the form U2(Prim H). We take advantage of the simple description of the free 2as-algebra in terms of planar trees to unravel the structure of the operad B-infini.