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Regular Orthomodular Posets

Borzyszkowski Andrzej M.

Fundamenta Informaticae

2019

Vol. 166, nr 1

15--28

Rozenberg and Ehrenfeucht has shown a duality between 2-structures (a.k.a. transition systems) and (elementary) Petri nets. The tool has been the notion of a region of a 2-structure, the regions then define a Petri net. Bernardinello et al. has observed that the regions of a 2-structure form an orthomodular poset and there is a similar relation between 2-structures and orthomodular posets. While in the theory of 2-structures we may ask if a 2-structure is full and forward closed, the analogous notion for orthomodular posets is their regularity. In the present paper we study the problem of closing a given orthomodular poset to a regular one. This is a dual problem to closing a 2-structure, which has been studied by the author earlier. Also, as in a seminal work of Rozenberg and Ehrenfeucht, one can be interested in a concrete representation, i.e. as a family of sets. We show here an appropriate construction for orthomodular posets too.

Comparability groups

Foulis D., Foulis D.

Demonstratio Mathematica

2006

Vol. 39, nr 1

15-32

A comparability group is a unital group with a compression base and with the general comparability property. The additive group of self-adjoint elements in a von Neumann algebra, and any Dedekind sigma-complete lattice-ordered abelian group with order unit are examples of comparability groups. We develop the basic theory of comparability groups, and show that an archimedean comparability group with the Rickart projection property can be embedded in a partially ordered rational vector space the elements of which admit a rational spectral resolution.

Meanders in orthoposets and QMV algebras

Dvurečenskij A., Pulmannova S., Salvati S.

Demonstratio Mathematica

2001

Vol. 34, nr 1

1-11

The notion of a meander of an ideal in lattices is generalized in two directions: to ideals in orthoposets and to ideals in QMV-algebras, and used to a characterization of subclasses of the above structures, namely Boolean orthoposets and QMV-algebras m which every ideal is closed under perspectivity, and to a characterization of Riesz ideals in orthoposets and perspectivity closed ideals in QMV algebras.