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The center of a generalized effect algebra

Foulis D. J., Pulmannová S.

Demonstratio Mathematica

2014

Vol. 47, nr 1

1--21

In this article, we study the center of a generalized effect algebra (GEA), relate it to the exocenter, and in case the GEA is centrally orthocomplete (a COGEA), relate it to the exocentral cover system. Our main results are that the center of a COGEA is a complete boolean algebra and that a COGEA decomposes uniquely as the direct sum of an effect algebra (EA) that contains the center of the COGEA and a complementary direct summand in which no nonzero direct summand is an EA.

The range of non-atomic measures on effect algebras

Khare M., Singh A. K.

Demonstratio Mathematica

2010

Vol. 43, nr 3

497-510

The present paper deals with the study of superior variation m+, inferior variation m¯ and total variation |m| of an extended real-valued function m defined on an effect algebra L; having obtained a Jordan type decomposition theorem for a locally bounded real-valued measure m defined on L, we have observed that the range of a non-atomic function m defined on a D-lattice L is an interval (—m¯ (1), m+(1)). Finally, after introducing the notion of a relatively non-atomic measure on an effect algebra L, we have proved an analogue of Lyapunov convexity theorem for this measure.

Relations between various continuity concepts in effect algebras

Khare M., Gupta S.

Demonstratio Mathematica

2008

Vol. 41, nr 3

507-518

The aim of the present paper is to establish relations between continuity concepts for a nonnegative extended real-valued function [...] defined on an effect algebra. Examples and counterexamples are given to illustrate various situations arising in this study.

Compression bases in effect algebras

Gudder S.

Demonstratio Mathematica

2006

Vol. 39, nr 1

43-54

We generalize David Foulis's concept of a compression base on a unital group to effect algebras. We first show that the compressions of a compressible effect algebra form a compression basis and that a sequential effect algebra possesses a natural maximal compression basis. It is then shown that many of the results concerning compressible effect algebras hold for arbitrary effect algebras by focusing on a specific compression base. For example, the foci (or projections) of a compression base form an orthomodular poset. Moreover, one can give a natural definition for the commutant of a projection in a compression base and results concerning order and compatibility of projections can be generalized. Finally it is shown that if a compression base has the projection-cover property, then the projections of the base form an orthomodular lattice