Symmetries in synaptic algebras

A synaptic algebra is a generalization of the Jordan algebra of self-adjoint elements of a von Neumann algebra. We study symmetries in synaptic algebras, i.e., elements whose square is the unit element, and we investigate the equivalence relation on the projection lattice of the algebra induced by finite sequences of symmetries. In case the projection lattice is complete, or even centrally orthocomplete, this equivalence relation is shown to possess many of the properties of a dimension equivalence relation on an orthomodular lattice.     

Projections in a Synaptic Algebra

A synaptic algebra is an abstract version of the partially ordered Jordan algebra of all bounded Hermitian operators on a Hilbert space. We review the basic features of a synaptic algebra and then focus on the interaction between a synaptic algebra and its orthomodular lattice of projections. Each element in a synaptic algebra determines and is determined by a one-parameter family of projections—its spectral resolution. We observe that a synaptic algebra is commutative if and only if its projection lattice is boolean, and we prove that any commutative synaptic algebra is isomorphic to a subalgebra of the Banach algebra of all continuous functions on the Stone space of its boolean algebra of projections. We study the so-called range-closed elements of a synaptic algebra, prove that (von Neumann) regular elements are range-closed, relate certain range-closed elements to modular pairs of projections, show that the projections in a synaptic algebra form an M-symmetric orthomodular lattice, and give several sufficient conditions for modularity of the projection lattice.     

The center of an effect algebra

An effect algebra is a partial algebra modeled on the standard effect algebra of positive self-adjoint operators dominated by the identity on a Hilbert space. Every effect algebra is partially ordered in a natural way, as suggested by the partial order on the standard effect algebra. An effect algebra is said to be distributive if, as a poset, it forms a distributive lattice. We define and study the center of an effect algebra, relate it to cartesian-product factorizations, determine the center of the standard effect algebra, and characterize all finite distributive effect algebras as products of chains and diamonds.     

Lie isomorphisms of reflexive algebras

A Lie isomorphism ? between algebras is called trivial if ?=ψ+τ, where ψ is an (algebraic) isomorphism or a negative of an (algebraic) anti-isomorphism, and τ is a linear map with image in the center vanishing on each commutator. In this paper, we investigate the conditions for the triviality of Lie isomorphisms from reflexive algebras with completely distributive and commutative lattices (CDCSL). In particular, we prove that a Lie isomorphism between irreducible CDCSL algebras is trivial if and only if it preserves I-idempotent operators (the sum of an idempotent and a scalar multiple of the identity) in both directions. We also prove the triviality of each Lie isomorphism from a CDCSL algebra onto a CSL algebra which has a comparable invariant projection with rank and corank not one. Some examples of Lie isomorphisms are presented to show the sharpness of the conditions.     

强Ockham代数与剩余格

首先讨论了Ockham代数与剩余格的关系,引入了强Ockham代数的概念,并讨论了它的基本性质．然后,将著名的风蕴涵和风算子推广到Ockham代数上,证明了添加广义R0蕴涵和广义风算子后的Ockham代数L成为剩余格的充要条件是L为强Ockham代数．最后给出若干重要例子,以此来说明强Ockham代数的条件是独立的．     

格蕴涵代数的左幂等元

为了研究命题真值取于格上的逻辑系统,文献[1]给出了格蕴涵供数的概念,文献[2-6]给出了格蕴涵代数的滤子,同态和性质（P）的概念,并讨论了它们的一些性质。本文在格蕴函代数中引入左幂等元的概念,讨论格蕴函代数中左幂等元的性质及由全体左幂等元所构成集合的代数结构,得到格蕴涵代数的分解定理：格蕴涵代数可以分解为由左幂等元构成左映射的像集合与对偶核的直和。     

Frobenius Poisson algebras

This paper is devoted to study Frobenius Poisson algebras. We introduce pseudo-unimodular Poisson algebras by generalizing unimodular Poisson algebras, and investigate Batalin-Vilkovisky structures on their cohomology algebras. For any Frobenius Poisson algebra, all Eatalin-Vilkovisky opera tors on its Poisson cochain complex are described explicitly. It is proved that there exists a Batalin-Vilkovisky operator on its cohomology algebra which is induced from a Batalin-Vilkovisky operator on the Poisson cochain complex, if and only if the Poisson st rue ture is pseudo-unimodular. The relation bet ween modular derivations of polynomial Poisson algebras and those of their truncated Poisson algebras is also described in some cases.     

L-型模糊格蕴涵代数

This paper introduced the concept of L-fuzzy sub lattice implication algebra and discussed its properties. Proved that the intersection set of a family of L-fuzzy sub lattice implication algebras is a L-fuzzy sub lattice implication algebra, that a L-fuzzy sub set of a lattice implication algebra is a L-fuzzy sub lattice implication algebra if and only if its every cut set is a sub lattice implication algebra, and that the image and original image of a L-fuzzy sub lattice implication algebra under a lattice implication homomorphism are both L-fuzzy sub lattice implication algebras.     

Monotone clones and the varieties they determine

A finite, nontrivial algebra is order-primal if its term functions are precisely the monotone functions for some order on the underlying set. We show that the prevariety generated by an order-primal algebra P is relatively congruence-distributive and that the variety generated by P is congruence-distributive if and only if it contains at most two non-ismorphic subdirectly irreducible algebras. We also prove that if the prevarieties generated by order-primal algebras P and Q are equivalent as categories, then the corresponding orders or their duals generate the same order variety. A large class of order-primal algebras is described each member of which generates a variety equivalent as a category to the variety determined by the six-element, bounded ordered set which is not a lattice. These results are proved by considering topological dualities with particular emphasis on the case where there is a monotone near-unanimity function.This research was carried out while the third author held a research fellowship at La Trobe University supported by ARGS grant B85154851. The second author was supported by a grant from the NSERC.     

Linear Maps Preserving the Closure of Numerical Range on Nest Algebras with Maximal Atomic Nest

Let be a maximal atomic nest on Hilbert space H and denote the associated nest algebra. We prove that a weakly continuous and surjective linear map preserves the closure of numerical range if and only if there exists a unitary operator such that for every or for every , where denotes the transpose of T relative to an arbitrary but fixed base of H. As applications, we get the characterizations of the numerical range or numerical radius preservers on . The surjective linear maps on the diagonal algebras of atomic nest algebras preserving the closure of numerical range or preserving the numerical range (radius) are also characterized. Submitted: January 3, 2001?Revised: December 2, 2001     

Universal state space embeddability of Jordan-Banach algebras

We study extensions of states between projection structures of JB algebras and generalized orthomodular posets. It is shown that projection orthoposet of a JB algebra admits the universal extension property if and only if the Gleason theorem is valid for . As a consequence we get that any positive Stone algebra-valued measure on projection lattice of a quotient of a JBW algebra without type direct summand extends to a positive measure on an arbitrary larger generalized orthomodular lattice.

     

SPECTRAL PROPERTIES OF POSITIVE ELEMENTS IN BANACH LATTICE ALGEBRAS

Abstract

We prove that if a unital Banach lattice algebra has sufficiently many one-dimensional elements and if its unit element has sufficiently many components then its positive elements have spectral properties analogous to those of positive operators on Banach lattices. In particular, if a positive element is irreducible (in the sense that (1—e)xe > 0 for all components e of 1 satisfying 0 ≠ e ≠ 1) and compact, its spectral radius is positive and its spectrum shows cyclic behaviour.     

Related structures with involution

We give a necessary and sufficient condition for an involution lattice to be isomorphic to the direct square of its invariant part. This result is applied to show relations between related lattices of an algebra. For instance, generalizing some earlier results of G. Czédli and L. Szabó it is proved that any algebra admits a connected compatible partial order whenever its quasiorder lattice is isomorphic to the direct square of its congruence lattice. Further, a majority algebra is lattice ordered if and only if the lattice of its compatible reflexive relations is isomorphic to the direct square of its tolerance lattice. In the latter case, one can establish a bijective correspondence between factor congruence pairs of the algebra and its pairs of compatible lattice orders; several consequences of this result are given.     

格蕴涵代数与MV-代数

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｏｒｄｅｒｔｏｒｅｓｅａｒｃｈｔｈｅｌｏｇｉｃａｌｓｙｓｔｅｍｗｈｏｓｅｐｒｏｐｏｓｉｔｉｏｎａｌｖａｌｕｅｉｓｇｉｖｅｎｉｎａｌａｔｔｉｃｅｆｒｏｍｔｈｅｓｅｍａｎｔｉｃｖｉｅｗｐｏｉｎｔ,ｗｅｈａｖｅｐｒｏｐｏｓｅｄｔｈｅｃｏｎｃｅｐｔｏｆｌａｔｔｉｃｅｉｍｐｌｉｃａｔｉｏｎａｌｇｅｂｒａｓｉｎ［１］ａｎｄｈａｖｅｄｉｓｃｕｓｓｅｄｔｈｅｉｒｓｏｍｅｐｒｏｐｅｒｔｉｅｓ．ＭＶ－ａｌｇｅｂｒａｓｗｅｒｅｉｎｖｅｎｔｅｄｂｙＣ．Ｃ．Ｃｈａｎｇ［２］ｉｎｏｒｄｅ…     

A note on ideals in synaptic algebras

The notion of a synaptic algebra was introduced by David Foulis. Synaptic algebras unite the notions of an order-unit normed space, a special Jordan algebra, a convex effect algebra and an orthomodular lattice. In this note we study quadratic ideals in synaptic algebras which reflect its Jordan algebra structure. We show that projections contained in a quadratic ideal from a p-ideal in the orthomodular lattice of projections in the synaptic algebra and we find a characterization of those quadratic ideals which are generated by their projections.     

标度广义效应代数和标度效应代数的结构

研究了标度广义效应代数与标度效应代数的代数结构,给出了比较完整的结果.通过引入全标度广义代数的概念,本文证明了区间[0,1)上的标度广义效应代数和单位区间[0,1]上的标度效应代数完全由单位区间上的阿基米德余模确定,标度广义效应代数恰同构于全标度广义代数的下集.若标度广义代数满足局部有限条件,则它同构于实数加法群的子群代数.满足(S)条件的标度效应代数同构于实数加法群的子群代数和全标度广义代数的字典序乘积的子代数.     

Effect-like algebras induced by means of basic algebras

Having an MV-algebra, we can restrict its binary operation addition only to the pairs of orthogonal elements. The resulting structure is known as an effect algebra, precisely distributive lattice effect algebra. Basic algebras were introduced as a generalization of MV-algebras. Hence, there is a natural question what an effect-like algebra can be reached by the above mentioned construction if an MV-algebra is replaced by a basic algebra. This is answered in the paper and properties of these effect-like algebras are studied.     

von Neumann 代数上保持混合Jordan三重积的非线性映射

本文证明了在没有中心交换投影的von Neumann代数上的一个双映射$\Phi$如果保持混合Jordan三重积, 则$\Phi(I)\Phi$是一个线性*-同构和一个共轭线性*-同构的和, 其中$\Phi(I)$是中心自伴元素且$\Phi(I)^{2}=I$. 同时给出了因子von Neumann 代数上保持混合Jordan三重积映射的结构.     

Embedding Distributions in Algebras of Generalized Functions with Singularities

 It is a known open question whether in Rosinger’s nowhere dense quotient algebra distributions can be embedded, ensuring consistency with partial derivatives. In this paper, we construct algebras of generalized functions with nowhere dense singularities similar to Rosinger’s algebra in which such an embedding of distributions is possible. Moreover, in some of the algebras, the embedding preserves the products of smooth functions. Further, we indicate that most of the properties of Rosinger’s algebra, in particular concerning the solvability of nonlinear PDEs, remain valid in the new algebras. Summarizing, we give a positive answer to the above-mentioned open question, if one allows a modification in the definition of the algebra that does not affect its basic properties.     

Coproducts of bounded distributive lattices: infinite distributivity

Summary It is shown that, if two bounded distributive lattices satisfy the join-infinite distributive law (JID), then their coproduct also satisfies this law. In 1986, Yaqub proved that generalized Post algebras with a finite lattice of constants satisfy JID, and stated that, in general, it is not known whether a generalized Post algebra satisfies JID when its lattice of constants satisfies JID. In this note, the statement is proved.